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Lecture Notes in Mathematics A collection of informal reports and seminars Edited by A. Dold, Heidelberg and B. Eckmann, Z0rich
203 Donald Knutson Columbia University in the City of New York, New York, NY/USA
Algebraic Spaces
Springer-Verlag Berlin.Heidelberg- New York 1971
A M S Subject Classifications (1970):
14-02, 1 4 A 1 5 , 1 4 A 2 0 , 14F20, 1 8 F 1 0
I S B N 3-540-05496-0 Springer-Verlag Berlin • H e i d e l b e r g - N e w Y o r k I S B N 0-387-05496-0 Springer-Verlag N e w Y o r k • H e i d e l b e r g • Berlin
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreemznt with the publisher. © by Springer-Verlag Berlin . Heidelberg 1971. Library of Congress Catalog Card Number73-164957.Printed in Germany. Offsetdruck: Julius Beltz, Hemsbach
PREPACE
The core of this book is the author's thesis, Algebraic written under Michael Artin at the ~assachusetts Technology.
Institute
Spaces,
of
The object there as here was to work out the foundations
la EGA for the theory of algebraic
spaces, and hence give the
necessary background for Artin's fundamental papers Al~ebraization of Formal Moduli I, II. While working on this book, College,
Columbia University,
Seminar at Bowdoin College, Foundation.
I was supported by M.I.T.,
Boston
and the Advanced Science Summer
sponsored by the National Science
To all these institutions,
I extend my gratitude.
My
special thanks goes to Professor Michael Artin both for many helpful discussions
and for his initial suggestion that I undertake
this project.
Donald Knutson
CONTENTS
Introduction
. . . . . . . . . . . . . . . . . . . . . . . . . .
Chapter
The
One:
Etale
I. G r o t h e n d i e c k 2. The
Zariski
3. The
Flat
Chapter
Two:
Topologies Topology
Topology
4. The E t a l e 5. E t a l e
Topology
Algebraic
Category
2. The
Etale
3. D e s c e n t
of S c h e m e s
Topology
and
7. P r o p e r
and P r o j e c t i v e
Three:
I. The
Completeness
2. The
Serre
5. D e v i s s a g e
Chapter
Four:
I. A c t i o n s 2.
3. C h o w ' s
91
Spaces
. . . . . . . . . . . . Spaces
Spaces
. . . . . . . . .
101
. . . . . . . . . . .
106
. . . . . . . . . .
113
Cohomology
Morphisms
. . . . . . . . . . . . .
139
. . . . . . . . . . . . . . . .
on N o e t h e r i a n
Lemma
. . . . . . . . . . .
Lemma
4. The F i n i t e n e s s
153 159
. . . . . . . . . . . . . . . .
165
. . . . . . . . . . . . . . . . . . .
Theorem
Group
169 173
. . . . . . . . . . . . . .
176
. . . . . . . . . . . . . . . .
177
of P r o j e c t i v e
Spaces
. . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . Theorem
153
. . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . .
powers
144
Locally
. . . . . . . . . . . . . . . . .
/ Extension
of a F i n i t e
120 129
Sheaves
Theorem
91
. . . . . . . . . . . . .
Spaces
The F i n i t e n e s s
Symmetric
. . . . . . . . . . . . . . . . .
and N i l p o t e n t s
4. C h e v a l l e y ' s
52
Topology
Criterion
3. S c h e m e h o o d
. . . . . . . . . . . . . .
59
Spaces
Quasicoherent Algebraic
38
72
and
Zariski
Algebraic
29
. . . . . . .
. . . . . . . . . . . . . . . . . . .
6. P o i n t s
Separated
the
Theory
. . . . . . . . . . . . .
of A l g e b r a i c
Constructions
29
. . . . . . . . . . . . . .
for A l g e b r a i c Sheaves
. . . . . . . . . . .
. . . . . . . . . . . . . . .
Relations
of A l g e b r a i c
4. Q u a s i c o h e r e n t
Chapter
and D e s c e n t
of S c h e m e s
Spaces
Theory
8. I n t e g r a l
of S c h e m e s
of S c h e m e s
Equivalence
I. The
5. L o c a l
Topology
I
. . . . . . . . . . . . . . . . .
185 192 202
VI Chapter
Five:
Formal
Algebraic
I. A f f i n e
Formal
2. F o r m a l
Algebraic
3. The
Theorem
4. A p p l i c a t i o n s 5. C o m p l e t i o n s 6. The
Index
Schemes
Spaces
. . . . . . . . . . . . . . . . . . .
Spaces
. . . . . . . . . . . . . . . . . .
of H o l o m o r p h i c to P r o p e r of M o d u l e s
Grothendieck
. . . . . . . . . . . . . .
Functions
Morphisms
Theorem
215
. . . . . . . . . . .
224 233
. . . . . . . . . .
241
. . . . . . . . . . . .
245
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Bibliography
204
. . . . . . . . . . . . .
of H o m o m o r p h i s m s
Existence
204
. . . . . . . . . . . . . . . . . . . . . . . . . .
252
257
INTRODUCTION
The notion the n o t i o n Elements w e give
of a l g e b r a i c
of scheme,
de G e o m e t r i e a tentative
of examples,
space
as p r e s e n t e d Algebrique
definition
and some
is a g e n e r a l i z a t i o n in A. G r o t h e n d i e c k ' s
(EGA).
In this introduction,
of a l g e b r a i c
indication
of
of the
space,
literature
a number on the
subject. The rest of the book foundations background scheme
with
theory
from EGA, various SGAD)
of the
theory.
and d e s c e n t
Seminaires
"etale a brief
who
of schemes",
reading
of sections spaces
arise
algebraic-geometric
Picard
Schemes,
impose
algebraic
analytic
Artin,
spaces.
development
I contains
the r e l e v a n t
This b a c k g r o u n d Topologies
Al~ebrique
is v a g u e l y
(GT), (SGA,
J.L.
familiar
can skip C h a p t e r
of is drawn and
the
SGAA,
and
Verdier,
feels h a p p y w i t h
1 and
of the
(but not proofs)
P. Deligne,
the reader w h o
topology
Algebraic certain
Grothendieck
T h e reader
i.e.,
theory.
de G e o m e t r i e
of G r o t h e n d i e c k ,
material,
Chapter
all the d e f i n i t i o n s
M. A r t i n ' s
M. Demazure.
is an o r d e r l y
with
and this
the p h r a s e
I except
for
5.
in the a t t e m p t objects:
to c o n s t r u c t
Hilbert
Schemes,
and m o d u l i
varieties;
and in the attempt
structure
on certain
given
objects,
such
to as
As
an example,
the H i l b e r t
we consider
Scheme H of a complex
H should be a scheme, the closed requirement
should
condition:
noetherian
scheme
Z of X X S
such
Hilbx(S) of l o c a l l y
points
of closed
Specifically,
the map
noetherian a variety
X has
mined
and
(-,H)
In this w a y we v i e w tain o b j e c t
we define
(XVII)
a Hilbert
a Hilbert
need not have
ot X, locally
from the c a t e g o r y
Scheme
if
is r e p r e -
Hilbx(-)
and
deter-
of X. of c o n s t r u c t i n g
of r e p r e s e n t i n g
Scheme
We
if and only
then H is u n i q u e l y
the p r o b l e m
In general,
of sets.
Scheme
if the functors
case of H i l b e r t
a Hilbert
a
subschemes
If Hilbx(-)
a certain
Schemes,
that if X is a p r o j e c t i v e
Scheme.
impose
for each
to the c a t e g o r y
i.e.,
as the p r o b l e m
In the s p e c i f i c shown
so w e
This
subvarieties
functor
are equivalent,
is the H i l b e r t
Ideally,
Z + S is flat and proper.~
schemes
by a scheme H,
HOm(schemes)
manner.
S, H i l b x ( S ) = [Set of closed
that
X.
S, H O m s c h e m e s ( S , H )
functor H i l b x ( - ) is r e p r e s e n t a b l e .
sentable
variety
of w h i c h p a r a m e t r i z e
H uniquely,
is a c o n t r a v a r i a n t
now define:
algebraic
for any scheme
be the set of families by S.
of c o n s t r u c t i n g
of X in a c o n t i n u o u s
does not d e t e r m i n e
parametrized
the
the closed
subvarieties
naturality
the p r o b l e m
however,
functor.
it has b e e n
variety,
then X has
an a r b i t r a r y
(see E x a m p l e
a cer-
3 below).
variety
To
find objects
which
we m u s t
look at a larger
Because
of the results
briefly
below,
be m e n t i o n e d
of the H i l b e r t Our
I.
that D o u a d y
Functor
first
of schemes
requirement
among
the Z a r i s k i
the
flat topology.
functor
comes
has
must know
Hence, that
the
On the other hand, functors,
every
functor
speces
~
given
(Sets)
a functor
at least
is u n i q u e l y
spaces
[XII].)
is:
the c a t e g o r y
subcate@ory
the etale
of schemes. topologies~
topology,
and
a scheme X,
satisfies
the
the sheaf
like Hilbx(-) , one
satisfies
from the u n i q u e n e s s
scheme
of a n a l y t i c
Grothendieck
In any of these,
should
the r e p r e s e n t a b i l i t y
from the t o p o l o g y
topology,
to r e p r e s e n t
(It
contains
full
several
X" = Hom(-,X) :(Schemes)
axiom.
shown
spaces and
of a l g e b r a i c
category.
for a l g e b r a i c
as a faithful
of schemes
others
has
described
the n o t i o n
larger
Schemes,
that of schemes.
(II,III,IV,V), that
of a l g e b r a i c
The c a t e g o r y
as H i l b e r t
than
in the c a t e g o r y
requirement
The c a t e g o r y
A second
likely
the a p p r o p r i a t e
here
serve
category
of A r t i n
it seems
space p r o v i d e s
might
the sheaf
involved
determined
axiom.
in r e p r e s e n t i n g
by its
associated
T h r o u g h o u t we d e n o t e by (- - -) the c a t e g o r y of all - - -. Also, we use the n o t a t i o n X" for the c o n t r a v a r i a n t functor r e p r e s e n t e d b y X.
sheaf.
A sheaf
is d e t e r m i n e d
and by the Y o n e d a X" ~ Y" there
is induced
are
full
(Affine
logy.
For
topology
In c o n t r a s t notions topology.
of p o i n t On
context
is not
fine
in which
much
take
as our
II.
category
topology,
is the etale functor X"
there
on the W e i l
topology.
his
spaces
that
(e.g.,
the Zariski topology
above-mentioned
spaces:
a Grothendieck
alqebraic
= HOm(Alg. Spaces) (-,X) :
is the
theorems.
for a l g e b r a i c
to the s u b c a t e ~ o r y
For e v e r y
the
in the etale
the etale
has
topology.
topology,
are i n d i c a t i o n s
requirement
restriction
flat
topo-
numbers,
analytic
are stable
Conjectures)
has p r o v e d
second
as the
Finally,
of a l g e b r a i c
whose
such
first
the etale
the c o m p l e x
the usual
scheme
we m u s t
on the c a t e g o r y
we choose
like
enough.
spaces,
over
topologies
Artin
Hence we
The
for schemes
and reduced
work
Thus
(Sheaves)
topology
reasons,
the other hand,
Grothendieck's topology
Grothendieck
is d e f i n e d
to finer
sheaves
m a p X ~ Y of schemes.
to a l g e b r a i c
several
On one hand,
schemes
imbeddings
this over
the a p p r o p r i a t e
of schemes.
this
by a u n i q u e
faithful
on affine
every m a p of r e p r e s e n t a b l e
Schemes) C___~> (Schemes) ~
To carry pick
Lemma,
by its v a l u e s
of schemes
space X,
(Algebraic
Spaces)
the ~
(Sets)
is a s h e a f and
and the t r a n s f o r m a t i o n
faithful
sets
on s c h e m e s
We might
property
But
aspect
construction
the Z a r i s k i
clear,
"gluing
w e can w r i t e U that V13.. = U i
diagram
of
space
to m i m i c
roughly,
that
any
of all s c h e m e s
(such
by the p r o p e r t y
of a f f i n e
w e go b a c k
described the open
schemes.
or
To m a k e
to the c a s e of s c h e m e s
xX U j ,
and separated,
we c a n
write
say)
of X by affine
by giving
just the
in
and schemes,
family
[Ui}
s u b s e t s V.. = U. D U. of X 13 l 3
V.. + U. and V.. -~ U.. x3 i x3 3
for the d i s j o i n t
union
More
abstractly,
of the U. 's. 1
R = U X × U for
the
Noting
disjoint
V..°s. is a c a n o n i c a l
R as an e q u i v a l e n c e projections
(Sheaves
w e need
is,
is d e t e r m i n e d
(quasicompact
data",
and the i m m e r s i o n s
There
geometry,
is an open c o v e r i n g
then X is c o m p l e t e l y
the
a full
topology.
[Ui -~ X } i = l , 2 , . . . , n
of
~
an a l g e b r a i c
the c a t e g o r y
on the s u b c a t e g o r y
If X is a s c h e m e
union
on
gives
topology).
theory w h i c h
of a sheaf)
Spaces)
just d e f i n e
to do m u c h
of s c h e m e
this a s p e c t m o r e
and the
and
or c o n s t r u c t i o n
as the g i v i n g
(Algebraic
in the etale
stop h e r e
to b e a sheaf. another
imbedding
X ~>X"
injection
relation
on U.
R * U X U which
identifies
L e t ~i and ~2 be
R ~ U and ~ : U + X the c o v e r i n g
map.
Then
the two in the
R
~U
~X
~2
~ : U ~ X is the c o k e r n e l schemes. image
There
of the m a p s ~i,~2
is also a c a n o n i c a l
is a c o m p o n e n t
of R,
in the c a t e g o r y
of
injection U ~ R whose
isomorphic
to U,
the d i a g o n a l
com-
~onent. R and U are b o t h separated) on a f f i n e "adding category
and
affine
schemes
thus X is d e s c r i b e d
schemes.
quotients
In this w a y
by an e q u i v a l e n c e
schemes
of r e a s o n a b l e
of a f f i n e
(if X is q u a s i c o m p a c t
relation
are c o n s t r u c t e d
equivalence
and
relations"
by to the
schemes.
But one m u s t be c a r e f u l
here.
The e q u i v a l e n c e
relation
->
R
U of affine
category E.g.,
schemes
of a f f i n e
schemes,
if X is a c o m p l e t e
affine quotient In s c h e m e
of R
spaces.
Tying
that the
Zariski
of sheaves,
a quotient
it m a y be the
variety
over
in the
"wrong"
a closed
field
quotient. k, the
is c i r c u m v e n t e d
U in the c a t e g o r y
in w i t h
functor X"
topology
but
this d i f f i c u l t y
of R
this
already have
U w i l l be Spec k, not X.
theory,
X as the q u o t i e n t
shows
may
of local
the c o n s i d e r a t i o n s
and that X"
is the quotient,
of the e q u i v a l e n c e
relation
taking
ringed
~bove,
is a sheaf on affine
by
schemes
one then in the
in the c a t e g o r y
R" ~ U"
The point that
all
of t h i s
equivalence
the i n f o r m a t i o n
and c o n s t r u c t i o n s
sheaves,
closed
subvarieties,
U modulo
gluing
data
Etale mation
descent
is e q u a l l y
relation
etc.--is
construction
is
on X - - q u a s i c o h e r e n t
determined
locally
on
on R.
theory
( C h a p t e r I)
determined
by
shows
giving
that
all
an a r b i t r a r y
the
infor-
etale
sur-
jective map ~:U ~ X with
R = U × U. H e r e X" is the q u o t i e n t X in the c a t e g o r y of s h e a v e s in the e t a l e t o p o l o g y .
of R" ~ U"
In e i t h e r
of t h e s e
R ~
cases,
~
we
~ × ~ is an e t a l e
see
that
conditions With for
III.
8 has
on X a r e all
algebraic
For
the
this
8
surjective
×U
map.
same properties
imposed
we
take
Applying
as ~ and
by making
in m i n d ,
diagram
~X×X
descent
so s e p a r a t i o n
restrictions
as a f i n a l
theory,
on 6.
requirement
spaces:
each
algebraic
s p a c e X,
covering
map U ~ X
such
is a s c h e m e ~ the m a p
is a c a r t e s i a n
_~U
A
X
where
there
the m a p s
R + U X U
that
R ~ U
there the
are
is a s c h e m e U fiber product
etale
is q u a s i c o m p a c t .
and
a
R = U × U X
surjective~
X i._ssl o c a l l y
and separ-
ated
if R ~ U × U is a q u a s i c o m p a c t
separated
These
if R ~ U X U is a c l o s e d
three r e q u i r e m e n t s
definition
of a l g e b r a i c
One might be ringed
sometimes
be
which
here
is given
to c o n s t r u c t
quotient,
(See E x a m p l e
immersion.
or less d e t e r m i n e
X 1 of R ~ U.
the w r o n g
not be U × U.
space,
tempted
space q u o t i e n t
more
i m m e r s i o n t and
the
formal
in II.l.l.
X as the local
Unfortunately,
in the sense
this can
that R m i g h t
1 below.)
X1 The q u a s i c o m p a c t n e s s to m a k e
the g e n e r a l
are g i v e n There
in III
foundations
neater.
requirement
Equivalent
requirements
in 1.5.12. are s c h e m e s
X
for w h i c h
is not q u a s i c o m p a c t
(so-called
We choose
these
throughout
is a t e c h n i c a l
to r e g a r d the b o o k
that
the d i a g o n a l
map X + X x X
non-quasi-separated
as p a t h o l o g i c a l all s c h e m e s
schemes).
examples
considered
and a s s u m e
are q u a s i -
separated. In C h a p t e r algebraic to s h o w i n g
II,
the formal
spaces begins
development
and the rest of the b o o k
that a l g e b r a i c
spaces
In the r e s t of this I n t r o d u c t i o n , in w h a t w a y s
algebraic
of the
spaces
are v e r y m u c h w e give
are n o t
theory of is d e v o t e d
like schemes.
examples
like schemes.
to show We
also
I~ T h r o u g h o u t w e use the n o t a t i o n A . n . m to d e n o t e p a r a g r a p h m of s e c t i o n n o f C h a p t e r A. W i t h i n A, A . n . m is w r i t t e n just n.m.
make
some
general
algebraic
space
comments
on the r e l a t i o n
of scheme
theory
to
theory.
Example
0.
Every
(quasise~arated)
Example
i.
An a l ~ e b r a i c
scheme
space w h i c h
is an a l g e b r a i c
is not
locally
space.
separated
and not a scheme. In this complex copies
example,
numbers.
we work with
schemes
L e t U be the scheme
of the affine
defined
obtained
line A 1 = Spec C[s],
over
the
by taking
two
A 2 = Spec C[t]
and
4;
identifying point
the p o i n t s
by p.
The real p o i n t s
w i t h p the center diagonal
part
ing the p o i n t
s = 0 and t = 0.
point.
of R,
and
of U then
Let R consist a scheme U'
We denote look
like
the common \
a figure
of one copy of U,
obtained
the
from U by d e l e t -
p.
In p i c t u r e s ,
R -> U i s
A 1
A2
A1'
A2 '
A1
'n" P
~ rr 2
R
nl and ~2 are ~i :A'i ~ Ai,
/
the i d e n t i t y
maps
i = 1,2 and ~ 2 : A ' i
on U c R and on U', + A3_i,
i = 1,2.
This
is an
T h r o u g h o u t the book, C d e n o t e s the c o m p l e x numbers, the rationals, and Z the ring of integers.
A2
I0
etale
equivalence
locally
which
is c l e a r l y
the a l g e b r a i c category
of R ~ U
in the c a t e g o r y
the
space
affine
X which
of sheaves
is n o t
l i n e A.
but not
a local
not
of t h e
l i n e A = S p e c C[s]
affine X has,
tangent
not
a scheme).
a double
of l o c a l
In
ringed
fact, except
point,
but
ringed
B u t R / U × U. A
is the q u o t i e n t
in p a r t i c u l a r
where
is q u a s i s e p a r a t e d
separated.
The quotient spaces
relation
X
of R
space looks
U
Thus
in the
(and h e n c e like
a copy
at the p o i n t
s = 0 where
a single point with
two
directions:
X:
v
S = 0 Example We schemes
2.
Locally
start with
Separated a general
and ~:U + X
an e t a l e
L e t R = U × U. Then, X q u o t i e n t of the e t a l e Now
let T b e
Alqebraic
SDaces,
construction. surjective
applying
descent
equivalence
a subscheme
L e t X and U b e
quasifinite theory,
relation
of X
and
map.
X is the
R ~ U.
i : T ~ X the
We write
immersion.
shows
U T = T × U and R T = R × U and a s i m p l e d i a g r a m c h a s e X X that RT = U T × U T a n d t h a t RT ~ U T i s an etale equiva-
lence
relation
T
with
quotient
T:
RT
.~ ->
UT
R
>
u
________>
-
) x
T
II
This
construction
works
in general,
with
X an a r b i t r a r y
alge->
braic
space defined
B e l o w w e use
b y an e t a l e
this to c a l c u l a t e
Going back
equivalence
to the c a s e w h e r e X is a scheme,
of X.
the q u o t i e n t
a new e q u i v a l e n c e
relation
on U. With
diagonal of RT, R' T.
the n o t a t i o n A m U T,
above,
and a n o t h e r
so a s u b s p a c e
R T consists component
of R.
and
In g e n e r a l valence
equivalence
.~ U
~
R'
U
Indeed,
constructed
the s u b s p a c e U T ~ U and a b i t of d i a g r a m
there
is a c a r t e s i a n
diagram
U T
X I
R' T is a s u b s p a c e
relation.
)
-.
f
of Let
i n d u c e d map:
)X
U has b e e n
->
the
>X'
f is not an i s o m o r p h i s m .
relation
R' T.
f:X' + X the c a n o n i c a l l y
R'
R
of two parts,
L e t R' c R be the c o m p l e m e n t
T h e n R' + U is a n e w e t a l e
X' be its q u o t i e n t
on
R ~ U.
some of the p r o p e r t i e s
of R ~ U, w e n o w m o d i f y R to o b t a i n R'
relation
)X
T
the new e q u i to b e
trivial
chasing
shows
12
where
the v e r t i c a l
of f,
f (X._UT) : (X'-UT) T h u s X'
looks
the s u b s p a c e
X be obtained Spec C[P,Q]
two s p e c i f i c
minus
the
the a f f i n e
let
P = i.
F o r the
over
along
= 0.
first,
the c o m p l e x
the line Q = 0.
T1 c X be the
line
Q'
line.
let
numbers,
L e t U be the same:
L e t U ~ X be the e t a l e
closed
i m a g e UT1 i s
= O) a n d c o n s i s t s The
~ C[P',Q'],
associated
not a s e p a r a t e d
P ~>P',
Q = 0 axis,
with
containing
The a s s o c i a t e d
of
two disjoint
Thus X'
the
2 1) of
copies
X' 1 is the plane,
subscheme
minus
the
is a scheme,
1
X' 2 is not
a scheme
the g e n e r i c p o i n t separated
which
(minus
is the line
the line Q'
X' 2 is the plane,
the line P = 1 r e p l a c e d
B u t X' 2 is a l o c a l l y
is
Spec C[P',Q']/(Q,
= Spec C[P',Q']/(p. = I)
c o v e r i n g U T 2 ~ T 2.
which
scheme.
let T 2 ~ X be the c l o s e d Then U T 2
subscheme
the line Q = 1 doubled.
and is c o n n e c t e d :
of X'2
an etale e x t e n s i o n
examples.
the line Q'
The inverse
Q = 0 axis, w i t h
Now
is an i s o m o r p h i s m .
2
Q = 1.
although
and the r e s t r i c t i o n
from the m a p of r i n g s C[P,Q]
We f i r s t
(minus
(X-T),
from the a f f i n e p l a n e
Spec C[P',Q']
O .--~> Q'
~
like X b u t w i t h
by removing
map obtained
are s u b s p a c e s
T.
We now give
line
arrows
by a nontrivial since no o p e n
of UT2
algebraic
minus
the double
subset
is an affine
space.
= 0)
scheme.
13
For
a second
convenient field
example,
to w r i t e
containing
x
start with
for the
q
q = p
we
n
affine
elements.
We
S p e c Z.
It w i l l
spectrum
of
the
can draw
S p e c R as
be
finite
follows:
Spec
Z : x2
Let
X =
E + Z[ ~ ] . etale
x3
x5
x7
(Spec Z - x2) . Let U = X
Spec Q
Consider
the r i n g
X (Spec Z[ ~-2]). Spec E
extension
T h e n U + X is an
covering. Given
a point
T = x
e X,
there
are
two p o s s i b i l i t i e s
P for U T = T × U. If the e q u a t i o n t 2 = 2 is s o l v a b l e in E / p T , X t h e n U T is a d i s j o i n t u n i o n of two c o p i e s of x . If w e p e r P form
the general
object
has
the
construction
above
case,
the n e w
form:
- . .
x3
which
in t h i s
x5
is a s c h e m e :
:
x7
X with
x
. - .
Spec Q
P
a single point
x
doubled. P
On (e.g., our
the o t h e r p =
hand,
5), U T is
construction
we
if t 2 = 2 is n o t
the s i n g l e p o i n t
x(p2)
in E / p ~ ,
and p e r f o r m i n g
get
X':
•
x3
solvable
X5
x7
•
•
X(p2)
Spec Q
14
which
is X w i t h a field e x t e n s i o n
at one point:
C l e a r l y not
a scheme. This X' i l l u s t r a t e s spaces.
Given an a l g e b r a i c
valence ringed
another p h e n o m e n o n
relation space
space Y, d e f i n e d by an etale equi-
S ~ V, w e can c o n s t r u c t
IYI by taking
tor
above,
I I applied
spaces.
X' has a n o n t r i v i a l
Hence
to X.
f > x yields
this functor
automorphism
local
of S ~ V in
For the a l g e b r a i c
is i s o m o r p h i c
to the map X'
local ringed spaces.
which becomes
IX'I
an a s s o c i a t e d
IYI to be the q u o t i e n t
the c a t e g o r y of local ringed X' c o n s t r u c t e d
of algebraic
Thus
space
the func-
an i s o m o r p h i s m is not full.
(the a u t o m o r p h i s m
the i d e n t i t y map of IX'I = X.
Hence
of
Similarly,
of X(p2)) I I is also
not faithful. O n e could ask w h e t h e r when restricted not know.
to separated
in nature
cient to r e s t r i c t 3.
al@ebraic
algebraic
But even if it were,
spaces occur
Example
the functor
Quotients space w h i c h
This example
spaces.
This we do
enough n o n s e p a r a t e d
(see below)
our a t t e n t i o n
I I is fully faithful
algebraic
and it w o u l d not be suffi-
to the separated
by Group Actions. is not a scheme.
case.
A separated nonsinqular Two theorems
is due to H i r o n a k a ( X V I I ~
of Artin.
and is taken from
M u m ford 0CKV~. All schemes c o n s i d e r e d numbers.
here are defined over the complex
15
Let V 0 be p r o j e c t i v e intersecting
normally
3-space
and Y1 and ~2 two conics
in e x a c t l y two p o i n t s P1 and P2"
i = 1,2, we c o n s t r u c t ~i by first b l o w i n g result.
up
~i'
Let V. be the open set in ~. of points l l
(V 0 - P3_i ) .
For
and then
~3-i
laying over
L e t U be obtained by p a t c h i n g V 1 and V 2 together
along the common open subset. U is a n o n s i n g u l a r
v a r i e t y on which,
over P1 and P2"
the two curves Y1 and 72 have b e e n b l o w n up in o p p o s i t e
order
U.
Let ~:V 0 + V 0 be a p r o j e c t i v e which p e r m u t e s P1 and P2" m o r p h i s m s:U ~ U w h i c h In this situation, nonprojective
variety,
and 71 and Y2"
IV.I.I below.)
of order 2
~0 induces
an auto-
is of order 2. H i r o n a k a has s h o w n ( X V I I I ) t h a t and that
there is no ~ u o t i e n t
the action of the group G = [i,~} (Specifically,
transformation
in the c a t e g o r y
there is no geometric Of course,
quotient
U is a of U by
of schemes.
in the sense of
there is a q u o t i e n t X a in the cate-
in the
16
gory
of a n a l y t i c
the q u o t i e n t But
this
spaces.
map.
is n o t
structure obtained
on X
a
~
the .
In
is not
a
source
Indeed,
by deleting
etale
if w e
not
variety
quotients
varieties
L e t X 1 be
of U 1 b y G.
~ has
the o p e n
laying
on w h i c h by
subspace
Then X 1 contains
all
fixed p o i n t s .
in f i n d i n g
let U 1 c U be
then U 1 is a p r o j e c t i v e
exist.
since
of the p r o b l e m
all p o i n t s
of p r o j e c t i v e
let ~ a :U ~ X a be
this c a t e g o r y ,
the o p e n
o v e r Y1
G = [i,~
finite
subspace
and Y2" acts,
groups
of X a w h i c h
the
algebraic
and
always
is the q u o t i e n t
famification
points
of
a :U ~ X a . Now and X 2 be
let U 2 c U be the
analytic
The m a p n 2 : U 2 gluing
we have
on a s c h e m e category Let one
U2,
modulo
where
subsets
and X a
freely,
of U 2 by G.
is o b t a i n e d
by
X 1 and X 2.
of a f i n i t e
the q u o t i e n t
now R =
map
~ U 2 be o EG E G. There
does
group not
the d i s j o i n t are
on each U 2 c R,
of U 2 to a(u). relation
an e x a m p l e
~ acts
is the q u o t i e n t
surjective
two o p e n
on w h i c h
G acting
exist
freely
in the
of s c h e m e s .
for each ~
identity
the
subspace
space which
~ X 2 is e t a l e
together
Thus
the o p e n
whose
quotient,
the a c t i o n
of G.
two maps,
and ~2
R ~ U 2 is then
union
R ~ U 2.
takes
a separated
if it e x i s t s ,
of c o p i e s ~i
u in the ~ etale
of U2,
is the th
copy
equivalence
is i d e n t i f i a b l e
as U 2
17
In the c a t e g o r y etale
equivalence
the s t r u c t u r e
of a l g e b r a i c
relations
X a has
(In fact
the s t r u c t u r e
complex
given
an a r b i t r a r y
group,
P. D e l i g n e
quotient
of a s e p a r a t e d
A number
has
always
theorem
certain
algebraic
Another
of finding
to the p r o b l e m scheme
on which
of
of e finite
space.
to finding
just
under
that
the action
of a l g e b r a i c
by a large
of a spaces.
can be
of schemes
has
the
interest.
of m o d u l i
quotients
quotients
of c o n s t r u c t i n g
is closed
space quotient.
theoretical
Seshadri
of the example
always
As above,
(unpublished)
space
finding
group,
varieties
in the c a t e g o r y than
of a
and a f i x e d - p o i n t - f r e e
shown
a ~uotient
a finite
the q u o t i e n t
is an a l g e b r a i c
algebraic
groups [XXVI).
aspect
that
in the c o n s t r u c t i o n
to the p r o b l e m
be reduced
space
recently
has m o r e
reduced
cases
algebraic
such quotients.)
there
exists
of p r o b l e m s
the p r o b l e m
analytic
three-dimensional
of p r o j e c t i v e
of taking
In fact,
This
the o r i g i n a l
the action
elgebraic
of a finite
group
Hence
of a s e p a r a t e d
the c a t e g o r y
the o p e r a t i o n
finite
so our X 2 above has
it is k n o w n
under
under
action
of s e p a r a t e d
exist
of a n o n s i n g u l a r
above,
variety, (I.e.,
quotients
space.)
As m e n t i o n e d
exists.
space.
the s t r u c t u r e
algebraic
projective
always
of a l g e b r a i c
quotient
spaces,
by
shown~XXXI 1 that group
can in m a n y
by
finite
groups.
above
is its
~pplication
Hilbert
group G acts
Schemes. freely,
If U 2 is a the q u o t i e n t
of
18
U 2 can be
formed
as a c e r t a i n
S c h e m e H of U 2. parametrizes an n - t u p l e
as above, can H.
the q u o t i e n t
spaces,
(~,6.2) :
theorem
Algebraization ditions spaces
space
spaces,
field or over
zero.
Then
tation
where
o v e r S.
space of
which
of A r t i n ' s
Picard
by an a l g e b r a i c
flat m o r p h i s m
of finite
domain. flat
locally
con-
application:
type over
Suppose
a
f is of
in d i m e n s i o n
functor P i C x / s spece
general
of a l g e b r a i c
f:X + S b e a p r o p e r
Dedekind
finite
is r e p r e s e n t e d
gives
another
S is l o c a l l y
For
type over k.
as an a p p l i c a t i o n
and c o h o m o l o g i c a l l y
the r e l a t i v e
is r e p r e s e n t e d
Let
then n e i t h e r
the f o l l o w i n g
the f u n c t o r Hilbx(-) of finite
If,
do not exist.
an a l g e b r a i c
We mention
an e x c e l l e n t
finite p r e s e n t a t i o n
Schemes
of
of G).
of G.
exist,
a sheaf on the c a t e g o r y
(~,7.3) :
of a l g e b r a i c
are the o r b i t s
(~,i.6,1.7)
is r e p r e s e n t a b l e .
Theorem
which
locally
Theorem
T consisting
(where n is the o r d e r
L e t X be
is p r o v e d
under which
T of U2, w i t h
Artin has proved
Then
C of the H i l b e r t
look at that p a r t of H w h i c h
Hilbert
however,
a field k.
subspace
of U 2 by G d o e s n ' t
in g e n e r a l
b y an a l g e b r a i c This
points
those n - t u p l e s
Theorem type over
subspaces
of d i s t i n c t
Hence
algebraic
Specifically,
closed
L e t C ~ H be
closed
(defined
in XVII)
of finite p r e s e n -
19
Example
4.
Modifications
to A n a l y t i c
Subspaces).
Application
Spaces.
A modification
is a p a i r
f:X' + X of a l g e b r a i c that
(Blowinq U p
the r e s t r i c t i o n
spaces
consisting and
of f to Y,
of a p r o p e r
a closed
s u b s p a c e Y c X such
fl I
morphism
:(X'-f-l(Y))
~
(X-Y),
X' _f-I (y)
is an i s o m o r p h i s m . There first
are
two p r o b l e m s
is the p r o b l e m
considered
of s t a r t i n g w i t h
in this context. a g i v e n X,
a modification
X' ~ X, Y c X, w i t h Y a p r o p e r
of X,
that X'
and such
arguments. that S,
is m o r e
The c l a s s i c a l
result
for any s c h e m e X, p r o p e r
a modification
surjective
this b e l o w
step
in the p r o o f
Grothendieck Hironaka's
0, and X'
over
X'
which
states base
scheme proper
problem
over
Theorem
and the
recent
result
(XIX),
is
in w h i c h
variety.
and a c l o s e d
f >X, Y c X, w i t h
IV.4.1,
A more
is to d e t e r m i n e ,
under what
We
a field k of c h a r a c t e r i s t i c
projective
a field k)
over S.
and it is a c r u c i a l
Theorem
of S i n g u l a r i t i e s
and a p r o p e r m a p Y' ~ Y, modification
subspace
separated
X' p r o j e c t i v e
T h e o r e m V.6.3.
is a n o n s i n g u l a r
(say, p r o p e r
lemma,
a noetherian
spaces,
of the F i n i t e n e s s
Resolution
The c o n v e r s e
and w i t h
for a l g e b r a i c
Existence
X is an i n t e g r a l
closed
to a l g e b r a i c - g e o m e t r i c
is C h o w ' s
over
and f i n d i n g
X' ~ X, Y c X e x i s t s w i t h X' ~ X b i r a t i o n a l ,
and p r o j e c t i v e ,
prove
amenable
The
given
s u b s c h e m e Y'
conditions f-l(y)
a scheme X'
= Y'
of X',
there e x i s t s and w i t h
the
a
20
restriction
of f to Y' ~ Y
The c l a s s i c a l
result
the g i v e n map. in this case
t h e o r e m w h e r e X'
is a n o n s i n g u l a r
curve
negative
In
on X' w i t h
(III),
Artin
modifications Theorem space of space. d-i
respectively,
Y'
bundle
Then
there
single point
and Y' and
of Y'
In other words, This
is a n a l o g o u s
for a n a l y t i c by Hironaka ology)
in the c a t e g o r y was
in X'
fications.
spaces
is the
and a s s u m e
following:
algebraic a closed
sub-
of d i m e n s i o n s
over k. ~
Let ~
d and b e the
is an ample b u n d l e
on
f:X' ~ X, Y c X, w i t h Y a
Y' m a y be c o n t r a c t e d Theorem
A previous who
of a l g e b r a i c
motivations
Satz
of this
that
This
in X.
8, p.
353)
sort w a s p r o v e d
(using our
of a s c h e m e
spaces.
as the n a t u r a l
to a p o i n t
(XVI,
result
showed
a subspace
one of the o r i g i n a l
algebraic
is p r o p e r
such d o w n w a r d
= y.
(unpublished),
in some cases,
case
are n o n s i n g u l a r ,
to G r a u e r t ' s
spaces.
and Y is a point.
a field k and Y' c X'
that Y°
f-l(y)
is a r a t i o n a l
showing when
A specific
is a m o d i f i c a t i o n
and
and Y'
L e t X' be a n o n s i n g u l a r
type over
A s s u m e X'
conormal
a theorem
can be made.
finite
surface
self-intersection,
has p r o v e d
(III,6.2) :
is the C ~ s t e l n u o v o
termin-
can be c o n t r a c t e d
theorem
of H i r o n a k a
for the c o n s i d e r a t i o n
category
in w h i c h
of
to do m o d i -
21
Using singular braic
Artin's
surface
theorem,
over
space but
not
we
the c o m p l e x
in X 0.
so t h a t points As
of C O .
known
an a l g e b r a i c tiple
this
times),
points Since
group
function,
of a
is an a l g e -
and C O an e l l i p t i c 3 and
and
given
inflection
2 3 2 of y z - x + z x = 0.) curve
for our
is g i v e n as
is p o s i t i o n e d
of the
an e l l i p t i c
structure
if C 0 ~
points
Q on C O w h o s e
C,
locus
XIII),
D O in the p l a n e ,
any i n t e g e r
the
set of p o i n t s
t h e n Q1 + Q2 +
For
over
c u t s C O at o n e
e.g.,
on its
intersection
which
t h a t C O is of d e g r e e
C O can b e
(see,
as above,
for any c u r v e
plane
at i n f i n i t y
(E.g.,
structure
tioned
assume
the l i n e
is w e l l
group
We
numbers
an e x a m p l e
a scheme.
L e t X 0 be the p r o j e c t i v e curve
can c o n s t r u c t
the
curve
the
a
C posi-
by
assuming
that
locus
of z e r o s
of
D O = [QI,Q2,...,Qq]
appear
has
appropriate
(where m u l number
of
"'" + Q q = 0 in the group.
n,
there
order
C O is u n c o u n t a b l e ,
are o n l y
in the
there
a finite
associated
is a p o i n t
P0
number
group
of
is n.
on C O of
infinite
order. Let X 1 be Let
C 1 be
point
the
surface
the p r o p e r
on C 1 o v e r P0"
so C 1 h a s
obtained
transform
of C O and P1 b e
C O in X 0 has
self-intersection
by blowing
up X 0 at P0" the u n i q u e
self-intersection
C l ' C 1 = 9 - 1 = 8.
C0"C 0 = 9
22
To continue, up X.I at Pi.
let Xi+ 1 be the surface o b t a i n e d b y b l o w i n g
L e t Ci+ 1 be the p r o p e r
the unique p o i n t of Ci+ 1 over P..
t r a n s f o r m of Ci and Pi+l
The s e l f - i n t e r s e c t i o n
1
C i + l . C i + 1 = Ci.C i - i = 9 - i. C o n s i d e r X10:
iiI (where the straight in the process).
lines are the e x c e p t i o n a l
Here Cl0.Cl0
the s e l f - i n t e r s e c t i o n normal
of C10
bundle,
above, a point
If we let ~
we have deg ~
degree on an elliptic curve
= i.
Z is a singular
f:Xl0
be a curve D on Z with z 0 f D.
projection
be the dual,
the
Applying Artin's Z, z 0 ¢ Z, with
theorem f(Cl0)
surface.
W e claim Z is not a scheme.
not i n t e r s e c t i n g
of
Any b u n d l e w i t h p o s i t i v e
is ample.
there is a m o d i f i c a t i o n in Z.
An i n t e r p r e t a t i o n
is that it is the degree of the
bundle of CI0 in XI0.
conormal
= -i.
curves p r o d u c e d
For it it were,
there w o u l d
f-l(D) c Xl0 w o u l d be a curve
CI0 , and the image D O of f-l(D)
under
the
XI0 ~ X 0 w o u l d be a curve in the p r o j e c t i v e p l a n e
= z0,
23
intersecting P0"
the o r i g i n a l
Let mP 0 = D O ~ C O .
elliptic curve C O only at the p o i n t Then
so P0 is a torsion point, Another fications
application
is to M o i s e z o n
Definition: (nilpotent
in the group
which
is a contradiction.
of A r t i n ' s
elements
in the s t r u c t u r e
components
a Moisezon
space
of X.
functions
and
Let d.l = dim C .l
on C i.
The
Space if the t r a n s c e n d e n c e
over C is d. for each i. l
For a general d i s c u s s i o n (XXII,XXIII,XXIV).
of these spaces,
N o t e that the t r a n s c e n d e n c e
is in any case at m o s t equal There
analytic
sheaf are allowed),
be the field of m e r o m o r p h i c
degree of K(Ci)
theorem on m o d i -
Let X be a c o m p a c t c o m p l e x
space X will be called
K(Ci)
general
Spaces.
let C l , . . . , C r be the i r r e d u c i b l e Let K(Ci)
law on C, mP 0 = 0,
see M o i s e z o n degree of
to d. (XXXII). 1
is a functor Algebraic
spaces ~ I
___~___2 (Complex A n a l y t i c
Spaces)
finite type over assigning space
to each a l g e b r a i c
(see 1.5.17).
space its "underlying"
By a simple extension
(XXX),
the r e s t r i c t i o n
proper
over C is fully faithful
analytic
spaces,
of this functor
the image including
analytic varieties.
of Serre's GAGA
to algebraic
and carries
analytic
spaces
them to c o m p a c t
at least the p r o j e c t i v e
24
Moisezon Moisezon
Ch.
l, Thm.
space X of d i m e n s i o n
i) showed
n, there
that
for a reduced
is a d i a g r a m
of irredu-
cible m o d i f i c a t i o n s X !
/\ y c X
where
X''
is a p r o j e c t i v e
if X is and of d i m e n s i o n all o f d i m e n s i o n By GAGA, jective
X''
variety.
non-reduced
space,
X'
f-l(y)
and
and Y,
is i r r e d u c i b l e f,,-l(y,,)
are
than n.
is the a n a l y t i c Artin
case
Theorem lence
less
X" D Y"
analytic n,
fl
(III)
and applies
(III,7.3) :
The
space
extends
Moisezon's
his general above
associated
theorem
functor
to a p r o -
result
to the
to obtain:
induces
an e q u i v a -
of c a t e g o r i e s : (Alg.
Spaces
proper
In other words, meromorphic
functions
over C)
every "is"
compact
~
(Moisezon
analytic
an a l g e b r a i c
Spaces)
space w i t h
space
enough
in a u n i q u e
manner. Having
constructed
a particular
then can try to show it is a scheme. line are p r o v e d
in this book:
algebraic
space,
Some r e s u l t s
one
along
this
25
l)
add s e p a r a t e d I f f:Y ~ X is l o c a l l y q u a s i r ~ n i u e ~ a n d X is a n o e t h e r i a n scheme,
then Y is a scheme.
is an immersion.
2)
If
3)
this a p p l i e s w h e n
f
(II. 6. ] 6)
f:Y ~ X is q u a s i a f f i n e
scheme,
E.g.,
so is Y.
or q u a s i p r o j e c t i v e
(II.3.8)
and X is a
(II.7.6)
If Y ~ g ? x
Z
is a c o m m u t a t i v e f affine, finite
4)
g finite
type,
Z a noetherian
surjective
then
separated
so is X.
If X is any a l g e b r a i c
With
the n o t a t i o n o f
space,
the g e n e r i c point. all p o i n t s
7)
(III.4.1)
algebraic
there
space
if X is integral,
one.
over S w h i c h
T h e n X is a scheme. L e t X be a curve,
red
open
sub-
U contains
variety,
r i n g and let X b e an
is an S - a l g e b r a i c
(A c o r o l l a r y
(V.4.9,10)
U contains
(V~4.4)
or a n o n s i n g u l a r
T h e n X is a scheme.
and X
(II.6.7)
If X is a n o r m a l
of c o d i m e n s i o n
space
is a d e n s e
L e t S = S p e c A, w h e r e A is an A r t i n algebraic
8)
5),
and of
(III.3.6)
s p a c e U c X w i t h U a scheme.
6)
scheme,
and h s e p a r a t e d
then X is a scheme.
If X is a n o e t h e r i a n is a scheme,
5)
diagram with
of
group.
5).
surface,
over
a field k.
26
We make
some c o m m e n t s
all the r e s u l t s general, The
is w h e n
a theorem
i:U ~ X w i t h
so in p a r t i c u l a r
p E i
there
an i n j e c t i v e
spaces.
uses
For
In mutandis.
the fact that
is an a f f i n e
and w i t h
map.
of e x t e n d i n g
over m u t a t i s
explicitly
(U)
scheme U
i an o p e n
algebraic
immersion,
spaces,
the
one can do is find U ~ X etale. S o m e of the p r o b l e m s w h i c h
For
to a l g e b r a i c
seem to c a r r y
a p o i n t p in a s c h e m e X,
and a m a p
best
theory
all of the r e s u l t s
exception
around
of s c h e m e
h e r e on the p r o b l e m s
instance,
sheaf
the G r o t h e n d i e c k - t o p o l o g i c a l
is not r e l e v a n t
be modified.
of II.4
shows
notion
of flask
on this n o t i o n m u s t
up h e r e
in the s o r i t e s
and the local E x t
functor
on
in V.5.
are e a s i l y m o d i f i e d .
A more theorems
and p r o o f s b a s e d
This problem
sheaf cohomology The proofs
arise can b e e a s i l y r e s o l v e d .
serious problem
use i n j e c t i v i t y
w e h a v e had to c o n s t r u c t Criterion
(III.2.5),
Extension
Lemma
o f open rather
Chow's
(III.l.l).
I I - V can be d i a g r a m m e d :
is that
subsets. different
Lemma The
a number
(IV.3.1)
logical
of the h a r d e r
For
this reason,
proofs
of the Serre
and the C o m p l e t e n e s s /
structure
of C h a p t e r s
27
Sheaf Criterion for Isomorphism
Weak Serre Criterion
Serre Criterion
(II.5.3)
Theorem(II.6.16)
(III.2.3)
(III.2.5)
Chevalley's Theorem
(III.4.1)
Chow's Lemma
(IV.3.1)
Completeness/Extension Lemma III.l.l
Devissaqe
(III.5.1)
~
~
The Finiteness Theorem
(IV.4.1)
Holomorphic Functions Theorem
Grothendieck
Of these steps,
the three mentioned
Existence
Theorem
(V.6.3)
above and II.6.16
have somewhat different proofs than in EGA.
The arguments
in the other steps are mostly from EGA and in fact most of Chapter V on formal algebraic spaces is practically a straight translation of EGA. Finally,
(The exception is V.4.4)
it should be noted that there have been other
candidates considered in the search for more general algebraicgeometric objects. (XXVIII)
These include the notion of Nash manifold
and Matsusaka's notion of Q-variety
(xxI).
Indeed,
(V.3.1)
28
in the case of varieties,
algebraic
spaces
are a special
have b e e n
considered,
case
of Q - v a r i e t i e s . More
general
ford's m o d u l a r since (e.g.,
algebraic Artin's
objects
topology spaces
(XXVII). seem
theorems).
W e exclude
to h a v e m o r e (But see XXXV.)
such
as M u m -
these
(for now)
geometric
structure
CHAPTER THE ETALE
l,
Grothendieck
2.
The
Zariski
3.
The
Flat
4.
The Etale
5.
Etale
i.
Grothendieck
(where
Topologies Topology
Topology
Equivalence
gory C consists families
TOPOLOGY
Topology
Definition
ONE
and D e s c e n t
A
of S c h e m e s . . . . . . . . . . . . . . . . . . . . . . . . . .
52
of S c h e m e s . . . . . . . . . . . . . . . . . . . . . . . . .
59
Relations ...........................
72
and D e s c e n t
(Grothendieck)
of a c a t e g o r y
covering
C = Cat
the
Theory
ToDoloqv T and
in C a t
~
on a c a t e -
a set C o v
~ called
r a n g e U of the m a p s
~ of
coverings
~i
is
fixed)
satisfying i)
If ~ is an i s o m o r p h i s m
2)
If
{U i ~ U}
each
i then
position 3)
If
6 e Cov the
then
T and
family
{~]
£ Cov
~.
[Vij + U I ] c C o v
{Vij ~ U}
obtained
T for by com-
is in C o y T.
[U i + U]
29 58
{Ui ----~ U ] i E I of m a p s
in e a c h
Theory ............
of S c h e m e s . . . . . . . . . . . . . . . . . . . . . . .
Topologies i.i:
OF S C H E M E S
E Coy
t h e n Ul × V e x i s t s U
T and V ~ U and
E Cat
{U i × V ~ V} U
~ is a r b i t r a r y 6 Cov
T.
I. 1
30
Definition
1.2:
w i t h products. F:C °pp + D.
Let T be a t o p o l o g y
A presheaf
A sheaf
on T w i t h
is a p r e s h e a f
If {U i ~ U~
(Exactness
here m e a n i n g
1.3:
In general,
interesting object
unless
X e C,
is a sheaf. Other [f~
this
of sets,
object
e C, w e w r i t e (group,
of sheaves
is not A0:
of ~i,~2.)
too
For every
of a b e l i a n
the
a singleton and
of groups,
values
in the
or etc.)
and an
to the elements
of F on X.
Finally,
an
groups.
is s o m e w h a t
theories
a sheaf
set
for a sheaf F
and refer
as sections
over
kernel
functor HOmc(-,X)
a sheaf F w i t h
= F(X)
the reader
and with
of m o d u l e s
GT and SGAA.)
is,
F(X,F)
etc.)
is exact.
sheaf X'.
(that
is a sheaf
notations
representable
or the c a t e g o r y
We assume these
the A x i o m
map,
category
sheaf
topology
f is a c o v e r i n g
etc.
Uj) U
For
groups,
abelian
F(U i ~ i,j
terminology:
of sets,
set
~
satisfies
the c o n t r a v a r i a n t
convenient
of this
the d i a g r a m
a Grothendieck
We write
in D is a functor
that ~ is the d i f f e r e n c e
it also
£ C o v ~ we say
X
~ ~2
values
satisfying
E C o v ~ then
F(U) - - ~ ~-i F(Ui)
and D a c a t e g o r y
familiar
of abelian
of rings.
with
sheaves
(See,
and
for example,
I.l
31
In keep
the
following
in m i n d a)
the
abstract
definitions,
two
standard
examples:
the G l o b a l
Topology
on
it w i l l
the c a t e g o r y
help
to
of t o p o l o g i c a l
spaces. Cat
T = the c a t e g o r y
of t o p o l o g i c a l
Cov
T = all
{Ui~i2u]
~i
b)
families
is an i m b e d d i n g
and w h e r e
U
the L o c a l
Topology
Cat
T = that
dings tive
on
category
e : U + X,
where
o f U. as an o p e n l
is c o v e r e d
by
the
each m a p
subset
images
o f U,
of the U . ' s . i
a topological
s p a c e X.
whose
are o p e n
and w h e r e
triangles
spaces
objects
maps
a : e l ~ 02
U1 -
s
imbed-
are c o m m u t a -
~ U2
X
Cov
~ = all
families
{U i
-> U] w h e r e
U
is c o v e r e d
X by ~.~: =
In
the
the
following
(C,Cov T) Definition
(under ~) for all
images
of the U o. 1
(through
a topology
on C,
1.5:
A class
if for a n y
[U i + U}
i, U
1
e S.
1.13)
let C be a c a t e g o r y
satisfying
the a x i o m A 0.
of o b j e c t s
S c C is s t a b l e
c Coy
c S if and o n l y
T, U
and
if
I. 1
32
Definition category
1.6:
D such
A closed
subcateqory
D of C is a s u b -
that
a)
D contains
b)
If
all
isomorphisms
U----~V
X ~ , y is a c a r t e s i a n Definition (under in C,
T) and
1.7:
diagram
A class
if D is a c l o s e d [Yi + Y}
E Cov
in C,
and
D of m a p s
subcategory
Tp if each
f e D,
then
f'
E D.
in C is s t a b l e and
for any
f. :X × Y. + Y. l l 1 Y
f:X + Y E D,
then
feD. Definition domain {Xi
(under
1.8:
A stable
class
T)
if
for any
~X ] E Cov
T,
f e D if and o n l y
Definition effective
1.9:
descent
F be
a sheaf.
that
for e a c h
A stable
if the
Suppose i,
f:X + Y
of m a p s
class
following
there
the s h e a f
E C, if
and
D of m a p s
6
C,
and
suppose
the m a p W . ~
1
F is r e p r e s e n t a b l e , (E.g., to be
U. & D.
1
the
say F = W"
in the g l o b a l
set of c l o s e d
family i,
f~i
e D.
of C s a t i s f i e s
of s h e a v e s
fiber product
on the
L e t ~ i ~ C o ~
U ~ 1
W
any
for all
holds:
is a m a p
is l o c a l
F-->U',
F = W[ u •
~en
and such
for s o m e
1
it m u s t
be
that
1
(and h e n c e
standard
imbeddings.)
ex~ple
the m a p W ~ U 1.4a
above,
6 D). take D
I .i
33
Definition ~of any
i.i0:
Let X
e C.
~ cofinal
set of c o v e r i n g s
X is a set
[[Xij ~ X } i e i . ] j ~ j c C o v T, such t h a t for 3 [U k ~ X ] k e K • C o v ~, t h e r e is a j E J and a m a p
family
n:I. ~ K, 3
and
X.. ~ U 13 n(i)
for e a c h
of
i.ii:
the
A cofinal
b)
For
set
X
¢ C.
A local
n(i)
such
that
[{Xij
~ (Xij)
each
E C,
commutative Xij
construction
~ on X
data: ~ Xli6i
each X.. ~ X a p p e a r i n g 13
object For
Let
following
a)
c)
a m a p Xi-3 ~ U
~ X = X.. ~ X. 13
Definition consists
i e I_, 3
~
and
]jE J of c o v e r i n g s of X. 3 in this c o f i n a l set, an
a m a p 0 (Xij)
~ Xij.
triangle Xi, j ,
X a map
~ (Xij)
~ 0 (X i, j,) (X i j) -
such .)
~ (X i,j,)
; Xi j
that
1 ---------- -)
Xi, j '
is c a r t e s i a n . A local an o b j e c t
construction
Y and
~ on X is e f f e c t i v e
a m a p Y ~ X,
Xi~a × Y = ~ (Xij) . X
In this
such
case,
that
if there
for each
we w r i t e
i,j
Y = ~ (X) .
exists
I. 1
34
Proposition on X.
1.12:
Let X e C and ~ be a local c o n s t r u c t i o n
L e t D be a stable class of m a p s of C satisfying
tive d e s c e n t
and suppose
that
in a family in the cofinal { (Xij) ~ Xij is in D.
for each
effec-
f..:X.. ~ X appearing 13 13
set associated
to 6, the map
T h e n ~ is e f f e c t i v e
(and hence the
map { (X) ~ X is in D). (To see this, represent,
1.13=
of C is stable
e f f e c t i v e descent) isfying P is stable 1.14:
W e will (
~
say a p r o p e r t y P of objects
if the class of all objects
for c o n v e n i e n c e
we need a p r e l i m i n a r y
a universal
satisfies
(all maps)
sat-
(etc.).
W e now drop our a s s u m p t i o n
Definition
of strict descent.)~
local o_n the domain,
the p r o b l e m of c o n s t r u c t i n g W e assume
the sheaf w h i c h ~ (X) should
and then apply the d e f i n i t i o n
Definition (maps)
first c o n s t r u c t
a topology
that C has
of 1.4 and c o n s i d e r for a given c a t e g o r y C.
fiber products.
First
definition.
1.15:
effectively
A family
{U i ~ U}iEI
epimorphic
objects W of C, and m a p s W + U,
family
of maps of C is
(UEEF)
if for all
and for all o b j e c t s V e C,
the following d i a g r a m of sets is exact:
HOmc(W,V)
~--~Hom (W X Ui,V) i C U
~ ~-~HOmc(W. . × z,3 U
A single map
f:V ~ U in C is a u n i v e r s a l
(UEE)
family
if t h e
[f] is a UEEF.
effective
(U i × Uj),V) U
epimorphism
I. 1
35
Definition The B - t o p o l o g y ciated
1.16:
Let B be a closed
on C, ~B
(also called
subcategory
the t o p o l o g y
of C.
on C asso-
to B) has Cat T
B
= C
C o v rB = A l l
families
and in w h i c h (One can check
that
{Ui-----~U } which each m a p ~i
this d e f i n i t i o n
are U E E F
is in B.
satisfies
the d e f i n i t i o n
i.i and the axiom A0). 1.17:
To get an i n t e r e s t i n g
on B m u s t be satisfied. stable
a list of p o s s i b l e
Definition object phic
(For instance,
in the B-topology.)
we give
1.18:
After
in general
a preliminary
and H O m C ( ~ , X ) has
Given
a class
unions
of C,
sum of the X° 's and l
initial
initial
for X not
isomor-
an object
of the class X.l - - w r i t t e n
if X is the c a t e g o r i c a l
disjoint
definition,
exactly one element.
[Xi}i¢ I of o b j e c t s
X. × X. is a strict i x 3
B is not
An object ~ of C is a strict
if for all X e C, H O m c ( X , ~ ) is empty
is the disjoint, u n i o n
some r e q u i r e m e n t s
requirements.
to ~,
objects
B-topology,
object
if the d i s j o i n t
of C.
union
We
of any
X of C
X = ~
for each
Xi-i,j
say C has (finite)
£ I,
(finite) set of
of C exists.
1.19: B of C m i g h t
W e now
list
satisfy
some
in order
axioms
that
to give
a closed
a nice
subcategory
topology.
I. i
36
SI:
Let
the d i s j o i n t
[X.~ ____t_> Y}iEI
be a set of maps
~ X. exists, and let ~:X ~ Y be the induced l i£I Then ~ e B if and only if for all i c I, ~i E B.
map.
union X =
(Thus if C has d i s j o i n t {U i + U]
$2: and only
unions,
in Cov T B can be r e p l a c e d
The r e s u l t i n g
the
of C for w h i c h
lack of indices
A map
any c o v e r i n g
by a c o v e r i n g
often m a k e s
f e B is a u n i v e r s a l
family map
arguments
[_~ U. + U. l iEI m u c h easier.)
effective
epimorphism
is then
just given by
if
if it is an epimorphism.
(Combining
Sl,S2,
"surjective"
maps
$3:
the B - t o p o l o g y in B.) f
X
Let
_~ Y
Z
be a c o m m u t a t i v e
diagram [fl
B-topology. and local
then
S3(b)
If g E B,
and
local
The global
then
unions,
and B satisfies
example
of all open
S 2 and $3, but B is not The s m a l l e s t on the d o m a i n
class
g ¢ B.
f 6 B.
on the d o m a i n
standard
B to be the class
satisfies
6 C o y TB,
If
then B is stable
taking
h 6 B.
S3(a)
(If C has d i s j o i n t
1.20:
in C with
B'
S 1 and S3(a),
in the B-topology.) of 1.4 is o b t a i n e d
imbeddings.
local
This
on the d o m a i n
containing
in its own B ' - t o p o l o g y
B which is
by
class in the closed
(by definition)
I.i
37
the class
of local
isomorphisms.
and S 3 and is the p r o t o t y p e wants when the local exactly
forming
those m a p s
this
fact w h i c h
map,
and
(See 4.1, The
4.5,
general
local
We
Lemma
of m a p s
one
of m a n i f o l d s ,
function
theorem,
criterion.
the d e f i n i t i o n
for the c a t e g o r y
It is of etale
of schemes.
e x a m p l e o f 1.4 has B as the class
in the 1.21:
for lack o f a b e t t e r
is used
again
and again
place,
a
(although
often
following. L e t C be
and D c C a c l o s e d
of
S 1 and S 3.
i n s e r t here,
argument which
implicitly)
a Jacobian
S1,S 2
4.6)
This B s a t i s f i e s
1.21:
In the c a t e g o r y
to give
topology
satisfies
of c l a s s
b y the i n v e r s e
is e x p l o i t e d
standard
all maps.
are,
satisfying
the etale
class
of the k i n d
topologies.
isomorphisms
This
a category with
subcategory.
Let
X
fiber p r o d u c t s
~
Y
be a com-
Z mutative
diagram
in C.
Suppose
f e D,
and the d i a g o n a l
map
A
Y~>
Y X Y is in D. Then Z Proof. The f o l l o w i n g
the c o m p o s i t e X
two d i a g r a m s
of the top lines
1 >< g
> i × Y
f y
g £ D. are c a r t e s i a n
is the m a p
and
g.
X × Y
->Y
f X 1 .
A
"~Y X Y Z
h X
-------~ Z f
1.2
38
1.22:
A
final comment.
of p r o o f s w i l l
be s i m p l i f i e d
are local on the o b j e c t s ranges
of the m a p s
and d e s c e n t theorem
involved.
conditions,
in some p a r t i c u l a r
The Zariski For
(with w h i c h
complete
d o n n e IX,X,XI, Definition be c o m m u t a t i v e Spec R, i) R°
of rings 2)
L e t R be
and p o s s e s s
is the f o l l o w i n g As a set,
This makes
can be
or the o r i g i n a l 2.1:
Spec
stability
to p r o v e
the
the g e n e r a l
statement.
we
the r e a d e r
treatments
in
of S c h e m e s
the sake of c o m p l e t e n e s s
definitions More
and then i n v o k e
or
is m e a n t
the a p p r o p r i a t e
it is s u f f i c i e n t
to get the full
Topology
what
and m a p s m e n t i o n e d
satisfy
case
a lot
or local on the d o m a i n s
of o b j e c t s
so that
chapters,
that the t h e o r e m s
In these cases,
and c o n c l u s i o n s
descent machinery
2.
by assertions
involved,
is that all the p r o p e r t i e s the h y p o t h e s e s
In the f o l l o w i n g
recall
the
is a s s u m e d
following
to b e
found in M u m f o r d source, a ring
a unit).
familiar).
XXV,
Grothendieck's (assumed
The p r i m e
DieuEGA.
throughout spectrum
to
of R,
object:
Spec R is the set of all p r i m e a contravariant
functor
ideals
of
from the c a t e g o r y
to that of sets. Spec R is a t o p o l o g i c a l
g i v e n b y ideals
of R.
space with
For an ideal
I,
the c l o s e d
the c o r r e s p o n d i n g
sets
I. 2
39
closed
set V(I)
is the set of all p r i m e
Spec is thus a functor
to the c a t e g o r y of topological
The topology can be d e f i n e d For any element modulo
Bourbaki,
equivalently
I.
spaces.
as follows:
f in R, we w r i t e Rf = {a/f n I a ¢ R, n = 0,1,2,...
the relation
integer k w i t h
ideals c o n t a i n i n g
a/f n = b / ~
fk(fma - fnb)
if and only if there is an
= 0 in R].
(For details,
see
Alg. Comm.)
The natural map R + R f ( a ~ Spec Rf ~ Spec R, w h o s e set V((f)) .
a/f 0) induces
an inclusion
image is the c o m p l e m e n t
of the closed
O n e can then show that a basis of open sets for
the t o p o l o g y on Spec R is given by all open subsets of the form Spec Rf, 3)
f e R.
The s t r u c t u r e
of rings assigning
sheaf of Spec R is the unique sheaf
to every open subset of the form Spec Rf,
the ring Rf. Definition logical
2.2:
A loca.__l ringed
space X with a sheaf of rings
space ~X'
(X, 0X)
~X,x ) .
A m_~
map of ringed
f: (XI~x) ~ topological
(Y, O y )
to nonunits.
(which we denote
o__f local rinqed
spaces such
x • X, the induced map of local rings nonunits
that the stalk
such
of the sheaf at any p o i n t x~X is a local ring
is a topo-
s p a c e s is a
that for every p o i n t ~Y,f(x)
~
~ X,x maps
I. 2
40
proposition space
2.3:
and Spec b e c o m e s
of rings
each
ring R, Spec R is a local
a contravariant
to the c a t e g o r y
is full and
of local
functor
ringed
ringed
from the c a t e g o r y
spaces.
This
functor
faithful. •
Definition dual
For
2.4:
The
of the c a t e g o r y
image of this
functor
(i.e.,
of rings)
is called
th___e c a t e g o r y
the c a t e g o r y
of affine
schemes
the Q~ affin____..e
schemes. W e note products, and
given by the tensor
finite
disjoint
But o b s e r v e ducts
that
that
of rings
ite p r o d u c t
unions,
the
By e x t e n d i n g
the c a t e g o r y
of schemes,
spaces,
of a f f i n e takes
one gets
schemes.
direct
disjoint
limits
Definition
later
still
to inverse
2.5:
An open
be true,
subspace
Y of X with
(Note Y need
not itself be an affine scheme
of the
form Z = Spec R/I w h e r e
pro-
schemes
of
to
of alge-
of d i s j o i n t however,
union
that Spec
limits.
X is an o p e n
Z of an affine
of rings.
of the u n i o n
of affine
notion
rings,
(Spec of an infin-
to the c a t e g o r y
subscheme
scheme
associated
take i n f i n i t e
compactification
the correct
It w i l l
not
fiber
products
unions.
the c a t e g o r y and
of the
finite
Spec does
is the S t o n e - C e c h
the Specs.)
braic
given by
functor
to i n f i n i t e
products
has
Y of an affine
the induced scheme.)
sheaf of rings. A closed
X = Spec R is an affine I is any ideal
scheme
of R.
sub-
scheme A
1.2
41
s u b s c h e m e W of X is an o p e n of X.
In these cases,
called
an o p e n
subscheme
of a c l o s e d
the a s s o c i a t e d
immersion,
subscheme
i n c l u s i o n Y ~ X is
Z + X a closed
immersion,
and W ~ X
an immersion. Definition
2.6:
A m o d u l e M over
a s h e a f of m o d u l e s ~ taking
F ( S p e c Rf,~)
over
a ring R g i v e s
the s t r u c t u r e
= M ® Rf.
rise
to
s h e a f on Spec R, b y
A general
s h e a f of m o d u l e s
F
R
on Spec R is c a l l e d some R - m o d u l e M. w e say ~ maps
is coherent.
that
containing
2.7:
subscheme
p such
for a l g e b r a s
S over R
type, (i.e.,
and c o h e r e n t
6 X,
there
ringed
space
is an open
(X, ~ x)
subset U c X
that
(U, .~vl ) is an a f f i n e scheme. U an a f f i n e s c h e m e is c a l l e d an a f f i n e
2.8:
A scheme
as a t o p o l o g i c a l
and w r i t e
"compact"
is a local
(Such a open
of X.)
Definition
has
is of finite
of q u a s i c o h e r e n t
A scheme
for e v e r y p o i n t p
map U ~ X with U
tion
Similarly
and M
for
of algebras.
Definition
compact
if it is of the form ~
If R is n o e t h e r i a n
o f rings R ~ S) w e s p e a k
sheaves
such
quasicoherent
the F r e n c h
meaning
a finite
quasicompact
space.
in e i t h e r
so for a s c h e m e that
it h a v e
case
if it is q u a s i -
(We r e l u c t a n t l y
"quasicompact"
subcovering.)
and s u f f i c i e n t
is q u a s i c o m p a c t
scheme
covering
covering
is a u t o m a t i c a l l y
to be q u a s i c o m p a c t a finite
tradi-
for the E n g l i s h
that e v e r y open
An affine
follow
it is n e c e s s a r y
by a f f i n e
schemes.
I. 2
42
Definition ~X-mOdule. there
2.9:
(X, ~ X ) be a s c h e m e
F is q u a s i c o h e r e n t
is an open
sequence
Let
~X I ~
if for e v e r y p o i n t p of X
s u b s e t U of X, w i t h p ~X J ~ F I
and F an
~ 0 of
¢ U,
and an e x a c t
~X-mOdules
(where
~X I
U
and
~X J denote
infinite
index
this c l e a r l y We
the sums of the m o d u l e sets I and J) .
agrees w i t h
say F is l o c a l l y
can choose
the open
some i n t e g e r
n.
the p r e v i o u s
definition
scheme,
2.6.
free if it is q u a s i c o h e r e n t
sets U above
n
the p o s s i b l y
W h e n X is an affine
is u n i q u e l y
and one
~n OX
so that
If F is a l o c a l l y
p o i n t p the n u m b e r
LI~X over
I ~ F I , for U U free sheaf, then for each
determined,
and is c a l l e d
P the rank
of F at p.
If F has
the same rank n
for e v e r y p o i n t P
p,
this n u m b e r
an i n v e r t i b l e
is c a l l e d
sion)
2.10:
A map
(respectively
~ U
is an open
of s c h e m e s
closed
if for any a f f i n e
f-l(u)
open
subscheme
immersion
an immersion)
abuse
of language,
we say Y is an open
a subscheme.
U of X,
immer-
the m a p closed
in the sense of 2.5. subscheme
immerBy
(closed
sub-
of X.
If X is a s u b s c h e m e X is a s u b s c h e m e
respectively
(respectively
respectively
subscheme)
f:Y ~ X is an open
immersion,
sion,
scheme,
If n = i, F is c a l l e d
sheaf.
Definition immersion
the rank of F.
of Y
and Y is a s u b s c h e m e
of Z,
then
of Z.
The
intersection
of two s u b s c h e m e s
But note
that
an a r b i t r a r y
monomorphism
is
in the
1.2
43
category
of schemes
Proposition products (Affine serves
2.11:
and d i s j o i n t
The c a t e g o r y sums.
inclusion
is full,
(etc.)
that
the classes
of schemes the sense
2.12:
A map of schemes
(surjective , bi~ective,
tive
of injective,
are s u b c a t e g o r i e s of 1.6).
we d e f i n e
has
fiber
functor
faithful,
and p r e -
of
g:Z ~ Y
(Schemes).
(In E.G.A.
the term
necessary map
for the n o t i o n s
if f is injec-
One
open
the induced open m a p s
and closed
open
(in
if for
f':X X Z ~ Z is Y is then a c l o s e d universally closed.
for u n i v e r s a l l y
are rather
inefficient
f is u n i v e r s a l l y
injective.)
when
it comes
something,
the t o p o l o g i c a l
Luckily
of u n i v e r s a l l y
maps
map
and u n i v e r g a l l y
to check
note
subcategories
we d e f i n e
is used
g:Z + Y.
should
say in the case of open
Similarly
radiciel
a given m a p
for every p o s s i b l e
it is s u f f i c i e n t
this,
bijective
The above d e f i n i t i o n s
it is a p p a r e n t l y
spaces.
bijective,
of u n i v e r s a l l y
universally
that
closed)
f:X ~ Y to be u n i v e r s a l l y
injective,
to p r o v i n g
f:X ~ Y is called
b u t not closed
To r e m e d y
a map
The class
subcategory
open,
as a m a p of t o p o l o g i c a l
e v e r y m a p of s c h e m e s
that
The
of schemes
fiber products.
injective
open.
not be an immersion.
schemes) C_~-~ (Schemes)
Definition
maps,
need
condition
one can show
injective
to check w h e n Z is the affine
since
(EGA 1.3.5)
and bijective, spectrum
of an
I. 2
44
algebraically open
closed
and closed,
general
field.
the c o n d i t i o n s
(see E G A II.5.6.3,
Definition if the induced
For
2.13:
are either
of u n i v e r s a l l y
more
subtle
or less
IV.14).
A map
diagonal
the n o t i o n s
of schemes
f:X ~ Y is s e p a r a t e d
m a p A:X ~ X X X is a closed immersion. Y if the n a t u r a l m a p X ~ Spec Z is separ-
A scheme X is s e p a r a t e d ated. An affine the d i a g o n a l
scheme
open
2.14:
~JX.l,
fl
:X. ~ Y is an open l
of schemes
on each
any scheme X,
(EGA 1.5.3.9) .
f:X ~ Y of schemes
for short)
Xi,
For
is a u n i o n
if X is the d i s j o i n t
of w h i c h
of union,
the r e s t r i c t i o n
immersion.
i Proposition
subcategory SI,
A map
sets (Z-open,
X =
X
separated.
m a p X ~ X × X is an i m m e r s i o n
Definition Zariski
is c l e a r l y
The c l a s s o f Z - o p e n
of the c a t e g o r y
S 2 and S 3(b) Definition
of schemes
2.15:
(but not 2.16:
of schemes
maps
is a closed
and s a t i s f i e s
S 3(a)) .
The Z a r i s k i
is the t o p o l o g y
axioms U
topology
associated
with
on the c a t e g o r y the class
of Z - o p e n
maps. Definition-Proposition of schemes i)
are stable X is l o c a l l y
2.17:
The
in the Zariski noetherian
of X by affine
schemes,
following
properties
topology:
(i.e., each
there
of which
is a c o v e r i n g is the s p e c t r u m
I. 2
45
of a n o e t h e r i a n locally
ring.
noetherian
An affine
if and only
scheme
Spec R is
if the ring R is
noetherian) . 2)
X is r e d u c e d affine
3)
schemes,
each
X is n o r m a l
(i.e.,
~
X,x
ring
property
there
dimension
are stable
x e X,
ring.
the
This
X is regular.) field k
of schemes each
(i.e.,
and a c o v e r i n g
of w h i c h
2.18:
The
following
is of Krull
properties
topology:
~uasicompact
(f:X + Y is q u a s i c o m p a c t
quasicompact
open
separated
local
and is i n t e g r a l l y
local
n over a ground
in the Zariski
is quasicompact.) 2)
the
n over k) .
Definition-Proposition
I)
expressed:
schemes,
x 6 X,
for each p o i n t
is a map X ~ S p e c k
of X b y affine
of m a p s
domain
~. is a r e g u l a r 2 % ,x
X is of d i m e n s i o n
of
field).
(i.e.,
is also
is the s p e c t r u m
for each p o i n t
in its q u o t i e n t
of X b y
elements).
is an integral
X is n o n s i n g u l a r local
is a c o v e r i n g
of w h i c h
no n i l p o t e n t
closed
5)
there
a ring w i t h
ring
4)
(i.e.,
subspace
U c Y,
if for every
f-l(u)
= U × X Y
I. 2
46
3)
quasiseparated
(f:X + Y
is q u a s i s e p a r a t e d
4)
m a p A : X + X x X is q u a s i c o m p a c t . ) Y universally injective
5)
universally
bijective
6)
universally
closed
7)
an i s o m o r p h i s m
if the
diagonal
D e f i n i t i o n - P r o p o s i t i o n 2.19: of m a p s
are s t a b l e
and local
The
following properties
on the d o m a i n
in the Z a r i s k i
topology : i)
locally
of finite
type
(f:X -~ Y is l o c a l l y o f finite
type if for any p o i n t x e X, subschemes x
6 U,
there
are affine
open
U = S p e c R c X and V = S p e c S c Y with
f(x)
e V,
fJ
:U -~ V and the a s s o c i a t e d m a p U S ~ R of rings m a k e s R f i n i t e l y g e n e r a t e d as an S-algebra. )
2)
locally
of
as above
except we
presented
3)
further
(the same d e f i n i t i o n
reauire
R to be f i n i t e l y
as an S-algebra.)
surjective
(note,
incidentally,
that
a surjective
m a p m a y not be an e p i m o r p h i s m
in the c a t e g o r y
schemes,
a Zopen map
jective
4)
finite p r e s e n t a t i o n
flat
and v i c e - v e r s a . if and o n l y
(see def.
3.1)
But
of
is sur-
if it is an epimorphism.)
1.2
47
5)
faithfully
flat
(see def.
6)
etale
7)
universally
8)
locally
quasifinite
locally
of
(see def.
f-l(p)
3.1)
4.1)
open
finite
(those maps
type and
= X X p has
f:X + Y w h i c h
for all p o i n t s
a discrete
underlying
are
p + Y, topological
Y
space.)
I
Definition-Proposition maps
satisfy i)
effective
Affine open
maps
2.20:
descent
In this case,
in the Z a r i s k i
U of Y,
sheaf
uniquely
determined
To p r o v e
affine m a p s
Spec
(g'A)
of
A=
~y-modules by A.
classes
of
topology:
if for every
affine
= U X X is affine. Y f*(~'X' A is a q u a s i and X and
We write
are effective,
for all schemes
f are
X = S~
A.
one needs
Z and m a p s
to
g:Z ~ Y,
= Z × X.) Y immersions.
2)
Open
3)
Closed
4)
Immersions
5)
Quasiaffine exists such
f-l(u)
if w e w r i t e
coherent
that
following
(f:X ~ Y is affine
subscheme
show
The
immersions
maps
(f:X ~ Y is ~ u a s i a f f i n e
a scheme W,
that
f = hg,
and h is affine.
and m a p s
g : X ~ W and h : W + Y
g is a q u a s i c o m p a c t A useful
if there
fact--not
open
immersion
necessarily
for
1.2
48
the proof here--is
a theorem of Deligne
(EGA IV.18.12.12) : quasicompact, point y e Y,
f-l(y)
crete.
f is quasiaffine.)
constructions
Then
on a scheme X
set of coverings
as a topological
2.21:
and for every space is disI
The following
are local
(where in each case the cofinal
is all coverings of X by affine schemes):
For a given quasicoherent the scheme S ~ ~(u)
2)
f:X ~ Y is separated,
locally of finite type,
Definition-Proposition
i)
Suppose
A°
sheaf of
~x-algebras
A,
(For an inclusion ~:U + X with affine U,
= Spec F(U,A).)
For any subspace Y of X, the reduced closed subspace of X, whose set of points
is the topological
closure of Y. 3)
For a given map f:Y + X, with separated, image. here
the scheme-theoretic
(By EGA III.l.4.10, (II. ~ 6
~x-algebra.
of ideals satisfying ~
~
~ X ~ f* ~Y"
and
closure of its
or the equivalent proof
), these conditions
is a quasicoherent
0 ~
f quasicompact
imply that f, ~ y Let ~
be the sheaf
the exact sequence The schem_____e-theoretic closure
of the image of f is defined
as Sp~ec ~
is reduced and f is a quasicompact gives the previous definition
2).)
.
If X
immersion,
this
I. 2
49
4)
For a g i v e n
closed
subscheme
of X,
its open
complement.
5)
For
a given open
subscheme
of X,
its r e d u c e d
closed
complement.
6)
The associated
reduced
b y the r e q u i r e m e n t subscheme for m a p s
2.22: make
the
Definition i)
that X r e d is a r e d u c e d
of r e d u c e d
schemes
quasicompactness
A quasicoherent
if it is l o c a l l y
s h e a f F on a n o e t h e r i a n
if for any p o i n t x
OXI
our p r e v i o u s
scheme
w e can also
and q u a s i c o m p a c t .
sequence
scheme.
hypotheses,
definitions:
s u b s c h e m e U c X, w i t h x ¢ U,
with
closed
to X.)
A s c h e m e X is n o e t h e r i a n
is c o h e r e n t
exact
(Defined
A.:
noetherian ii)
of X, X r e d.
of X and the m a p X r e d ~ X is u n i v e r s a l
Applying
following
scheme
Note
~
finite
G X J ~ F + 0.
definition
that a l o c a l l y
is n e c e s s a r i l y
e X,
scheme
there is an o p e n sets I and J and an This
clearly
for a n o e t h e r i a n
agrees
affine
free s h e a f on a n o e t h e r i a n
coherent.
1.2
50
Definition i) type
B.:
Let
f:X ~ Y be a m a p
f is of finite
type
of schemes.
if it is locally
of finite
and q u a s i c o m p a c t . ii)
f is of finite
of finite p r e s e n t a t i o n ,
presentation
if it is l o c a l l y
quasicompact,
and the induced
A
m a p X + X × X is q u a s i c o m p a c t . Y iii) f is finite if f is affine, and
f, ~ X
as an
~y-module
iv)
f is q u a s i f i n i t e
and q u a s i c o m p a c t . now states
that
Y is noetherian,
is coherent. if f is locally
(Deligne's a quasifinite
theorem
quasifinite
2.20(5)
separated
map
above is q u a s i -
affine.) v)
f is q u a s i s e p a r a t e d
if the induced
map
X + X × X is q u a s i c o m p a c t . Y Proposition
2.23:
for any q u a s i s e p a r a t e d is quasicompact) two p r o j e c t i o n ~roof. only
Let X be a q u a s i c o m p a c t scheme Y
any map
use
× Y (Spec Z) A l s o the
m a p s X × X ~ X are q u a s i c o m p a c t .
if the m a p X ~ Spec
assertion,
Then
the m a p Y ~ Y
f:X ~ Y is q u a s i c o m p a c t .
The m a i n p o i n t
is a p u l l b a c k
(i.e.,
scheme.
of this map 1.21,
is that X is q u a s i c o m p a c t
Z is q u a s i c o m p a c t . so q u a s i c o m p a c t .
the c o m p o s i t e
the q u a s i s e p a r a t e d n e s s .
if and
Each X × X ~ X For
the m a i n
m a p X ~ Y + Spec
Z, and •
1.2
51
Proposition finite
type,
finite maps,
2.24:
maps
The classes
of m a p s
of finite p r e s e n t a t i o n ,
and q u a s i s e p a r a t e d
maps
of s c h e m e s
finite maps,
are s t a b l e
of quasi-
in the Z a r i s k i
topology.
I
Assumption will
assume
2.25:
Lemma
technical
reasons
(see II.l.9)
from n o w on that all the s c h e m e s w e deal w i t h
quasiseparated. can b e seen
For
That
from the 2.26.
this
is not too s e r i o u s
following
we are
a restriction
lemma:
L e t S be a s e p a r a t e d
U an S - s c h e m e w i t h U + S l o c a l l y
noetherian
of finite
scheme
type.
and
T h e n U is
quasiseparated. Proof.
U × U must be
is an immersion. Proposition in the Z a r i s k i a coherent o f X,
locally
2.27:
The
s h e a f on X.
following
is a local
L e t X be a n o e t h e r i a n For any a f f i n e
let M be the R - m o d u l e
L e t I be the i n t e r s e c t i o n
~ ( S p e c R)
and U + U × U
H e n c e U ~ U × U is q u a s i c o m p a c t .
topology.
for w h i c h m M ~ M
noetherian
of f i n i t e
open type
of all m a x i m a l
This
construction scheme
subscheme for w h i c h ~
and F Spec R = FI Spec R
ideals m of R
( e q u i v a l e n t l y M ® R / m ~ 0).
= Spec R/I.
I
local c o n s t r u c t i o n
Put is e f f e c t i v e
I. 3
52
and w e w r i t e
~ (X) = Supp F, c a l l e d
is then the r e d u c e d
closed
only
E X
those p o i n t s
x
the s u p p o r t
subspace
for w h i c h
of F.
of X c o n t a i n i n g
Supp F all and
the stalk of F at x is n o n -
zero.
J 2.28:
schemes Here
but rather
S is taken
S-scheme Maps
In g e n e r a l
one d e a l s
all s c h e m e s
f:X + Y of S - s c h e m e s X
over
to be a s e p a r a t e d
is a s c h e m e X w i t h
triangles
not w i t h
f
the c a t e g o r y
a fixed b a s e
noetherian
a map X + S
are r e q u i r e d
o f all
scheme
scheme.
S.
An
(the s t r u c t u r e map). to fit into c o m m u t a t i v e
~_Y
S S i n c e S is q u a s i c o m p a c t separated,
quasiseparated,
and separated,
quasicompact,
sense i f and o n l y if the s t r u c t u r e Having made all m e n t i o n
these c o n v e n t i o n s ,
of S and just write,
an S - s c h e m e X is
etc.,
in the
absolute
m a p X ~ S is separated, we will
e.g.,
usually
X × Y
etc.
suppress
for the p r o d u c t
X × Y of S - s c h e m e s . S
3.
The Flat
Topology
of S c h e m e s
W e m a i n l y deal h e r e w i t h of a f f i n e schemes
schemes.
is i n d i c a t e d
the flat t o p o l o g y
The generalization in 3.12.
on the c a t e g o r y
to the c a t e g o r y
The results
quoted
here
of all consti-
I. 3
tute
53
the b u l k
of B o u r b a k i ,
VI,VIII,
and E G A IV.2.
descent,
see (V~.
Alg.
schemes,
f is flat if any,
Ch.
For an e l e m e n t a r y
Definition-Proposition m a p of a f f i n e
Comm.,
3.1:
Let
all,
(VIII),
discussion
f*:Spec
or e q u i v a l e n t l y
hence
1
SGA
60-61,
of flat
S ~ Spec R be a
f:R ~ S a m a p of rings.
of the f o l l o w i n g
equivalent
condi-
tions hold: i)
f,:(R-modules)
any e x a c t induced
sequence
sequence
~ :
~
(S-modules)
is exact.
I.e.,
for
0 ~ M' ~ M ~ M '° ~ 0 of R - m o d u l e s ,
~ ~ S:
the
0 ~ M' ® S ~ M ~ S ~ M" ® S ~ 0 is
R
R
R
R
exact. 2) type,
For
any exact
the i n d u c e d
sequence
sequence
~of
R-modules
of finite
~ ® S is exact. R
3) type,
For
any m o n o m o r p h i s m
the i n d u c e d m a p
M' ~ M of R - m o d u l e s
S ® M' + S ® M R
4)
For any ideal
of finite
is a m o n o m o r p h i s m .
R
I of R,
I ® S ~ S is injective.
(In
R
o t h e r words,
I ® S ~ IS.) R f is f a i t h f u l l y flat
any,
hence i')
all,
(written
of the f o l l o w i n g
F o r any s e q u e n c e
if the induced
sequence
fflat)
if f is flat and
equivalent
conditions
hold:
~ : 0 ~ M' ~ M ~ M" ~ 0 of R - m o d u l e s , ~
S is exact,
so is
[ .
R
2')
For
the s e q u e n c e
any s e q u e n c e ~® R
~ of R - m o d u l e s
S is exact,
so is
of finite
type,
if
I. 3
54
3')
For
any m a p M' ~ M of R - m o d u l e s ,
M' ® S ~ M ® S is i n j e c t i v e , R
if the
induced
map
so is M' ~ M.
R
4')
For
any ideal
I of R,
5')
For
any R - m o d u l e
M,
f-l(Is)
= I.
if M ® S = 0,
t h e n M = 0.
R
6')
For
q of S w i t h 7')
any p r i m e
f-l(q)
= p.
f*:Spec
epimorphism 3.2: R-module,
ideal
(I.e.,
in the c a t e g o r y
and
the
Conversely,
satisfies
diagram
the
of all
M e R
ideal
S ~ S p e c R is s u r j e c t i v e . ) effective
schemes. flat m a p
and M is a n y
(s ® s) N
suppose
"usual
we
are g i v e n
an S - m o d u l e
N
S ®N R cocycle
of S ® S ® S - m o d u l e s
R
is a p r i m e
sequence
an S ® S - i s o m o r p h i s m R ~: N~ S+ R
which
f*:Spec
If R ~ S is a f a i t h f u l l y then
there
S ~ S p e c R is a u n i v e r s a l l y
M ~ M ® S R is exact.
p of R,
condition":
the
following
commutes
R ~o ® i ....
N ® S ® S R R
S®N®S R R
/ l®~
J S®S®N R
where
8 is " t e n s o r i n g
8(n ® s I ® s 2)
=
with
R
~ in the m i d d l e " - - i . e . ,
(i ® s I ® i)
• dl(~(n
® s2) , s c a l a r m u l t i -
I. 3
55
plication
in S ® S ® N, and dl(S ® n) = s ® 1 ® n. R
R
Then there is a unique R - m o d u l e M with N = M ~ S.
It
R
is this property,
together w i t h the c o r r e s p o n d i n g
for m a p s of R-modules, statements
is the basis
of the d e s c e n t
in this section.
Proposition a) a closed
which
statement
3.3:
The class of flat maps of affine
subcategory
of
(Affine Schemes)
SI, S 2 and S3(a ) of 1.19 b)
and satisfies
3.4:
is axioms
(but not S3(b)).
An open i m m e r s i o n of affine schemes
Definition affine schemes
schemes
is f l a t . "
The flat t o p o l o g y on the c a t e g o r y
is the topology
of
a s s o c i a t e d with the class of
flat maps. Proposition affine
schemes
3.5:
The following p r o p e r t i e s
are stable
in the flat topology:
i)
Universally
injective
2)
Universally
bijective
3)
Universally
closed
4)
Finite
5)
Finite presentation
6)
Finite
7)
Etale
8)
Quasifinite
9)
Isomorphism
type
of maps of
I. 3
56
Proposition stable
and local
schemes
:
3.6:
following
on the d o m a i n
i)
Surjective
2)
Flat
3)
Fflat
4)
Universally
Proposition satisfies
The
Proposition
in the
The class
descent
3.8:
Let
tively
3.9:
3.10: by the class the d e s c e n t indicated
theorems
by the
f is u n i v e r s a l l y
is flat
(respec-
(Quasicoherent
Sheaves
on X)
faithful). of schemes
determined
is too fine to be able to g e n e r a l i z e
following Let
schemes,
open.
if Y is noetherian.
on Y) ~
and
The r e m e d y
for this
situation
proposition: f:X ~ Y be
of finite p r e s e n t a t i o n .
{Xi]iE I of affine
schemes,
functor
3.3-3.7.
3.10:
Then
schemes
topology.
on the c a t e g o r y
of flat m a p s
Proposition locally
exact
The t o p o l o g y
of affine
f:X ~ Y of schemes
sheaves
(respectively
flat
if and only
f_flat) if the induced
is exact
on affine
f:X + Y be a m a p of affine
A map
f*:(Quasicoherent
of all m a p s
in the
X is n o e t h e r i a n
Definition
topology
are
•
flat and of finite p r e s e n t a t i o n . Furthermore
flat
of m a p s
open
3.7:
effective
classes
and
Then
faithfully there
for each
exists
flat and a family
i E I, a m a p X.I ~ X,
is
1.3
57
m a k i n g X.
a disjoint
union
of
(Zariski)
open
subschemes
of X;
1
a family and
[Yi]i• I of affine
for each
each
schemes,
i E I, an open
i • I, a map
indexed
immersion
f.:X. ~ Y. such 1 1 1
Y. ~ Y; 1
and
finally
[Xi}
is a c o v e r i n g
of X in the Zariski
topology
ii)
[Yi}
is a c o v e r i n g
of Y in the Zariski
topology
For each
iv)
Each
Proof. f is locally
i • I, the
following
X. 1
~ X
Y. 1
>Y
f. is f a i t h f u l l y 1
type
implies
• Y,
that
subsets
x • U' c X and y • V c Y, with x x
tion
of
f
3.8,
the image
moment. CX
to
For
Ux' i s
locally
f(U~)
is open
any point
v
of V v so an open
subset
an open c o v e r i n g
compact
so there
finite
in V
E Vx,
and v • V v c Y w i t h
one can c h o o s e V v c Vx.
forms
of
of Vx. of V . x
is a finite
commutes
x
the c o n d i t i o n
there
.
f(U a) c V . x x
The r e s t r i c -
and
flat
are
affine
open
set V v l
so by
sets
and U v nonempty.
f ( U v)
is
an open
Clearly subset
The set of all such Vv, Since
open
Fix U' and V for the x x
c Vv, each
that
are affine
presentation
there
f(Uv)
Again,
diagram
flat.
For each x • X, y = f(x) of finite
for
that
i)
iii)
Uv
by the same set I,
V ,Joo
x
is
affine,
'Vvn such
it
that
v • Vx,
is
quasi-
the
1.3
58
union of the images of the c o r r e s p o n d i n g Let U x be the d i s j o i n t
Uvl'''''Uvn
cover Vx.
union of Ux', U v l ' ' ' ' ' U v n
Now
let the index set I be the c o l l e c t i o n
of p o i n t s
and for each i e I, X. and Y. the U. and V. c o n s t r u c t e d 1 1 1 1 It is clear
from the c o n s t r u c t i o n
Proposition
3.11:
above.
that i-iv are satisfied. •
The class of maps of schemes w h i c h
are flat and locally of finite p r e s e n t a t i o n c a t e g o r y of the c a t e g o r y of schemes
is a closed sub-
and satisfies
axioms S 1
and S 2 .
•
Definition schemes
of X,
3.12:
The flat t o p o l o g y
is the topology
a s s o c i a t e d with
on the c a t e g o r y of the set of maps,
flat
and locally of finite presentation. O n e can now p r o v e d e s c e n t proving
for this topology by
them for the flat t o p o l o g y on affine schemes
the Zariski
t o p o l o g y on all schemes
Proposition
3.13:
following p r o p e r t i e s rive,
theorems
and then applying
are stable:
u n i v e r s a l l y bijective,
universally
quasifinite,
finite,
etale,
universally closed,
quasicompact,
the
injec-
finite
type,
separated,
and the p r o p e r t y of b e i n g an isomorphism.
The following p r o p e r t i e s the domain:
3.10.
In the flat t o p o l o g y on schemes,
of maps
finite presentation,
and for
surjective,
of maps
flat fflat,
are stable and local on and u n i v e r s a l l y
open.
1.4
59
The open
following
immersion,
properties
affine,
of m a p s
closed
satisfy
immersion,
effective
immersion,
descent:
and
quasiaffine. The
following
are local
Spec of a q u a s i c o h e r e n t closure
of the
Our definition
The Etale
with
x e U,
f(x)
6 V,
R ~ S satisfies the form S
4.1:
that other
on the c a t e g o r y
to the s o - c a l l e d
A map
there
that
fppf
o
f:X ~ Y of schemes
is an affine
f(U)
open
flat
types
topology
open
subscheme
subscheme
Condition:
if for
U = Spec
map of rings
S is an R - a l g e b r a
O
the d e t e r m i n a n t
of the J a c o b i a n
S c X
V = Spec R c Y w i t h
" X n / ~ ( f l ( X l , . . . , X n) ,.--,fn(Xl ,...,x n)) where
of
of schemes.
is etale
c V and the a s s o c i a t e d
the J a c o b i a n
R [ X I,
in the
of Schemes
and an affine
such
the
flat of finite p r e s e n t a t i o n ) .
Topology
any p o i n t x e X,
morphism,
of schemes
take w a r n i n g
defined
corresponds
Definition
separated
subscheme.
should
have b e e n
(fppf = faithfully
4.
the s c h e m e - t h e o r e t i c
noetherian.
The r e a d e r
topologies
in the flat topology:
of algebras,
is a stable p r o p e r t y
locally
3.14: flat
of a c l o s e d
following
topology:
sheaf
image of a q u a s i c o m p a c t
open c o m p l e m e n t The
constructions
matrix
(~fi/)
is a 3
of
I. 4
60
unit in S.
(The d e r i v a t i o n s
of course
are formed as R - d e r i v a -
tions in the ring R[Xl,...,Xn] ) . A consequence
of this d e f i n i t i o n
is that a map of rings
R + S is etale if and only if S is an R - a l g e b r a of the form S = R[XI,
.
]/( n ~ ..,X n" .fl,...,fm )
m, w h e r e
the ideal in S
g e n e r a t e d bY the n × n m i n ° r s °f the d e t e r m i n a n t I ~ f i / ~ j ~ i S x
the unit ideal. for separable is a field,
In particular,
algebras
by the usual J a c o b i a n
over a field,
if R ~ S is etale and R
then S is a finite p r o d u c t
field e x t e n s i o n s Alternative
criterion
of finite separable
of R. Definition
4.2:
the s m a l l e s t closed s u b c a t e g o r y
The class of etale maps is of the c a t e g o r y of schemes
which I)
Includes
2)
Is stable and local on the d o m a i n
topology 3)
all etale maps
in the Zariski
and Satisfies
axiom S 1 of 1.19.
(Note that we could replace maps of finite type, schemes
f:X ~ Y with X, Y affine.
satisfying
or w i t h
i")
1 with
i')
includes
the J a c o b i a n criterion.)
includes
all etale
all maps of affine •
1.4
61
Alternative unramified,
Definition
if the induced
A m a p of schemes fied,
and
locally
Proof. tation.
is etale
Consider
of schemes
and
unramified.
which
those m a p s
of finite
is d e t e r m i n e d type.
type
and the g e o m e t r i c
fibers
union
condition
f:X ~ Y over
of affine
of the r e s i d u e the
Alternative etale
By Mumford
are
finite
any p o i n t
scheme Y0'
of Y'
every p a i r
of m a p s
affine
defined
the class
satisfy
XXV,
p.
of
a map
points.
to the a s s e r t i o n
The
2) and
if it is flat
sets of r e d u c e d
separable
flat,
of just
436,
y • Y is a finite
of finite
field
equivalence
that
the
disjoint extensions of this
and
is E G A IV.17.4.1.
Definition
if for every
each
if and only
is e q u i v a l e n t
condition
and
b y the s u b c a t e g o r i e s
field of Y at y.
unramified
is
of finite p r e s e n -
of finite p r e s e n t a t i o n ,
two classes
is etale
spectra
locally
of etale m a p s
these
f:X ~ Y of finite
fiber of
is c l e a r l y
are l o c a l l y
Since
each
second
f:X ~ Y of schemes
m a p X ~ X × X is an open immersion. Y if and only if it is flat, u n r a m i -
the class
3) of 4.2,
This
A map
of finite p r e s e n t a t i o n .
A n etale m a p
maps
4.3:
4.4:
We
scheme Y',
say
the
f:X ~ Y
and every
by a n i l p o t e n t
g,h m a k i n g
•
ideal
following
is formally
closed of
~y,,
commute
suband
I. 4
62
h X ~
Y0'
g Y 4-------
y,
there is a unique map q:Y' ~ X with qi = h and fq = g. O n e can then p r o v e f:X ~ Y of schemes finite p r e s e n t a t i o n Alternative scheme.
(EGA IV.17.6.1,
is etale
17.3.1)
if and only if f is locally of
and formally etale.
Definition
Then a map
4.5:
Let Y be a locally n o e t h e r i a n
f:X ~ Y is etale if and only if f is locally
of finite type and the following c o n d i t i o n p o i n t x E X, w r i t e y = f(x), /k ~X,x
that a m a p
0x
,x
holds:
For every
the local ring of X at x,
its completion,
residue a free
6?y,y the local ring of Y at y, k the A C~y,y, ~Py,y its completion. Then ~ X , x is
field of A ~y,y module
and
~X,x
~®
k is a field,
~Y,Y separable
extension
(For a proof,
of k.
(Thus
and a finite /%
~X,x
is a finite
~y,y-algebra.)
see EGA IV.17.6.3)
In particular, a s e p a r a b l y closed
if X and Y are schemes of finite type over field k, and f:X ~ Y is any k-map,
then f
is etale if and only if for any closed p o i n t x e X, the induced map of c o m p l e t e morphism.
local rings
y, f(x) ~
QX,x
is an iso-
(This is c l e a r l y a c o r o l l a r y of the above.
p r o o f can be found in Mumford,
XXV, p. 353.)
Another •
63
1.4
Proposition I) category
4.5~:
Etale maps
of schemes
form a c l o s e d
satisfying
axioms
subcategory
Sl,
S2,
of the
S3(a ) and S3(b )
of 1.19. 2)
An etale map
presentation. Proof.
An o p e n Most
immersion
of this
from the c o r r e s p o n d i n g and S3(b)
are simple
definition
diagram
associated
4.6:
with
the Z a r i s k i
The
from the above.
for the
chases,
S 2 follows
flat topology.
using
S3(a)
the a l t e r n a t i v e
and
topology
flat
topologies.
4.7:
of schemes.
Then
schemes,
for each
union
of
i) ii) iii)
open
[Yi}
a family
immersion
surjective
map
and
X. a dis1
set I,
finally
and
for
for each
that
is a c o v e r i n g
of X in the Zariski
is a c o v e r i n g
For each
between
of X; a family
by the same
Y. + Y; l
topology.
[Xi]i~ I of affine
subschemes
indexed
f. :X. ~ Y. such l 1 1 {Xi}
in s t r e n g t h
f:X + Y be an etale
exists
schemes,
i c I, an open
is m i d w a y
of schemes
is the etale
i c I, a map X. ~ X, m a k i n g l
(Zariski)
{Yi}ie I of affine
i 6 I, a m a p
Let
there
on the c a t e g o r y
of etale m a p s
this
Proposition
and
topology
the class
By 4.5(2),
each
of finite
is etale.
is clear
assertion
locally
4.3.
Definition
joint
is flat and
i e I, the
of Y in the Zariski following
diagram
topology topology commutes
I. 4
64
X. ----'2X l
I Y.1
iv)
Each
~
Y
f. is etale l
hence
of finite p r e s e n t a t i o n ,
open,
and q u a s i f i n i t e . ) Proof. 4.8:
theorem
Exactly Just
applying affine
4.7, w e show
schemes,
results
etale m a p s
universally
result Using
a p r o o f of a d e s c e n t
now decomposes
for the Z a r i s k i
that
of affine
schemes.
(and
•
the t h e o r e m
and h e n c e w e are done.
the c o r r e s p o n d i n g affine
quasicompact,
as in the flat topology,
the t h e o r e m
for etale m a p s
flat,
and a f f i n e
as in 3.10.
for a r b i t r a r y
F i r s t w e show
surjective
schemes,
topology,
holds
to q u o t e
for flat m a p s
most
of
the t h e o r e m
it is s u f f i c i e n t
technique,
and then
etale maps
To show
(if it is true:)
this
into two parts.
of
of the f o l l o w i n g
are immediate.
Proposition are s t a b l e
4.9:
in the e t a l e
The
following
properties
of s c h e m e s
topology:
i)
locally
2)
reduced
3)
normal
4)
nonsingular
5)
of d i m e n s i o n
noetherian
n over
a ground
field k
I. 4
65
i) is proved
Proof. 4), and
5) are proved
Proposition schemes
Zariski
in the etale
separated
3)
universally
injective
4)
universally
closed
5)
of finite
6)
of finite p r e s e n t a t i o n
7)
finite
8)
universally
9)
quasifinite
i0)
being
ii)
quasiseparated
of maps of
bijective
an isomorphism
in both
the flat and
Apply 4.8.
4.11.
are stable
properties
type
these are stable
topologies.
3),
topology:
2)
Proposition schemes
The following
quasicompact
All
2),
•
I)
Proof.
in 4.8 above.
in SGA 1.9.
4.10:
are stable
as indicated
The
following
properties
and local on the domain
topology: i)
surjective
2)
flat
3)
fflat
of maps of
in the etale
I. 4
66
4)
universally
5)
etale
6)
locally of finite p r e s e n t a t i o n
7)
locally of finite
Proof. statements
i), 2),
4.5(1).
Proposition
follow from the c o r r e s p o n d i n g
4.12:
open i m m e r s i o n s
2)
affine maps
3)
closed
4)
immersions
5)
q u a s i a f f i n e maps
6)
immersions
in the etale
of m a p s of
topology:
immersions
i) through
Proposition
5) is p a r t
The following p r o p e r t i e s
i)
topologies.
topologies.
As for 6) and 7), see EGA IV.II.3.16. •
satisfy effective descent
Proof. Zariski
3), and 4)
type
for the flat and Zariski
of a s s e r t i o n
schemes
open
of reduced
5) are e f f e c t i v e
6) follows
4.13:
closed
subschemes
in the flat and
from 3) and 4.9(2).
The following
•
are local c o n s t r u c t i o n s
in the etale t o p o l o g y on a scheme X:
l) (x) = s p ~
For a q u a s i c o h e r e n t
sheaf A of
~-algebras,
A. 2)
a quasicompact
The s c h e m e - t h e o r e t i c separated map
c l o s u r e of the image of
f:Y + X,
1.4
67
subspace
3)
The open
4)
The reduced
closure
5)
The reduced
closed
Proof. logies.
4),
category
Xred,
5), and
3) hold
complement
of X.
of X.
of an open
of a b e l i a n
sheaves
There
and A z
of a field.
F 6 A E on X,
it can h a p p e n
nonzero,
even w h e n X is a point.
see
exposition
on schemes, [GT]
and
For q u a s i c o h e r e n t L e t X be a scheme topology
on X).
of rings,
which
A sheaf F of
of the etale
sheaves,
sheaf
Hq(X,F) (Ab))
is
cohomology
o f the r e s u l t s
the s i t u a t i o n
~X
its s t r u c t u r e
Then
~X
extends
also d e n o t e
~X-mOdules v
r:A E + A Z
of above
[SGAA]).
and
we
the
for an a b e l i a n
of F(X,-) :A E ~
and p r o o f s
AE)
if X is a point,
that the c o h o m o l o g y
functor
•
(respectively
functor even
In p a r t i c u l a r ,
as the d e r i v e d
and below,
above.
on X in the Zariski
(defined
sheaves
and flat topo-
(respectively
an isomorphism,
the s p e c t r u m
scheme.
from 4.12(6)
is a r e s t r i c t i o n
is not in general
(For a general
reduced
in the Zariski
6) follow
L e t X be a scheme
topology.
abelian
subscheme
of a s u b s p a c e
the a s s o c i a t e d
i) through
4.14:
which
of a closed
of X.
6)
etale)
complement
%,
sheaf
uniquely
is simpler. (in the Zariski
to give
a sheaf
on X in the etale
is q u a s i c o h e r e n t
in the etale
topology. topoloqy
I. 4
68
if t h e r e
is an e t a l e
cokernel
of a m a p
fact
then
~y I ~
is t h a t
(The SGA VIII
functor
happens
f*F is the
sheaves
Hence
specifying in the
A pleasant
(Quasicoherent
(Quasicoherent
on X w i t h o u t
same p h e n o m e n o n
we
the
flat
sheaves
on X in
refer
to
topology.
topology--see
for d e t a i l s . )
the d e r i v e d (Abelian
4.15:
functors
Sheaves
an a b e l i a n following
sheaf fact:
Proof. X}
L e t X be
a scheme,
of the g l o b a l
sheaf
on X.
Then
If X is an a f f i n e
scheme,
is a c o v e r i n g
(for
of X w i t h
~
F(X,-) :
(Ab).
Let
F can be c o n s i d e r e d
topology
theory
functor
topology)
on X in the e t a l e
By descent
and H i ( x , - ) d e n o t e
section
on X in the e t a l e
a quasicoherent
{Ui
~
that
~y-modules.
is an i s o m o r p h i s m .
sheaves
proposition
be
~yJ of free
topology)
topology)
quasicoherent
f:Y ~ X such
the r e s t r i c t i o n
on X in the e t a l e the Z a r i s k i
covering
flat
and we h a v e Hq(X,F)
as the
= 0~ q > 0.
coverings),
each U 1 affine, ,
F
if
the q
th
V
Cech and
cohomology sheaf
q > 0. erings holds q > 0.
of X w i t h
F, v a n i s h e s
Hence
Hq(X,F)
are c o f i n a l for
regard
for q > 0.
In s y m b o l s ,
= 0. q > 0 s i n c e
in the c l a s s
any a f f i n e
to the c o v e r i n g
such
of c o v e r i n g s .
scheme U mapping
etale
[U i ~ X}
Hq(x,{ui],F)
affine This
etale
= 0, cov-
of c o u r s e
to X, H q ( U , F )
= 0,
I. 4
69
W e now invoke generalizes implies
Cartan's
easily
Let
and F a q u a s i c o h e r e n t
functor
sheaf
of the
topology)
+
sheaf
functor
Rqf,(F)
where
f,:
(Abelian
Sheaves
which
Since
by an etale Rqf,(F)
contain
on X in the etale
on Y in the etale
is the sheaf
topology).
etale
covering
is zero,
lemmas
affine
covering
by affine
U(u,~qf,(F))
scheme U m a p p i n g
of Y by schemes
schemes,
etale
is d o m i n a t e d
the a s s o c i a t e d
sheaf
and r e s u l t i n g to show
that
proposition algebraic
will
spaces
be used always
subschemes.
4.17.
etale map.
(So,
p o i n t p E X,
to the p r e s h e a f
for q > 0.
1.5 and II.6 open
associated
= H q ( u X X,F) . Y so F(U, ~ q f , ( F ) ) = H q ( u × X,F) = 0 Y for q > 0. H e n c e -~qf,(F) is a
is zero on every
Lemma
f-l(p) .
the q th d e r i v e d
sheaves
for an etale map U -~ Y,
every
The next sections
schemes
Then F can be c o n s i d e r e d
L e t Rq f, d e n o t e
U × X is affine Y by the above p r o p o s i t i o n 4.15
to Y.
This •
on X.
For U affine,
presheaf
topology.
= 0 for q > 0.
Proof. ~qf,(F)
Grothendieck
in X V . I I . 5 . 9 . 2
f:X + Y be a m a p of affine
on X.
(Abelian
Then Rqf,(F)
proof
= 0. q > 0.
4.16:
as an a b e l i a n
whose
to an a r b i t r a r y
that Hq(X,F)
Corollary
Lemma,
Let
f:Y + X be a q u a s i c o m p a c t
in p a r t i c u l a r ,
let n(p)
f is quasifinite.)
be the number
Then n:(points
of X) ~
separated
of p o i n t s
(Integers)
in the
is upper
For each fiber semicon-
in
I. 4
70
tinuous
(in t h e
is an o p e n n(p)
sense
subspace
is c o n s t a n t ,
finite
Lemma
of X.
then
and
Let
of n o e t h e r i a n
o
~
m]
s u b s e t U of X,
of f to Y × U ~ U is a X
schemes Y
be
found
4.19:
the r e s t r i c t i o n
Proof.
Then
The
is a f f i n e .
there
first
Consider Here
point
o f X and n
Since
every
o
we
finite presentation
scheme.
and
Then
a map X ~ X
quasicompact
there
o
and
and
is
such separ-
separated.
an e t a l e q u a s i c o m p a c t
is
open
of
nonempty
6 X, finite
a dense
assertion first
n(p)
~
n
o
subset + i]
= no]
and e t a l e .
is l o c a l
on X so w e
the case where
apply Lemma
open
subscheme
U of X
f:Y × U ~ U is a f i n i t e e t a l e X U can b e t a k e n to b e a f f i n e .
= n (o)._ x , where
e X and n(p)
U X Y -~ U X
o
of
f:Y ~ X b e
If X is q u a s i c o m p a c t ,
I P
~ X
o
etale
Let
such
irreducible.
a map
is an a f f i n e
map.
that
•
f:Y ~ X b e
separated
U = {p
I n(p)
See EGA IV.8.9.1.
Proposition
I P
{P
If Y ~ X is e t a l e q u a s i c o m p a c t
o
Proof.
[P
for s o m e o p e n
the r e s t r i c t i o n
suppose X
that Y = X X Y • o X ated, Y ~ X° can
X
If
i n t e g e r m,
See EGA IV.18.2.8. 4.18.
of s c h e m e s
map.
for e v e r y
etale map.
Proof.
a map
that
4.17. n(-)
is the
o
open
be
the
function
of X c o n t a i n s
is a d e n s e
assume
X is n o e t h e r i a n
Let x
is empty.
can
generic of 4.17.
Xo,
Hence subset
and
of X w i t h
1.4
71
For
a general
irreducible open be
components
For
satisfying the c a s e
4.18
to
quasicompact set w i t h Y U ~ X is The fact
X,
of X.
let XI,...,X n be
Then by
s u b s e t s U 1 ~ X I , . . . , U n ~ Xn,
found
Lemma
noetherian
theorem
of g e n e r a l
find and
the
a map
Let U
x U ~ U finite o o X o a dense open subset
that
and w e
o
= ~
can
if i /
+ X
etale.
schemes Y
o
j, c a n
take U = U 1 U
s c h e m e s X, w e
of n o e t h e r i a n
separated.
above procedure,
U i n Xj
affine
o
second
the
the m a x i m a l
be
can use + X
o
a dense
Let U = U
any q u a s i c o m p a c t
scheme
contains
× X. X finite
o
a dense
o
etale
open
and Y × U ~ U is X of the p r o p o s i t i o n f o l l o w s
statement
... U U n.
sub-
Then etale. f r o m the
affine
open
subscheme.
I
Proposition subscheme
4.20:
Let X be
by a n i l p o t e n t
defined
a scheme
ideal
and X
o f (~X"
o
c X a closed
Consider the
functor (Schemes
etale
o v e r X) + Y ~
This
functor
is an
Proof.
See
Y×X X
equivalence
SGA
1.5.5,
(Schemes
etale
o v e r Xo)
o
of categories.
1.8.3.
I
I. 5
5.
72
Etale
Equivalence
Definition lence
relation
tion maps
Relations
5.1: on U.
Let U be
a set
We
~i
write
R + U × U + U,
equivalence
relation
in
and R c U × U
and ~2
and s a y R ~ U the c a t e g o r y
for the
an e q u i v a -
two p r o j e c -
is a c a t e g o r i c a l In
o f sets.
a general
->
category
C with
a cateqorical
equivalence
Z E C, H O m c ( Z , R ) relation
fiber products,
relation
~ HOmc(Z,U)
in the c a t e g o r y
a diagram
X
equivalence
is t h e n u n i q u e
effective gorical
quotient
If R + U v ~ U is
for
all o b j e c t s equivalence
the c a t e g o r i c a l
relation
R ~ U
isomorphism
R ~ U
quotient
of a
if U ~ X = C o k ( R ~ U). and U ~ X is an
is e f f e c t i v e
and R = U × U. X is a c a t e g o r i c a l e q u i v a l e n c e
any m a p
in C is c a l l e d
of sets.
up to u n i q u e
epimorphism.
if
is a c a t e g o r i c a l
A m a p U + X in C is c a l l e d categorical
on U
U
R
if it has
a cate-
U ~ X
of C, w e d e f i n e
S = R
relation
×
(V>
in C,
and
and n o t e
(uxu) that
S ~ V
equivalence
is a c a t e g o r i c a l relation
equivalence
relation,
the
induced
on V. ->
Proposition
5.2:
relation.
Then
Also
is a u n i q u e
there
U ~--~ R
the
Let R
U × U is
induced map map
U be
a categorical
6:R ~ U × U
i:U ~ R s u c h
the u s u a l
diagonal
that map
equivalence
is a m o n o m o r p h i s m . the c o m p o s i t e ~ : U ~ U × U.
I. 5
73
Proof.
Immediate
Definition and T =
5.3:
(C,Cov T)
from
|
the d e f i n i t i o n s .
L e t C be
a category
a Grothendieck
with
topology
fiber products
on C s a t i s f y i n g
the
->
a x i o m A 0 of 1.3. relation
A diagram
U
R
if it is a c a t e g o r i c a l
and e a c h m a p n
is a c o v e r i n g
in C is a T - e q u i v a l e n c e
equivalence
map
of T.
relation
Note
then
in C
that
in
1
the c a t e g o r y
of s h e a v e s
equivalence
relation.
R ~ U if U
~ X"
T-quotient R ~ U
is e f f e c t i v e
5.4: the
Let R
of s h e a v e s quotient
it is u n i q u e
quotient
is a c a t e g o r i c a l
U be
precise:
effective
a T-equivalence
relation.
In t h i s
If a
isomorphism.
exists.
relation
R" ~ U"
category,
of
U'.
if a T - q u o t i e n t
on C.
U" + F e x i s t s
of R"
up to u n i q u e
(or to be v e r y
equivalence
of sets
R. ~ U"
A m a p U ~ X of C is a T - q u o t i e n t
relation)
associated
o n C,
is a c a t e g o r i c a l
exists,
T-eauivalence
of sets
as a
Consider
in the c a t e g o r y a categorical
(and so in p a r t i c u l a r
R"
= U"
× U').
->
If F is r e p r e s e n t a b l e , tive
say F = X',
as a T - e q u i v a l e n c e
X is a c a t e g o r i c a l R ~ U may
have
quotient.
a categorical
The q u o t i e n t
sheaf
follows:
For
by g i v i n g
a covering
7 i e U" (X i)
relation
that
with
the
->
U
is e f f e c U + X,
if F is n o t
then
representable,
quotient.
X in C,
family
if R
T-quotient
even
F of R" ~ U"
any o b j e c t
such
But
i.e.,
can be c o n s t r u c t e d
an e l e m e n t
{X i ~ X)
following
7
£ F(X)
is g i v e n
of X and e l e m e n t s condition
holds:
as
I. 5
74
(X.l XX X3') and U" (Xj) ~ U" (X i X× Xj)
U" (X.l) ~ U yi,7j
in U
ering
family
induce elements
(X i X Xj) and we require the pair <~i,y 3.> to be X in R" (X. × X.). X 3 Two elements ~,~ e F(X) are the same if there is a cov[Yi ~ X] where,
writing
~i,~i
for the images of
~,~ in U(Yi) , the pair <~i,~i > is in R(Yi). To see that this defines the presheaf covering (Since,
quotient
families
there is a unique
of R" + U"
IX i ~ X],
for ~,~ e ~
the quotient
for all X e C, and
~(X)
+ -~(Xi) i (X), if ~ is equivalent
element <~.,~.> 1
sheaf axiom
Then
sheaf F, let '~ be
is injective! to ~ on each Xi,
E R(X i) for each i
l
for R gives <~,~>
and the •
e R(X)
so ~ is equivalent
to
on X.) Thus
the
quotient
plus-construction above
just gives
applied
~(X)
~
map of sets
sheaf U × U is identical isomorphic
to ~
{X i ~ X] implies
is an injective
F of R
this i n w o r d s
Note also that ing family
sheaf
U is
just once.
"~g+, t h e
Our construction
in this particular
-~(Xi)
usual
injective
case.
for every cover-
that the natural map ~ ( X ) for all objects X°
to the p r e s h e a f U × U.
to U X U so R _~ U × U. F
Hence
+ F(X) the
R is clearly
I. 5
75
5.5:
W e now restrict
c a t e g o r y with
fiber p r o d u c t s
induced by a closed and S 3(b)
to the case w h e r e C is a
and T = (C,Cov T) is a B - t o p o l o g y
subcategory
B of C satisfying
Sl, S3(a)
of 1.19.
The content Y o n e d a Lemma, on ~ ,
attention
F(X)
of 5.4 can be rephrased
which
asserts
that for all X in C, and F a sheaf
= HOmsheaves(X',F).
R" ~ U" and ~ 6 F(X),
Thus if F is the q u o t i e n t of
there is a c o v e r i n g map Y + X of X and
a map 71:Y ~ U such that the following
~1i Y" × Y"
-
X
-~
--
~
Y" ~
commutes:
11 ---~--~X"
2
IV
3"i~.
(where "commuting"
simply now using the
11
for the left hand side of the d i a g r a m m e a n s
that each square Ylni = 11i(~i >< 71 ) commutes.) O b v i o u s l y y is u n i q u e l y d e t e r m i n e d
by the maps Yl and
X YI" G i v e n the two elements ~i~ 6 F(X), ~ = ~ iff there Y is a c o v e r i n g map %0:Y ~ X and c o m m u t a t i v e d i a g r a m s • Y-
•
-)x
.
y.~?
lo J o U"
--) F
X"
io U'~)
F
I. 5
76
and
a map
Y"
~
R ° such
that
y"
R both
y°
._____._~? U"
and
R
--------9U"
commute. Proposition
a)
5.6: If ~:U
~ X
is
a covering
map
of
the
topology
T,
->
then U ~:U
× U -> U X ~ X. b)
lence map
Let
is
a T-equivalence
Conversely,
relation
with
if R + U
T-quotient
R = U × U. X Proposition 5.7:
relation
is
U ~ X,
an
with
T-quotient
effective
then U ~ X
is
T-equivaa covering
and
V ~ U be
any map
I Let in B
R ~ U and
e q u i v a l e n c e r e l a t i o n on V. relation
where
and t h e r e i s
"commutative"
S = R
a T-equivalence ×
(V × V)
~(u×u)
Then S ~ V i s
relation. the
induced
a T-equivalence
a "commutative" s q u a r e
s
~
v
R
.~
u
in
be
this
situation
means
that
f~l'
= ~i f''
I. 5
and
77
f~2'
U + X.
= ~2 f' .
and
induced
Also,
S ~ V is
map Y ~ X
Proposition
if R ~ U
effective
is e f f e c t i v e
with
with
~-quotient
~-quotient
V ~ Y,
the
is in B. 5.8:
l
Let D be
an e f f e c t i v e
descent
class
in C.
->
Let R
->
U ~ X.
U be
an e f f e c t i v e
Suppose
there
T-equivalence
is a " c a r t e s i a n "
s
R
where
"cartesian"
fn 'i = n i g effective
-~
. 7
in this
is c a r t e s i a n . T-equivalence
and h : Y ~ X
Then
~
the u n i q u e
h 6 D and
the
9
T-quotient
in C
U
situation
relation. map making
f~'
diagram
with
v
Suppose
square
relation
= ~f
means
f • D.
that
each
Then
S ~ V is an
Let V + Y be the
following
is c a r t e s i a n .
square
the T - q u o t i e n t diagram
commute:
I. 5
78
->
Proof. forward each
~.'
L e t F be S"
is
Then
for D,
the
V"
S
e a c h ~i
V is
5.9:
Let
a ~-equivalence
is the c a t e g o r i c a l
induced
U"
L e t U" ~ F b e
V"
the
of s h e a v e s
6 C.
Then
×U"
- - D
U
consider
f a c t o r s V" -~ U" -~ F.
I relation
to an e f f e c t i v e quotient
let V" ~ F b e
in the c a r t e s i a n
any
diagram
V
-'~ F
× U" is r e p r e s e n t e d b y an o b j e c t W in C and X is in B, and the m a p W ~ V × U is in D. First
of
effective
categorical
and
V"
Proof.
of
map,
relation.
quotient
a T-equivalence
map R + U × U belongs
in the c a t e g o r y with V
R ~ U be
is a s t r a i g h t -
is a c o v e r i n g
× F. Applying the notion X F is r e p r e s e n t a b l e . L e t Y" = F.
of R"
of s h e a v e s
so
Since
relation
= U"
c l a s s D in C.
map
map
which
descent ->
is an e q u i v a l e n c e
argument•
the q - s h e a f
suppose
V
->
a covering
Proposition
and
S
categorical
~ V'.
descent
That
the
special
the m a p W ~ V
case where
x U" = V" X U" × U" = V" X R" F U" U" so i s r e p r e s e n t a b l e . T h e m a p V" X U" -~ V" × U" is t h e n F V" ~ R" + V" X (U" × U') so is in D. A l s o , V" × U" ~ V" U" U" F is V" × R" -~ V ° = V" × U" so is a c o v e r i n g map. U" U" F o r a g e n e r a l ~:V" ~ F, t h e r e is an o b j e c t W in C and a covering
map W_
~-~ V
T h e n V"
V" + F
and
a commutative
diagram
I. 5
79
X W" V"
W"
'
R" (applying
By map V"
the
5.5).
special
in D.
_>7 W"
- - ~~
Consider
W"
X U"
Using
W"
~
----------3 V"
descent
is r e p r e s e n t e d
W" X U" F
W" is c a r t e s i a n
the
class
D of C.
R" + U"
-
induced
map
is r e p r e s e n t e d
a map
of D,
by
a
the m a p
in D.
0
V"
× U" F
-
~
V"
R + U × U belongs
L e t U" + F b e
in the c a t e g o r y
i n t o F.
of C-objects
X U"
× U" + V" is r e p r e s e n t e d b y a c o v e r i n g m a p . l F 5.10: L e t R ~ U be a ~ - e q u i v a l e n c e r e l a t i o n to an e f f e c t i v e
the c a t e g o r i c a l
quotient
descent
o~
of s h e a v e s .
L e t V'.z ~ F and V" 2 ~ F be sheaves
of s h e a v e s :
so V"
Proposition where
~
diagram
property
by
F
V" x U" F
× U" ~ W" × U"
the s t r i c t
× U" ~ V" × U" F Also,
7
the c a r t e s i a n
X W" . F
~? V
U"
D
W"
case,
~
any
two m a p s
of r e p r e s e n t a b l e
T h e n V" 1 × V" 2 is r e p r e s e n t a b l e , and F V" 1 × V" 2 is in D. i n d u c i n g V" 1 × V" 2
the m a p
I. 5
80
Proof.
As
in p r o p o s i t i o n
w h e r e V" 1 ~ F and V •2 ~ F A similar We
then
are g o i n g
to apply
all of this
and d e f i n e
etale
the o p e r a t i o n The
relations
the rest of this
equivalence
relations
to the etale
~ U" ~ F "
examples
of a l g e b r a i c
show
there
case.J
topology
of the c a t e -
of taking q u o t i e n t s
in the c a t e g o r y
section,
the case
for the g e n e r a l
the c l o s u r e
is n o n t r i v i a l - - i . e . ,
equivalence
V'2
the c a t e g o r y
(II.3.14)
relations.
operation
For etale
under
equivalence
suffices
(in II.l.l)
is in a sense
g o r y of s c h e m e s
first p r o v e s
factor V" 1 * U" ~ F '
argument
spaces which
closure
one
descent
of s c h e m e s
etale
5.9,
of
that this
are n o n e f f e c t i v e of schemes.
w e give some
and some c r i t e r i a
facts a b o u t
for s h o w i n g w h e n
they are effective. Proposition relation
5.11:
L e t R + U be
an etale
(by w h i c h p h r a s e w e w i l l m e a n
tion on the c a t e g o r y
C, w h e r e
topology).
Then
U × U is separated,
and h a s d i s c r e t e The map component
rela-
of schemes,
S u p p o s e U is a s e p a r a t e d locally
of finite
scheme. type,
fibers.
i:U + R is an immersion~
of R
a r-equivalence
C is the c a t e g o r y
and T is the e t a l e the m a p R ~ 6 ~
equivalence
(which we call
identifying U with
the d i a ~ o n a l
part
of R).
a
81
1.5
Proof. ated
and
Since
has d i s c r e t e
projection tative
R ~
on the i
th
U × U
is a m o n o m o r p h i s m ,
fibers.
Let Pi:U
component,
it is s e p a r -
× U + U denote
i = 1,2.
Consider
the
the c o m m u -
diagram 6 -~--~) U × U
R
U Since
U is
separated,
is s e p a r a t e d Also,
Pi
locally
~. is l o c a l l y 1
also
of
1.21,
relation
T h e map R
~6)
U X U
finite
type
since
it is e t a l e
and,
(U X U)
× (U × U) is a c l o s e d U A p p l y i n g 1.21, R - - ~ U × U is
type.
type.
i:U ~ R is i identifies
Propositio n
of
U X U ~
finite
of f i n i t e
Finally, so b y
separated.
so ~. is s e p a r a t e d . i
s i n c e U is s e p a r a t e d , immersion
is
5.12:
of s c h e m e s
a section U with Let
with
of
etale
a component U be
R
the
Then
)U,
o f R.
an e t a l e
U separated.
map R ~ l
equivalence
the
following
are e q u i v a l e n t :
immersions relation
i)
R
2)
R ~ 6 ~ U X U is q u a s i a f f i n e
3)
R - -8~ ) U × U
4)
For
V ~ U,
on V,
8 ~ U × U is of
finite
is q u a s i c o m p a c t
all q u a s i c o m p a c t if S
then
->
each
type
V denotes ~. ' is not
schemes
V,
the i n d u c e d only
etale
and o p e n equivalence but
l
compact
(hence
of
finite
type
and q u a s i f i n i t e ) .
also quasi-
I. 5
82
Finally,
if R is q u a s i c o m p a c t ,
of finite ditions
type over
are s a t i s f i e d
an immersion, tions
a ground
e.g., field,
if R and U are b o t h then
automatically.
and U × U is l o c a l l y
these
Also,
schemes
equivalent
con-
if 8 is k n o w n
noetherian,
then
to be
the c o n d i -
are satisfied. Proof.
by
i) <--> 3).
R 6>
i) <=>
By
U × U is l o c a l l y
of
finite
type
5.9.
discrete
fibers.
the finite affine.
2).
5.9,
By D e l i g n e ' s
type h y p o t h e s i s
Conversely, 3) <=>
4).
jections
Also,
implies
has
this w i t h
U × U is q u a s i -
quasicompact.
U X U is q u a s i c o m p a c t .
with V q u a s i c o m p a c t ,
relation since
(2.20(5)),
that R ~ 6 >
R ~
immersion
equivalence
map.
implies
Suppose
S ~ V × V is q u a s i c o m p a c t compact
theorem
quasiaffine
Let V ~ U be an open the i n d u c e d
R _~67 U X U is s e p a r a t e d a n d
on V w h i c h
and S ~ V
is etale by
it is the p u l l b a c k
V is q u a s i c o m p a c t
so
V × V ~ V are q u a s i c o m p a c t .
of a q u a s i -
(by 2.22a)
Each n
5.6.
the p r o -
':S + V is a l
composite
of q u a s i c o m p a c t
Conversely, open c o v e r i n g subschemes operation
suppose
of U by
of U.
This
of taking
maps,
hence
condition
4) holds.
the c o l l e c t i o n collection
finite
unions
quasicompact. Let
[Vi]
be the
of all q u a s i c o m p a c t
{Vi]
is c l o s e d
of subschemes.
under Hence
open
the the
I. 5
83
collection
{V i × V i ]
that R ~ g U that
forms
an open c o v e r i n g
× U is q u a s i c o m p a c t ,
for each
of U × U.
it is s u f f i c i e n t
To check
to check
i, R
X (V i × V i) ~ (V i X V i) is q u a s i c o m p a c t (UXU) since q u a s i c o m p a c t n e s s is stable in the Z a r i s k i topology.
H e n c e w e can assume
that U is q u a s i c o m p a c t .
We
are now reduced
->
tO showing ~.
etale
that
if R
->
U is an e q u i v a l e n c e
and q u a s i c o m p a c t ,
then R 6 ~
relation
with
each
U X U is q u a s i c o m p a c t .
l
Consider
now
the c o m m u t a t i v e
triangle
R--
8
:~U
×U
U
where
Pl is the
separated, sion.
R ~69
Finally, applying
1.21
(U x U) × (U × U) U i m m e r s i o n is q u a s i c o m p a c t .
if R is q u a s i c o m p a c t to the c o m p o s i t e
noetherian
If one w i s h e s with
quotient
ators"
U and
condition
so Pl is
is a closed Applying
immer-
1.21,
we
U × U m u s t be q u a s i c o m p a c t .
to v i e w
"relations"
set of generators,
And
any i m m e r s i o n
into
scheme m u s t be q u a s i c o m p a c t . an etale
U ~ X as a n a l o g o u s
4) above
and U × U is separated,
R--~--6> U × U ~ Spec Z, we
that R ~ U × U is q u a s i c o m p a c t . locally
U is separated,
hence U × U ~
A closed
are done;
first projection.
R,
equivalence
could
be v i e w e d
there
are only
as saying a finitely
a •
relation
to a s p e c i f i c a t i o n
for c o n s t r u c t i n g
see
of "gener-
an "algebra" "among many
U
R
any
X,
finite
relations."
I. 5
84
Let R
5.13:
tion w i t h
U be an e f f e c t i v e
quotient
U
X.
R =U
Then
there
8--~
XU X
etale
is a c a r t e s i a n
schemes,
stable
immersions, only
in the etale
quasicompact
if ~ e D.
the m a p 4)
immersion
Thus
defined
conditions
of
schemes.
One can also
immersion
(EGA 1.5.3.9)
separated
schemes
lence
relations
conditions
show
immersions,
maps), on X
that
(conditions
on
if A is a closed Thus
the
to q u a s i s e p a r a t e d
for any scheme
so our r e s t r i c t i o n that
closed
6 e D if and
if A is q u a s i c o m p a c t . correspond
of
on ~:R ~ U X U.
X to be s e p a r a t e d
5.12
entails
R ~ U,
(e.g.,
to c o n d i t i o n s
and X q u a s i s e p a r a t e d
equivalent
so for any class D of maps
quasiaffine
separation
already
,
topology
maps,
are e q u i v a l e n t
• We have
map
diagram
n X ~
XXX surjective
rela-
uxu
I
X ~ is an etale
equivalence
X, A is an
in2o~to
for all e f f e c t i v e
R ~ U × U satisfies
the
quasi-
etale
equiva-
conditions
of
5.12. Proposition relation
where
5.14:
each
R ~ U is effective. is given
map
L e t R + U be an etale
equivalence
~. 1
affine.
is
and
U is
If R and U are affine,
by X = S p e c ( K e r ( F ( U ,
Proof.
finite
XVII.212.5.1
Then
the q u o t i e n t
U ~ X
~ U ) ~ F(R, ~ R ) )) . •
I. 5
85
Proposition relation
5.15:
satisfying
the a f f i n e
spectrum
separable
Proof.
of
of a field K.
Then
Then
spectrum
the c o n d i t i o n
since U is the s p e c t r u m
union of K
of a f f i n e (applying
a quotient etale
Hence
Hence X has
there
exists
a quotient
(and h e n c e K is
of
5.12,
of a field,
R is q u a s i c o m p a c t .
R m u s t be a finite
separable
field
extensions
each ~. :R -~ U is finite 1
~ : U ~ X exists.
(5.5).
S u p p o s e U is
of L).
4)
s p e c t r a o f finite 4.1) .
5.12.
equivalence
of a field L
field e x t e n s i o n
Using
an e t a l e
the c o n d i t i o n s
U + X w i t h X the affine a finite
U be
Let R
~ is an e p i m o r p h i s m o n l y one p o i n t
so b y
(5.1)
and b y 4.9,
and X must
b e reduced.
•
Proposition Zariski
subscheme
of U and S
Then
for
Suppose
each
S
V is
Let i
on U
in the
equivalence
rela-
1
a)
R -~ U i s
effective.
V the i n d u c e d
Let
etale
V be
an
equivalence
open
rela-
effective.
[Ui]i61 be
6 I,
each R.
Proof.
is local
U be an etale
Let R
Suppose
b) and
(Effectivity
R ~ U × U quasicompact. a)
tion.
5.16:
Topology.)
tion w i t h
5.14,
R i ~ U.1 t h e
U.
a Zariski induced
is effective.
->
1
is
clear.
open c o v e r i n g equivalence
Then R
of U
relation.
U is e f f e c t i v e . ->
To prove
b),
we
first
need
a
I. 5
86
Lemma. S = R
Let W~_~-~U
× W × W UXU
and
be
an o p e n
suppose
S"
~ 2
immersion
the q u o t i e n t
W"
n
and
V of S
W exists:
-) V"
]o
1 R"
-_~- )
U"
-~
F
rT. Suppose
" 2 ( ~ i -l(g°(W)))
Proof
of L e m m a .
the n a t u r a l by
= U.
T h e n V" -~ F is an i s o m o r p h i s m •
We must
m a p V" (X) ~ F(X)
constructing
show
•
products
affine
is an i s o m o r p h i s m .
an i n v e r s e m a p F(X)
Homsheaves(X',F). S" x = S" × X,
for all
-~ V" (X) .
°
schemes We do
Let
X,
this
e 6 F(X)
°
•
= °
Let V X = V
× X, U X = U" × X, W X = W × X, F F F and R" x = R" × X. B y 5.10, all of t h e s e f i b e r
are r e p r e s e n t a b l e
sheaves.
Then
there
is a d i a g r a m
of s c h e m e s :
which
Sx W
"
~p
Rx
~ $
hypotheses:
with
X is
quotient
the
WX
--~
VX
Ux
(as a s h o r t d i a g r a m
original tion
SX
chase
) X shows)
RX ~ UX i s
an etale
U x ~ X, W X ~_~~ U x
induced
equivalence
Nx * Vx a n d n 2 ( ~ 1 - 1 ( ~ ( W x ) ) )
satisfies
equivalence
is an o p e n
relation
= UX"
all
with
the
rela-
immersion,
quotient
O n c e we show Vx ~ X i s •
an i s o m o r p h i s m ,
gives
the
by projection
inverse map
on t h e
first
(of s h e a v e s )
factor
X" ~ V X = V
a map ~:X" ~ V ' ,
°
•
x X F
I. 5
87
i.e.,
an e l e m e n t
m a p F(X)
-> V" (X) w h i c h
In short, assume
that
surjective
in our
clear.
The
same point special
We
An
a covering
relation
for
since W ~ V
that
To
R.
of V
this makes
• where
of
which
is e f f e c t i v e .
is it
is o n e - o n e
are done.
situation,
U. ->
show V + F
two p o i n t s
which
on e a c h
the lemma,
is e t a l e
identical--and
so we
subsets,
1
hypotheses
etale map
to the o r i g i n a l
of U by open
eauivalence
be
V" (X) -> F(X).
U = ~ 2 ( ~ ! -I (~(W)))
is an i s o m o r p h i s m ,
now r e t u r n
the r e q u i r e d
V ~ F is etale.
in F m u s t
that V + F is onto.
onto points
Then
to c h e c k
hypothesis
~ gives
to the n a t u r a l
original
it is s u f f i c i e n t
to the
duced
can
e ~
inverse
and W ~ F is etale,
going
is
we
is
This
F is r e p r e s e n t a b l e .
is o n e - o n e ,
clear
~ c V'(X) .
and (Lemma)
[U i ~ U] the
L e t V.
1
inbe
1
->
the
quotient
of
R. 1
subscheme
U.. '~
Then by
W. * U w i t h
W'.
1
with
the
lemma there
is
an o p e n
1
= U"
l
x V" i and we c a n F
replace
U. 1
W.. 1
[V i ~ F} pullback
is an e f f e c t i v e
[W i ~ U}
is),
and
for
epimorphic each
i,j
family there
diagram
w'.
×w'. i
U •
-
-~
v'.
]
xv" m
F
t
$
,
w. 1
)
v. l
(slnce
its
is a c a r t e s i a n
I. 5
88
(since
W'. = V'. × U') . l i F
V'. × V" + V" is i F j i Similarly,
V'.
Hence
W. X W. + W. is i U 3 i
represented
scheme
V° a n d V. a l o n g l 3
the
open
Consider
R ~ U where
each
the
numbers.
spaces.
We
Now is
an
suppose
Rh
N W
of V
so .
Then
of
are
the
is
etale
locally
open
gluing
each •
finite
A
Then
(image
subset
W
over
as
analytic
as R h -~ ~ Uh
spaces
immersion.
relation
type
structures
analytic
diagonal
an
by
subscheme.
equivalence
of
have
associated
In R h
obtained
an
is an
an o p e n
V. X V.. i F 3
R and U
R -~ U X U
there
is
by
R
h
~ U
h
Uh
X
o f U h -~ R h)
is
of U h X U h with
= ~ ~ W.
Let of
case
and U
these
immersion. so
which
subscheme
the
of R
write
a component,
subscheme
X V" . + V " is r e p r e s e n t e d 3 J the
complex
an o p e n
immersion
i
F is
5.17:
by
an o p e n
p
E Uh
be
6 A C Uh
(p,p)
is H a u s d o r f f ,
any point. X U h which
there
is
Then
W
doesn't
an o p e n
is
an o p e n Rh
meet
neighborhood
neighborhood
- A. U
Since
of p
Uh
in U h w i t h
P U
× U c W. P P h ~uh R to U
Thus C U
h
the is
restriction the
trivial
of
hhe
equivalence
equivalence
relation
relation
In p a r t i c u l a r ,
-~
U
p
U
P ~ the
induced
equivalence
relation
on U
is P
effective. Hence course)
by
U h has open
a covering
subsets
U
(in
the
on which
analytic the
topology,
induced
of
equivalence
P relation
is e f f e c t i v e .
The
reasoning
in
the
theorem
above
. p
is
I. 5
89
clearly shown
applicable
the
to the
following
case
of a n a l y t i c
spaces,
so w e h a v e
proposition. ->
Proposition relation
of
numbers,
with
be
the
schemes
Let R
locally
U be
of
an e t a l e
finite
equivalence
relation
T h e n R h ~ U h is e f f e c t i v e
equivalence
type over
the m a p R ~ U × U an i m m e r s i o n .
induced
spaces.
5.18:
the c o m p l e x Let
Rh ~ U h
in the c a t e g o r y in t h e
category
of a n a l y t i c of analytic
spaces.
I
Proposition relation there
satisfying
exists
induced
5.19:
equivalence Let
R + U be
the e q u i v a l e n t
a dense
Proof.
Let
open
u
conditions
subscheme
relation 6 U, V
S
an e t a l e
V of U,
equivalence of
such
5.12. that
Then the
V is e f f e c t i v e .
an o p e n
affine
subscheme
of U
con-
U
taining
u,
and
S
~ U
V
~
the
E a c h m a p n.lu is q u a s i f i n i t e so, b y p r o p o s i t i o n
induced
equivalence
relation.
U
(applying
4.19,
there
the
induced
our
hypothesis
is a d e n s e
open
of
5.12)
subscheme ->
V
' of V U
such
that
equivalence
relation
S
U
' U
V ">
' U
-->
is
finite.
By
5.12,
S
' U
that u ~ V
U
'
J
L e t V be
V ->
' is e f f e c t i v e .
Now
it m a y b e
U
but
u is in the c l o s u r e
the
union
of
all
the V
of V '
V
U
' is d e n s e
in U
and,
U
locally Hence,
on V, by
effective.
the
5.16,
induced
equivalence
the i n d u c e d
relation
equivalence
is e f f e c t i v e .
relation
on V
is I
I. 5
90
Corollary
5.20:
Under
the h y p o t h e s i s
also q u a s i c o m p a c t ,
V can be chosen
affine
by
of
5.19,
if U is
so that V and S are b o t h ->
(and hence,
Proof. second
5.14,
the q u o t i e n t
T h e same as the p r o o f
statement
of 4.19.
above,
of S
V is affine).
now a p p l y i n g
the I
CHAPTER
ALGEBRAIC
TWO
SPACES
l,
The C a t e g o r y
2.
The E t a l e T o p o l o g y
3.
Descent
4.
Quasicoherent
5.
Local Constructions ................................
120
6.
Points
and the Z a r i s k i
Topology ....................
129
7.
Proper
and P r o j e c t i v e
Morphisms ....................
159
8.
Integral
i.
The C a t e g o r y We
of A l g e b r a i c
Theory
separated
91
Spaces . . . . . . . . . . . . .
101
Spaces ................
106
andCohomology ................
115
for A l g e b r a i c
Algebraic
Spaces
of A l g e b r a i c
.........................
w i l l be suppressed)
base and
topology.
As in C h a p t e r
separated,
and
scheme
of s c h e m e s S
over
(all m e n t i o n
take this c a t e g o r y w i t h I, all s c h e m e s
a given of w h i c h the etale
are a s s u m e d
for any s c h e m e X, w e w r i t e X" sheaf.
144
Spaces
the c a t e g o r y
noetherian
representable
..................
of A l g e b r a i c
Sheaves
start w i t h
Spaces
quasi-
for the a s s o c i a t e d
II.l
92
Definition
i.i:
An A l g e b r a i c
A : ( S c h e m e s ) °Pp ~
S p a c e A is a functor
(Sets)
such that
a)
A is a sheaf
b)
(Local R e p r e s e n t a b i l i t y ) and a m a p
in the e t a l e
topology There
exists
of s h e a v e s U" ~ A such
s c h e m e s V,
and m a p s V" ~ A,
the
that
a s c h e m e U, for all
(sheaf)
fiber
p r o d u c t U" × V" is r e p r e s e n t a b l e and the m a p A U" X V" ~ V" is induced b y an etale s u r j e c t i v e A of schemes. c)
(Quasiseparatedness) Then
map
L e t U" ~ A b e as in p a r t b°
the m a p of s c h e m e s
i n d u c i n g U" × U" ~ U" × U" A
is q u a s i c o m p a c t . A m a p U" ~ A s a t i s f y i n g called
a representable
algebraic 1.2: which
The
is full,
factors
through
is a full For
spaces
maintain algebraic
and c)
etale
functor
~
left exact
imbedding
between
After
2.5)
will
A morphism
be
of
of functors. of sets
on schemes)
(by the Y o n e d a
of a l g e b r a i c
(through D e f i n i t i o n
space X'.
(Sheaves
and left e x a c t
the c a t e g o r y
a distinction
of A.
transformation
(Schemes)
faithful,
of this d e f i n i t i o n
covering
is a n a t u r a l
faithful
the m o m e n t
b)
spaces.
(Schemes) it w i l l
Lemma)
Hence +
there
(Algebraic
be c o n v e n i e n t
a s c h e m e X and its a s s o c i a t e d
that p o i n t w e
identify
the two and
spaces). to
II. 1
93
just w r i t e X for b o t h Proposition a)
objects.
See 2 . 6
1.3:
L e t A b e an a l g e b r a i c
a representable s e n t i n g U" × U ' .
for details.
space,
e t a l e covering. Then
R .
U a scheme,
and U" ~ A
L e t R be the scheme
U is
an
etale
repre-
equivalence
relation
A
in the c a t e g o r y
of s c h e m e s
of R" ~ U ' .
particular,
In
equivalence U + V,
relation
R ~ U is
in the c a t e g o r y
L e t R -> U be an a r b i t r a r y
in the c a t e g o r y quasicompact.
of schemes•
effective
quotient
as
of schemes,
separated
which
type over
base
scheme
S--see
space A,
unique
up to u n i q u e
s h e a v e s U" ~ A, =U
satisfying
etale
Suppose
(A r e q u i r e m e n t
if R and U are of finite
R
if
is the c a t e g o r i c a l
an
etale
with quotient
then A = V'.
b)
braic
and A
equivalence
relation
the m a p R ~ U × U is is a u t o m a t i c a l l y
the u b i q u i t o u s
1.5.12.)
Then
there
satisfied
noetherian is an a l g e -
isomorphism,
and a m a p
p a r t b of d e f i n i t i o n
i.i, w i t h
of
×U A Proof. a)
I n d e e d U" ~ A is a u n i v e r s a l l y
in the c a t e g o r y
o f sheaves•
U" ~ G be any m a p of sheaves
so that
U" ~ A
G
epimorphism
To see the e f f e c t i v i t y ,
commutes R"
effective
the
following
let
diagram
II. i
We
94
are r e q u i r e d
To c o n s t r u c t a natural By
find
a map
fashion
the Y o n e d a
G(X)
to
of
a m a p ~0:A + G s u c h
sheaves,
for e v e r y
Lemma,
A(X)
= Homsheaves(X;G) .
We will ition
construct
i.i,
there
it
that ~
is s u f f i c i e n t
s c h e m e X,
a m a p A(X)
= #.
to give,
in
~---~--x >G(X).
= H o m s h e a v e s ( X ' , A ) and Let
~ be
a map
a m a p ~x(~) :X" + G. is a s c h e m e V
s c h e m e s V + X so t h a t V"
= U"
of s h e a v e s , Using
and
an e t a l e
× X'. A
Hence
~:X"
+ A.
p a r t b of d e f i n surjective
we have
the
map
of
follow-
ing d i a g r a m :
(v x v)"
=r% v" ~ >
x"
X
R"
"'~
U"
) A
G where
the
two m a p s
(V × V)
+ V" + G are
eaual.
sheaf
axiom
for G,
there
V" -~ X" + G = V" + G. we
have
constructed
The is n o w
fact
The
second
the
quotient b)
The
first
This
U.
is
just
A(X)
assertion
o f a) from
a reatatement
in G(X)
and
so
+ G(X).
is a u n i v e r s a l
is i m m e d i a t e
R
the
~ G with
is an e l e m e n t
the r e q u i r e d
assertion of
is a u n i q u e m a p X" --
This map
t h a t U" + A
clear.
Now using
~x (~)
x
effective now
epimorphism
follows
the d e f i n i t i o n
of 1 . 5 . 9 .
from 1.5.5. of V
~s
I
II. 1
95
Proposition
1.4:
Let A 1 and A 2 be algebraic
U I" ~ A 1 and U 2" ~ A 2 be representable g and h be maps
such
spaces
etale coverings.
that in the following
and Let
diagram
TT (U 1 × Ul)" A1
~
~
U I"
~2
rr[ (U 2 X U2)" A2
with ~h = f~. this way
proof.
Conversely,
for some choice
Proof. Let
representable
rr
rf2¢
hn I = ~Ig and h~ 2 = ~2g.
Of course
U 2" -----~> A2
Then there every map of Ul, U2,
the converse
f:A 1 + A 2 be given etale coverings
so the composite
p A1
is a unique map
f:A 1 ~ A 2 is induced
in
g, h. is the only part requiring
and U 2" ~ A 2 and V I" ~ A 1 be H o m s h e a v e s(v I',A 2) = A 2(V I)
map ~:V I" ~ A 2 e A 2(VI) •
A 2 is the quotient
sheaf of
(U 2" × U2" ) ~ U 2" so by the construction A2
sheaves,
y ¢ A2(Vl)
section
f:A 1 -~ A 2
of quotient
is given by a covering U 1 + V 1 and a
h e U 2" (U I) such that the two images
U 2" × U 2" (U 1 × U I) coincide. A2 V1
of h in
This gives h:U I" ~ U2".
the map U I" + V I" ~ A 1 is also a representable of A I.
The existence
mapping
property
of the map g follows
of U 2" × U 2" A2
Clearly
etale covering
from the universal •
II. 1
96
Proposition disjoint
sums
exist--i.e., spaces,
1.5:
exist. for
In the c a t e g o r y Also
any p a i r
the s h e a f
of a l g e b r a i c
fiber p r o d u c t s of m a p s
fiber product
spaces,
of a l g e b r a i c
spaces
A ~ C and B ~ C of a l g e b r a i c
A × C is an a l g e b r a i c
space.
B
Proof.
The
Given maps proposition
1.4
assertion
of d i s j o i n t
of algebraic to
find
Y" ~ B,
and Z" + C,
induced
by m a p s
spaces
representable
so t h a t
the m a p s
X ~ Z and Y ~ Z.
We
sums
A ~ C,
is clear. B + C, we use
etale
coverings
X" ~ A,
A + C and B ~ C are then have
a diagram YXY B
/ B
A
9 C
--
X
>
Z ~
XXX A
Let and X"
× z C
F be
the s h e a f A × B. T h e t w o m a p s X" × Y" -~ X" + A C Z" × Y" ~ Y" ~ B i n d u c e a m a p X" X Y" ~ F. Then Z" Z"
(X" X Y') Z"
× F
(X" × Y') Z"
=
(X" X X') A
~ (Z" X Z') C
(Y" X Y ' ) . B
II.l
97
(This is c e r t a i n l y
true if X', Y', Z', A, B, C are sets,
and to check a s t a t e m e n t category, functor,
one can r e p l a c e
X F
(X" X Y') ~ Z"
~i x n i(x" × X') A
~ (Z" × Z') C
~i × ~i((Xl'X2)' (YI'Y2)) (X" × Y') Z"
(X" × X') A which
and
= (xi'Yi)
and h e n c e
(X" × Y') Z"
×
are given by
are etale.
(X" X Y') Z"
(X" × X')
The
is the map
~ (Z" X Z')
(Y" × Y')
H e n c e we h a v e an etale e q u i v a l e n c e
(x" × Y') X (X" × Y') ~ (X" X Y').
chase shows indeed F.
Z" F Z" that the q u o t i e n t Applying
Definition a representable R" = U" X U'.
1.3b),
1.6:
A simple d i a g r a m Z" of this e q u i v a l e n c e r e l a t i o n is
F is an a l g e b r a i c
Let A be an algebraic
etale covering.
R ~ U × U is a closed
immersion.
•
space and U" ~ A
L e t R be the scheme
for w h i c h
if the map
A is separated
if
immersion.
for the locally separated
finite type
space.
W e say that A is l o c a l l M separated
R + U X U is a q u a s i c o m p a c t
Note
The two m a p s
(Y" x Y') ~ X" X Y', B
(Y" × Y') ~ B
is q u a s i c o m p a c t .
relation
(X" × Y') Z"
X (X" X Y') ~ F Z"
~>~ (Z" × Z') C
in an a r b i t r a r y
each object by its r e p r e s e n t a b l e
and c h e c k in the c a t e g o r y of sets.)
~. :(X" × Y') l Z"
map
about inverse limits
(over the u b i q u i t o u s
case,
if U is locally of
noetherian
separated b a s e
scheme S) and R + U x U is an immersion,
then R + U × U is
automatically
(See 1.2.27)
a qrasicompact
immersion.
II. 1
98
Prpposition
1.7:
a representable R"
L e t A be an a l g e b r a i c
etale
covering.
Let
R be
space
the
a n d U" ~ A
scheme
for w h i c h
= U"
× U'. L e t X and Y be s c h e m e s and X" ~ A and Y" + A A b e a r b i t r a r y m a p s of s h e a v e s . T h e n the s h e a f X" × Y" is A r e p r e s e n t a b l e , say by a s c h e m e W , W " = X" × Y', and t h e m a p A W ~ X × Y is q u a s i c o m p a c t . If
furthermore
a closed tively
immersion),
a closed
Proof. Case and Y"
R ~ U X U is an i m m e r s i o n
consider
The
T h e n X"
so is r e p r e s e n t a b l e .
X"
× (U" × U') U" Case
The
X" ~ A and Y" × Y" A
fact t h a t
2.
(see 1.5.12) . X Y" U"
= X"
etale
representable,
the m a p s
and
U"
× Y" ~ Y"
are
to
the m a p s U"
etale
covering,
diagram
× U" × Y" A U"
R + U
= X"
is an e t a l e
implies
× R" × Y" U" U °
equivalence
that R ~ U × U
X"
arbitrary. U"
X X" A
representing
surjective.
× X" ~ A and U" A
f a c t o r X" ~ U" -~ A
× Y" = X" × R" × Y" A U" U" is q u a s i a f f i n e .
X" + A and Y" + A are
is a r e p r e s e n t a b l e
cartesian
Hence
× Y"
~ A
= X" × U" U"
and R + U × U is q u a s i c o m p a c t ,
is q u a s i a f f i n e
(respec-
two c a s e s .
two m a p s
~ U" + A.
relation
is an i m m e r s i o n
immersion).
We
1.
then W + X × Y
(respectively
× Y" A
We
+ A.
and U"
U"
now
S i n c e U" + A × Y" A
× X" ~ X" A apply case
Consider
the
are
and one
99
II. 1
(u" xx')
x (u" xY')
A
A
(U" X X') A
~ x" x Y "
A
A
L
1
X
(U" X Y') A
-p X" X Y"
The b o t t o m
line of this d i a g r a m
surjective
m a p of schemes
sented
by a q u a s i a f f i n e
of q u a s i a f f i n e quasiaffine compact,
map
the left hand
m a p of schemes.
the right
of schemes.
hand
Since
by an etale
side
is r e p r e -
By e f f e c t i v e
side
descent
is r e p r e s e n t e d
a quasiaffine
map
by a
is q u a s i -
we are done.
Corollary a) etale
maps,
and
is r e p r e s e n t e d
I
1.8:
L e t A:(Schemes) °pp +
topology.
V" ~ A be maps condition
(Sets)
be any sheaf
L e t U and V be schemes of sheaves
of D e f i n i t i o n
satisfying
l.lb).
in the
and U" ~ A and
the local
L e t R and S be
representability the schemes
representing compact a closed
U" × U" and V" × V" T h e n R ~ U × U is q u a s i A A (respectively a q u a s i c o m p a c t immersion, r e s p e c t i v e l y immersion)
(a q u a s i c o m p a c t separation
if and only
immersion,
conditions
of the p a r t i c u l a r
if S ~ V × V is q u a s i c o m p a c t
a closed
immersion).
on an a l g e b r a i c
choice
Thus
the
space A are i n d e p e n d e n t
of r e p r e s e n t a b l e
etale
coverings
of A.
II. I
100
b)
L e t A b e an a l g e b r a i c
and g be two e l e m e n t s X" ~ A.
Then
Y ~ X such maps
there
of A ( X ) - - i . e . ,
exists
h ' : G ~ X"
Y + X is q u a s i c o m p a c t . tively
separated),
immersion
Detail
in the d e f i n i t i o n tion
1.3
volved
Indeed with
this p o i n t
had
mination
The map
separated
(respec-
I
T h e q~]asiseparatedness space has b e e n used
assumption for P r o p o s i -
of fiber p r o d u c t s ,
and the c l a s s
a basic
of q u a s i a f f i n e
maps
descent. descent
c l a s s e s m i g h t h a v e b e e n used.
assumed
are i m m e r s i o n s
this a s s u m p t i o n
out n o n q u a s i c o m p a c t At
fh' = gh'
for
is that all of the m a p s R + U × U in-
w e could have
R ~ U × U involved
class
1.9:
The point
effective
particular,
is u n i v e r s a l
immersion).
for the e x i s t e n c e
effective
Other
two m a p s
from the last p r o p o s i t i o n .
are q u a s i a f f i n e ,
satisfies
a closed
of a l g e b r a i c
and h e n c e
construction.
f
and a m a p o f s c h e m e s
m a p h:Y" ~ X"
satisfying
Let
then the m a p Y + X is a q u a s i c o m p a c t
Immediate
Technical
f and g are
If A is l o c a l l y
(respectively
Proof.
and X a scheme.
a scheme Y,
that the induced
of s h e a v e s
space
only
that
all the m a p s
(quasicompact
it w o u l d
In
or not).
not be n e c e s s a r y
to r u l e
schemes. in the t h e o r y
to be made.
One
effect
of w h i c h p a t h o l o g i c a l
a choice
of e f f e c t i v e
of this c h o i c e
examples
to allow
descent
is the d e t e r and disallow.
II.2
101
We've over and and
noted
(I.5.11)
a noetherian (I.5. 12),
base
scheme
for any R ~ U ,
locally
of
S are always
if U ~ U
R ~ U × U is an immersion,
So at the leastj not
that s c h e m e s
finite
type
quasiseparated
is l o c a l l y
noetherian
R ~ U × U is q u a s i a f f i n e .
the p a t h o l o g i c a l
examples
involved
are all
locally noetherian• Hence
the d e c i s i o n
case has b e e n m a d e the s t a t e m e n t s
to r e s t r i c t
to the q u a s i s e p a r a t e d
on o t h e r m o r e p r a g m a t i c
3.13,
6.2,
and
6.7 true
grounds--we
for a l g e b r a i c
have
spaces
in general.
2.
The Etale
Topoloqy
Definition spaces.
2.1:
of A l g e b r a i c
Let
Using proposition
coverings
U I" ~ A 1 and U 2
m a p h:U 1 ~ U 2.
We
say
Spaces
f:A 1 ~ A 2 be a m a p 1.4,
choose
~ A 2 so that
f is e t a l e
of a l g e b r a i c
representable
f is induced b y a
if such c o v e r i n g s
found w i t h h etale.
(It is then a simple m a t t e r
that any h'
f m u s t be etale.)
inducing
if f is e t a l e surjective
and a c a t e g o r i c a l
f will
can b e
to show
f is e t a l e
epimorphism.
also be r e f e r r e d
etale
Such
surjective an e t a l e
to as a c o v e r i n g map.
I I. 2
102
Proposition-Definition f:X ~ Y,
f is etale
associated
map
surjective)
2.2:
(etale
spaces
sense.
algebraic
spaces
is a closed
algebraic
spaces
satisfying
of I.i.19.
The
associated
braic
spaces
is the etale
cise,
the global
etale
any m a p of schemes
surjective)
of a l g e b r a i c
in the above
For
if and o n l y
f':X" + Y"
The class
subcategory axioms
SI,
topology topoloqy
topoloqy
if the
is etale
(etale
of etale m a p s
of
of the c a t e g o r y
of
S2,
S3(a ) and S3(b )
on the c a t e g o r y (or sometimes,
on the c a t e g o r y
of alge-
to b e p r e -
of a l g e b r a i c
spaces) . For topoloqv
a particular
algebraic
space A,
the
(local)
etale
T on A has: Cat T = that c a t e g o r y maps
whose
objects
are etale
B ~ A,
and w h o s e
morphisms
commutative
triangles
B l a B
C o v T = all
families
[B i + B]i6i
are 2
with
~_/ B. ~ B surjective. 1 i EI 2.3: etale
It is clear
covering
B ~ A, with covering there
from
the d e f i n i t i o n s
U" ~ A is e x a c t l y
B a representable
sheaf.
m a p A + C in the etale
is a scheme U,
the same
that a r e p r e s e n t a b l e
as an etale
Conversely,
topology
and a r e p r e s e n t a b l e
for any
of a l g e b r a i c etale
covering
spaces,
covering
II. 2
103
U" + A w i t h covering. union
the c o m p o s i t e To c a r r y
U" ~ C also a r e p r e s e n t a b l e
this one step
of all the open
affine
subschemes
p o s i t e W" ~ C is a r e p r e s e n t a b l e Proposition any etale
covering
to an etale schemes. coverin~
a full
W" + B w i t h W
As m e n t i o n e d ~
Then
covering.
left exact,
compatible
with
its global
etale
union
an affine
of affine
etale
of
taking X ~ X "
(algebraic
every
functor is full,
the c a t e g o r y spaces).
topologies
This
in both
extends
spaces
for an a l g e b r a i c
uniquely
in its global space A w i t h
imbedding
is d e t e r m i n e d
of schemes.
by
is
of schemes
in
to a sheaf on the etale affine
topology. etale
and R" = U" X U', F(A) = Ker (F(U) ~ F(R)). A conversely, every sheaf on the c a t e g o r y of a l g e b r a i c topology
with
categories.
sheaf F on the c a t e g o r y
topology
faith-
of schemes
U" ~ A,
category
spaces,
space B can be refined
the d i s j o i n t
the
identifying
of a l g e b r a i c
(By setting,
Spaces)
the etale
Furthermore,
the etale
the com-
Hence we have
of a l g e b r a i c
w i l l be c a l l e d
in 1.2,
(Algebraic
subcategory
category
of U.
A ~ B of an a l g e b r a i c
covering
let W be a d i s j o i n t
of B.)
(Schemes) and
etale
In the c a t e g o r y
(Such a c o v e r i n g
2.5:
ful,
2.4:
further,
etale
its r e s t r i c t i o n
covering
O f course, spaces to the
in
II. 2
104
In particular, s h e a f of rings
~
the c a t e g o r y (where
mines
~,
rings
on the c a t e g o r y
denote
the sheaf For of
~
~*,
of n i l p o t e n t s
of
a particular
of A, d e n o t e d
every
representable
spaces w h i c h
sheaf of w e also
of units
~,
to a l g e b r a i c
extend
of
~,
and N,
spaces.
space A, the r e s t r i c t i o n
topology
of A is the s t r u c t u r e
sheaf
and
as such extends
of a l g e b r a i c
functor
space A is a sheaf
spaces.
on the c a t e g o r y
on schemes
uniquely
to a sheaf
In other words, of a l g e b r a i c
every
spaces
is
(the axiom A0).
2.6:
The time has come
the symbols
algebraic
spaces
the associated algebraic
is a scheme is an affine
the symbols
sheaves
on either
We consider
of a l g e b r a i c
if it lies scheme
to sort out our notation.
S,T, U, V, X,
and
spaces.
a subcategory
schemes.
the s t r u c t u r e
the sheaf
algebraic
topology,
on the c a t e g o r y
a sheaf
the s t r u c t u r e
%.
Finally, in the etale
to give
algebraic
etale
has
R) = R for any ring R d e t e r -
of a l g e b r a i c
Similarly,
to the local
now on,
~(Spec
and this extends
~.
of schemes
if X lies
S',
T',
denote
etc.,
the c a t e g o r y
the c a t e g o r y
spaces
in this
... will
arbitrary
will
denote
of schemes
of schemes
and say an a l g e b r a i c
subcategory.
From
Similarly,
in the s u b c a t e g o r y
or
to be space X X
of affine
II. 2
105
Definition if X has map
2.7:
a covering
An
algebraic
space X is q u a s i c o m p a c t
W + X with W a q u a s i c o m p a c t
f:X + Y of a l g e b r a i c
spaces
scheme.
is q u a s i c o m p a c t
etale m a p w + Y, w i t h W a q u a s i c o m p a c t
scheme,
A
if for every W × X is Y
quasicompact. X is locally
noetherian
W ~ X w i t h W a locally
noetherian
if it is q u a s i c o m p a c t Proposition algebraic 2.9: has
spaces
schemes.
compact
etale
The class
is stable
covering
union
(2.4)
If X is q u a s i c o m p a c t to be affine.
can both be c h o s e n W e will covering
later
every
hence
R ~ W w i t h W affine, and,
hence and
algebraic
space X
union of affine
affine.
separated,
spectra
that every
Thus
of an etale
by 1.5.12,
quasifinite
to be affine
[Xi ------>X], with
•
R a quasicompact
If X is n o e t h e r i a n
show
of
we can take W to be a finite
schemes,
scheme,
maps
topology.
W + X with W a disjoint
and q u a s i c o m p a c t ,
chosen
X is n o e t h e r i a n
of q u a s i c o m p a c t
space X is the q u o t i e n t
of an affine
covering
noetherian.
in the etale
above
of affine
algebraic
relation
scheme
2.8:
is an etale
scheme.
locally
If X is q u a s i c o m p a c t ,
disjoint
lence
and
As m e n t i o n e d
an etale
if there
the maps
a quasiequivaopen
sub-
~i,~2
and of finite
type.
R and W can b o t h be and separated, of n o e t h e r i a n
algebraic
R and W rings.
space X h a s
each X i a q u a s i c o m p a c t
a
algebraic
II. 3
106
space
and each ~i an "open
(Prop. "keep
3.13)
Thus
algebraic
algebraic
are
Descent
Theory
theory
for A l g e b r a i c
theory
assert
that the e x t e n d e d
notion
when
Extension
topology.
algebraic
spaces b y
representable Thus w e h a v e will
speak
spaces, spaces
etale
over
the n o n q u a s i c o m p a c t fashion
Spaces
to extend
notion
a number
In each
case w e
is c o m p a t i b l e
of d e f i n i t i o n s implicitly
with
Let P be a stable p r o p e r t y Then P extends taking, covering
spoken o f
of r e d u c e d
normal
can c o n c e n t r a t e
the o r i g i n a l
to schemes.
3.1:
the etale
since
which
in a Z a r i s k i - t o p o l o g y
to our context.
applied
schemes"
spaces".
ones.
We now use d e s c e n t in scheme
spaces,
just b u i l t
from the q u a s i c o m p a c t
3.
from b e i n g
algebraic
spaces
of a l g e b r a i c
any study of the p a t h o l o g i e s
spaces
on q u a s i c o m p a c t
immersion
U + X, P'(X) noetherian
algebraic
algebraic
to a stable p r o p e r t y
for any a l g e b r a i c
locally
spaces,
of schemes
spaces,
in
P'
of
space X, w i t h
if and o n l y algebraic
spaces
nonsin~ular
and n - d i m e n s i o n a l
if P(U). and
algebraic algebraic
a field k.
Extension
3.2:
local
on the domain.
braic
spaces w h i c h
L e t D be a stable Then D extends is local
class
of m a p s
to a class
on the d o m a i n
D'
of schemes,
of m a p s
and stable.
of alge-
W e define,
II. 3
107
for any map f:X + Y of a l g e b r a i c for some r e p r e s e n t a b l e
spaces,
D'(f)
if and only if
etale c o v e r i n g U ~ Y and for some repre-
sentable etale c o v e r i n g V ~ U × X, the induced map of schemes Y V + U is in D. H e n c e w e can speak of m a p s of a l g e b r a i c spaces being
sur~ective,
etale,
flat,
faithful!y
flat, u n i v e r s a l l y
l o c a l l y of finite p r e s e n t a t i o n ,
open,
locally of finite type,
and locally quasifinite. Definition
3.3:
A map
f:X + Y of algebraic
of finite type if it is locally of finite f is o f finite p r e s e n t a t i o n tion, pact.
quasicompact,
spaces
is
type and q u a s i c o m p a c t °
if f is locally of finite p r e s e n t a -
and the induced m a p X + X × X is q u a s i c o m -
f is q u a s i f i n i t e
if f is locally q u a s i f i n i t e
and q u a s i -
compact. Proposition which
3.4:
The classes
of m a p s of algebraic
are of finite type, maps of finite p r e s e n t a t i o n ,
q u a s i f i n i t e m a p s are stable in the etale P r o p o s i t i o n 3.5: which
Let
Proof.
I
f:X + Y be a map of algebraic Then
spaces f is
open. All three classes
of maps
involved
locally on the d o m a i n in the etale topology, consequence
and
topology.
is flat and locally of finite p r e s e n t a t i o n .
universally
spaces
of the c o r r e s p o n d i n g
assertion
are d e f i n e d
so this is a
for schemes
1.3.8.1
II. 3
108
3.6: locally scheme
From ChapterIII
noetherian theory,
products.
on, w e w i l l
or n o e t h e r i a n
these categories
But,
as in s c h e m e
deal p r i m a r i l y
algebraic
spaces.
are n o t c l o s e d
theory,
we have
with
As in
under
fiber
the H i l b e r t
Basis Theorem: Theorem which
is
3.6:
Let
(locally)
noetherian.
f:X + Y b e
of finite
T h e n X is
Proof.
Exactly
a m a p of a l g e b r a i c
type.
(locally)
S u p p o s e Y is
Basis
Corollary (locally) or g is
as in schemes,
we reduce schemes
easily
to the
and apply the
Theorem.
3.7:
Let
noetherian (locally)
(iocally)
noetherian.
case w h e r e b o t h X and Y are a f f i n e usual Hilbert
spaces
•
f:X + Y and g:Z + Y be m a p s of
algebraic
of finite
spaces.
type.
Suppose
Then X × Y Y
is
either
f
(locally)
noetherian.
•
Extension schemes
3.8:
satisfying
Let D be a stable effective
stable
effective-descent
spaces
in the
spaces
is in D'
descent.
class D'
f o l l o w i n g way: if and only
c l a s s of m a p s Then D extends
of m a p s
A map
f:X ~ Y of a l g e b r a i c
if for any s c h e m e Y'
affine morphisms,
to a
of a l g e b r a i c
and m a p
Y' ~ Y, X × Y' is a scheme and X × Y' ~ Y' is in D. Y Y H e n c e w e can talk about o p e n i m m e r s i o n s , closed immersions,
of
quasiaffine
morphisms
immersions, and r e d u c e d
I I. 3
109
closed
immersions
closed
subspace,
ticular
open
morphism so is X.
immersion,
3.9:
quasi-separated,
separated,
3.10:
The classes
topology.
algebraic
quasiseparated,
spaces
separated,
is
respectively
schemes
representable
of a l g e b r a i c
Also,
respectively
The
first
maps
of q u a s i - s e p a r a t e d ,
locally
if the m a p
first
the case w h e r e
etale
x
is clear.
everything
~
L
""~
U
x S
~.,
For
is taken.
Then
X
X
respectively etc.
the second,
Y is Spec Z, or any s e p a r a t e d
covering.
U
space X and
f:X ~ Y, X is q u a s i -
separated,
there
diagram U
locally
are stable
f is q u a s i s e p a r a t e d ,
statement
over w h i c h
spaces
for any a l g e b r a i c
space Y and any map
if and only
Proof.
base
is a scheme.
map X ~ X × X is q u a s i c o m p a c t , respecY immersion, r e s p e c t i v e l y a closed immersion.
and s e p a r a t e d
in the etale
consider
then
if the induced
Proposition
separated
affine
is a scheme,
f:X ~ Y of a l g e b r a i c locally
in p a r -
spaces w h i c h
immersion,
and Y
of a scheme
respectively
a quasicompact
separated
immersion,
A map
Note
of a l g e b r a i c
morphism,
a subspace
Definition
separated
closed
also use the w o r d s
and subspace.
is a m a p
or q u a s i a f f i n e Thus
We will
subspace,
that if f:X ~ Y
is an o p e n
tively
of schemes.
--~XxX S
U
Let U ~ X be a
is a c a r t e s i a n
II. 3
with
110
the r i g h t
separated
hand
side
algebraic
etale
s p a c e Y,
surjective.
apply
For
I.i.21
an a r b i t r a r y
to the c o m p o s i t e
X ~ Y ~ S.
•
Proposition spaces.
Let
fg = Iy. etale,
If
3.11:
g:Y + X b e
Apply
Proposition an a f f i n e
etale
a section
I.i.21
3.12:
of
f, i.e.,
g is a c l o s e d
to the c o m p o s i t e
L e t X be
covering.
~ is an a f f i n e
etale
covering
algebraic
a map
satisfying If
immersion.
map.
finite
fg.
an a l g e b r a i c
space
and n : U ~ X
If X is s e p a r -
If X is q u a s i c o m p a c t ,
there
U an a f f i n e
type,
hence
scheme
is an
and ~ etale,
quasicompact
and q u a s i -
finite.
•
Proposition there
is a f a m i l y
immersion
f is
T h e n ~ is s e p a r a t e d .
~:U + X with
and of
of
a map
immersion.
ated,
separated,
f:X ~ Y be
f is s e p a r a t e d ,
g is an o p e n
Proof.
Let
and
3.13:
L e t X be
of m a p s
{X i ,
i-7 X}i£i
each X i q u a s i c o m p a c t ,
d X. -~ X is e t a l e s u r j e c t i v e . l i eI each of the X. is a scheme. i Proof.
an a l g e b r a i c
Let W + X be
W =
union
let R. -- W. X W. = R i i-i 2%
×
e a c h %0i an o p e n
that
the m a p
covering
of a f f i n e
Then
with
X is a s c h e m e
an e t a l e
J W. the d i s j o i n t 1 ieI c o v e r i n g s e x i s t b y 2.4.
such
space.
if and o n l y
of X with
schemes
W..
Such
1
L e t R = W >< W, X
(w >< W)
(W. >< W.). 1 l
and
for e a c h
i e I,
T h e n R. W. is i ~ 1
if
111
II.3
is
an e t a l e
there
is
equivalence
an algebraic
relation. space
we
claim
this
in
a diagram
map
is
X. w i t h l There
W. ~ X. a n d R. = W. X W.. 1 1 1 1 1 X. 1
an o p e n
the
given
R. 1
--
--~
R
----------~ W
By
affine
is
an
and
etale
induced
immersion.
information W. l
1.3b,
Let name
covering
map
us the
X. + X l
first
and
sum
relevant
up maps:
.__-------~ X. i
TT
7
X
-rq W ~ X map
is
of
a representable
schemes.
A
etale
covering
simple
diagram
Also,
W. ~ W l
is a m o n o m o r p h i s m .
chase is
so W × X. ~ W is a 1 X s h o w s t h a t X. ~ X l
etale
so X. + X l
is e t a l e .
Hence
W × X. ~ W is an e t a l e m o n o m o r p h i s m , h e n c e an o p e n 1 X immersion. ( I n d e e d , W x X.1 = ~ l ( ~ 2 - 1 ( W i ) ) a l s o s h o w i n g t h a t X W x X. + W m u s t b e an o p e n i m m e r s i o n . ) H e n c e X. ~ X is an 1 i X open immersion. Each The
X. is q u a s i c o m p a c t l
family
a covering
[X i
2X]
of W.
Hence
equivalence
[
;
relation
x. x x .
i,jeI o f X. and l
X o f X.. 3
[-] ~ K keI
is
of
X
since
e a c h W. is q u a s i c o m p a c t . l
a covering is t h e
algebraic where
each
of X
quotient
since of
the
[W i ~ W}
is
etale
spaces X. × X. is l 3 X
an o p e n
subspace
112
II. 3
If X is a scheme, Conversely,
if each
scheme
obtained
common
open
o p e n s u b s p a c e X.l is a scheme.
of the X. is a scheme, l
by gluing
3.14:
of a l g e b r a i c
each X. and X. t o g e t h e r i 3
R
= U
along
Let R ~> T[2
U be a c a t e g o r i c a l
spaces with
space Z and etale
the •
each m a p ~. etale 1
i n d u c e d m a p R -~ U X U q u a s i a f f i n e . algebraic
then X is the
s u b s a p c e X. X X.. i X 3
Proposition relation
every
Then
surjective
there
equivalence and the
is a u n i q u e
m a p U ~ Z such
that
X U. Z
Proof. spaces,
In the c a t e g o r y
let Z be the q u o t i e n t
L e t V + U be a r e p r e s e n t a b l e Then V" X V" Z = R" ~
of s h e a v e s sheaf. etale
of sets on a l g e b r a i c
T h e n R" = U" × U" Z
covering
of U b y a scheme V.
= V" × U" X U" × V" = V" × R" X V" U" Z U" U" U" (V" × V'). H e n c e V" × V" -~ V" × V"
(u" x u ' )
is q u a s i -
z
affine
so V" X V" ~ V" is an e t a l e e q u i v a l e n c e r e l a t i o n of Z schemes, w h o s e q u o t i e n t exists b y 1.3b. It is a s i m p l e d i a g r a m chase
to show that
this q u o t i e n t
m u s t be Z.
•
II.4
4.
113
Quasicoherent Definition
structure coherent
Sheaves
4.1:
and C o h o m o l o q y
L e t X b e an a l g e b r a i c
s h e a f of rings.
A s h e a f of
if for some c o v e r i n g
the i n d u c e d the usual
s h e a f of
map
~X-mOdules
i'F,
algebraic
free of rank
(respectively
locally
free of rank r).
(coherent)
Notation following
sheaves
4.2:
notation
for the
in
(~X-mOdules),
AP x =
(Abelian p r e s h e a v e s
and AB3*.
For n o e t h e r i a n that
4.3:
(Abelian on X),
Qcs X
of
(Coherent
sheaves
Sheaves
on X),
on X),
(Abelian
groups).
is an a b e l i a n
category
satisfying
m a p s QCS x ~ MS x ~ AS x are exact.
for a family as
the
and A b =
C 3 are in CS X then
~F. can be c o m p u t e d l
W e use
categories:
the m a p CS x + Q C S x is e x a c t
where
of q u a s i -
~X-mOdules.
space.
on X), CS x =
if X is q u a s i c o m p a c t ,
(= coproducts)
A morphism
and has
for all 0 + C 1 ~ C 2 + C 3 ~ 0 e x a c t
if two of CI, C2, Also,
AS x =
The n a t u r a l X,
following
say F is c o h e r e n t
if i*F is c o h e r e n t
F ~ G is a m a p
sheaves
MS x =
Proposition
r)
L e t X be an a l g e b r a i c
(Quasicoherent
property
its
F is q u a s i -
is q u a s i c o h e r e n t
space X, w e
locally
AB5
~X
sense.
(respectively
QCS x =
and
i:U ~ X, w i t h U a scheme,
~U-mOdules,
For a n o e t h e r i a n
coherent
space
the third
then Q C S x has [Fi]
the
in QCSx,
is also. arbitrary
of q u a s i c o h e r e n t
the sum of the F. in the c a t e g o r y l
sums sheaves, of
I1.4
114
abelian p r e s h e a v e s 4.4: ab f
L e t f:X ~ Y.
:ASy ~ AS X.
The
left adjoint ~ (F) = fab
write
on X.
"same"
(F) ab ® ~ X "
functor
for F an ~ X - m O d u l e ,
Following
and ~
f, :MS X ~ M S y also has a
standard p r a c t i c e we will
but the reader should k e e p
the two are q u i t e different.
exact w h i l e ~
(For instance,
is in general only right exact.)
if f is etale,
then for F an
Proposition
4.5:
and G a q u a s i c o h e r e n t is quasicoherent, ~y-modules
I
Then f,:AS x + ASy has a left adjoint
:MSy ~ MS x w h e r e
f* for both fab
m i n d that
(See [GT], p. 49)
Let
~X-mOdule,
lab(F)
Then
f*G
of
map of algebraic Proof. to be affine.
= ~
(F) = FIX.
(that is, ~
~X-mOdules.
fully flat iff f is flat and for any F e QCSy, 4.6:
Let
The assertions Since
Then
spaces (G)) of
f is faithf*F = 0 iff F = 0 . 1
f:X ~ Y be a q u a s i c o m p a c t
spaces and F E QCS x.
is
O f course
f is flat iff f* takes exact sequences
into exact sequences
Proposition
fab
f:X ~ Y be a map of algebraic
sheaf on Y.
in
separated
f,F e QCSy.
are local on Y so we can take Y
f is quasicompact,
W ~ X with W an affine scheme.
Also,
X has an etale c o v e r i n g
since X is separated,
W × W is an affine scheme. Thus we have a d i a g r a m X qT~ n W X W --~ W -------~>X x
Y
II.4
115
The fact that F is a sheaf on X says that there is an exact sequence F ~ ~,(FIw)
(7
~
~i)
Jl
W × W X
f, is left exact so f,F ~ f,n,FIw ~
(f ~ ~ i ) , F I w x w X
is exact. schemes,
But the two maps so they p r e s e r v e
f~ and f n ~
l
are m o r p h i s m s
quasicoherence,
and FIW X W are quasicoherent.
and w e k n o w
of affine that F]W
Since Q C S y ~ MSy is exact,
X this p r o v e s
that f,F is q u a s i c o h e r e n t .
Proposition coherent
4.7:
Let X be an affine scheme
sheaf on X.
is the c o h o m o l o g y Also,
Then Hq(X,F)
of F c o n s i d e r e d
and F a quasi-
= 0, q > 0 , ~ w h e r e Hq(X,F) as an abelian
sheaf),
if f:X ~ Y is a map of affine schemes,
then the higher d i r e c t w h e r e Rqf,
1
is here
images Rqf, F of F are zero,
the q
th
derived
and F e QCSx, for q > 0,
functor of the functor
f,:AS x ~ ASy. Proof.
See 1.4.16.
Corollary spaces.
4.8:
Let
• f:X ~ Y be an affine map of algebraic
Then f,:QCS x ~ QCSy is exact.
Also,
for any F e QCSx~
Rq f, (F) = 0, q > 0, w h e r e Rq f, is the q th right d e r i v e d of the functor
f,:AS x ~ ASy.
functor
II.4
116
Proof. hence
X,
Both
assertions
to be affine
Proposition T h e n QCS E has
schemes.
4.9:
enough
injectives
= 0, q > 0, w h e r e
abelian
sheaf.
union
Let
of affine
has
schemes
an exact
schemes,
of F into
To show cohomology, form
f*F
c QCS U.
exist.)
f.I is an injective
is a m o n o m o r p h i s m (by the sheaf
an i n j e c t i v e
injectives
it is s u f f i c i e n t
Let
union Since
in QCS E.
Since
f,I.
axiom:)
so there
object
of QCS E.
of f,
f,f*F ~
in QCS E are acyclic to show
for I an i n j e c t i v e
commutes
and c o h o m o l o g y
cohomology
of I vanishes,
summand
J of I.
this
But is
Hence
Indeed,
of such with
so m u s t
for abelian
for injectives
in QCS U.
ment
of any d i r e c t
with U a disjoint
Then
envelopes
summand
abelian
map,
in QCSE,
of I as an
(U is a d i s j o i n t
tive J in QCS E is a d i r e c t agove,
space.
injectives.
that
f.I
algebraic
~ the c o h o m o l o g y
injective
there
is then clear. I
for any I i n j e c t i v e
of f*F.
is a m o n o m o r p h i s m
QCS E has enough
and
and F e QCS E.
left adjoint,
a monomorphism
the
this
envelope
so such
f, is left exact, F ~ f,f*F
assertion
f:U ~ E be a c o v e r i n g
I be an i n j e c t i v e affine
The
L e t E be a s e p a r a t e d
Hq(E,I)
Proof.
are local on Y so w e can take Y,
of
any injec-
an I, by the argu-
direct
sums.
the abelian
If the cohomology
II.4
117
So let I be i n j e c t i v e The c o m p o s i t e a spectral
for U ~ E our c o v e r i n g map.
functor AS U + AS E ~ A b gives,
for any F e ASu,
s e q u e n c e E 2 P q = H P ( E , R q f , F) = HP+q(U,F).
F be our i n j e c t i v e b y 4.7, HP+q(u,I) Definition The
in QCSu,
sheaf I.
= 0, p+q > 0.
4.10:
(quasicoherent)
the d e r i v e d
By 4.8, Rqf, I = 0
cohomoloqy
functors
for q > 0 and
H e n c e HP(E,f,I)
Let X b e a separated
L e t now
= 0, p > 0 . 1
algebraic
groups Hi(x, - ) , i ~
of the left exact global
space.
0 of X are
section
functor
F:QCS x ~ Ab, F w ~ 2 F ( X , F ) . Proposition
4.11:
For each i ~
0, there is a c o m m u t a t i v e
diagram QCS x
n
inclusion
/
Ab
~
AS X
In other words,
the abelian sheaf c o h o m o l o g y
the q u a s i c o h e r e n t Proof. natural spectral
sheaf c o h o m o l o g y
is exact.
Hence
sequence of a c o m p o s i t e
this case since b y 4.9, X is acyclic
on QCS x-
The d i a g r a m c l e a r l y commutes
inclusion
coincides with
this
for i = 0, and the
follows
functor,
which
from the usual applies
an injective q u a s i c o h e r e n t
for the abelian cohomology.
in
sheaf on R
II.4
118
Definition separated
4.12:
algebraic
Let
f:X ~ Y be a q u a s i c o m p a c t The
functor
f,:QCS x ~ Q C S y is
left exact and its higher direct
images,
Rqf.,
right derived
spaces.
map of
q -~ 0, are the
functors.
Proposition
4.13:
is a commutative
(With the assumptions
of 4.12)
There
diagram Rqf,
QCS x
>
QCSy
~
u,~ ASy
Rqf, AS x
where
the bottom
lines
Proposition
4.14:
separated
algebraic
are the derived Let
spaces
Then there is a spectral Let g:Y ~ Z be another
and F a q u a s i c o h e r e n t
quasicompact
E2Pq = RPg.(Rqf.F)
= RP+q(gf),F.
spaces
and F e QCS x.
that f,G = F. Then H q(x,F)
Let
= HP+q(X,F) • spaces
sequence •
f:Y + X be an affine map of algebraic
Suppose
(In particular, = H q(Y,G),
map of algebraic
is a spectral
map of
sheaf on X.
sequence E2Pq = HP(Y,Rqf,F)
Then there
4.1~:
ll of f.:AS x + A S y . 1
f:X + Y be a quasicompact
with Z separated.
Corollary
functors
q > 0.
there is a sheaf G c QCSy such this holds
if F = f.f*F.) I
119
I1.4
Proposition
4.16:
Let X be an algebraic
elements of the abelian cohomology naturally
in a one-one
invertible
space.
group HI(x, ~ )
fashion to isomorphism
The
correspond
classes of
sheaves on X.
Proof.
It is a standard
fact that in any Grothendieck
topology with a final object X, the twisted
forms of an
abelian sheaf F on X are classified by the set Hl(X,Aut F ) . I Proposition space,
4.17:
Let X be a quasicompact
and [Fi]ie I a filtered
sheaves on X. Proof. "noetherian"
inductive
algebraic
system of quasicoherent
Then Hq(X,Li_~__. Fi) = ~ Hq(X,Fi) . I I If X is quasicompact, its etale topology is in the sense of [GT], II.5,
from w h i c h this
theorem is quoted. Proposition map of algebraic spaces.
I
4.18:
Let f:X ~ Y be a quasicompact
spaces.
Construct
Let n:Y' + Y be a flat map of algebraic
a cartesian diagram
X'
~
y,
~
~'
f'
->x
,I f ~
Then ~*f, = f,'~'*:QCS x ~ QCSy~
>y
(I.e., the natural
tion f,'~'* + ~*f, is a natural equivalence.) q > 0, ~*R q~.
=
separated
Rqf,1 ~'* :QCS x + QCSy,.
transforma-
Indeed,
for all
I I. 5
120
Proof. be
The
assertion
is local
an affine
scheme.
It is local
be an affine
scheme.
Using
it is local
on X so we can
In the c a t e g o r y
5.
Local
using
apply
in I.l,
the class
involved of affine
take Y'
of the p r o o f
take X to be an affine schemes,
the a s s e r t i o n
1.4,
for a number
of q u a s i a f f i n e
and c o v e r i n g s schemes)
of local
maps
by affine
to
of 4.6,
scheme. is c l e a r . ~
for the cofinal 5.1:
Spec.
m a p ~:Y ~ X,
= Spec ~*A.
construction. that
The
any affine m o r p h i s m
descent
(or d i s j o i n t
set of c o v e r i n g s
~x-algebras.
induced
the map ~
for the strict
For
is denoted
A + X is an affine
f:Y + X is of the
unions
involved. space
any c o v e r i n g
By E 6 A II.i.5.2 ~(X)
as
always
L e t X be an a l g e b r a i c
sheaf of
let ~(Y)
constructions,
of c o n s t r u c t i o n s ,
schemes
and A a q u a s i c o h e r e n t
Note
so we can
the t e c h n i q u e
the t e c h n i q u e
Construction
local
on Y'
take Y to
Constructions
We now outlined
of affine
on Y so we can
this
~
is a
A.
morphism
and
form Y = Spec A ~ X = S ~
~ 2 %
where
A = f, ~ y .
on X,
and clear
(Throughout
if ~
a quasicoherent
a closed
subspace
describable.
all a s s e r t i o n s
are local
for X affine.)
In particular, (i.e.,
here,
of X,
c
~X
subsheaf
is a sheaf
of ideals
of ~ X ) then Spec
and every c l o s e d
subspace
~X
on X is
of X is so
121
II. 5
Also, we have
for A = ~ x [ T 1 , . . . , T n ] ,
S~
A =/A x n ' "affine
Proposition
5.2:
Let
and A a q u a s i c o h e r e n t
and Y'
= Spec
and X' ~ X g i v e
Then
the n a t u r a l
Let
f:X ~ Y be
spaces
with Y
f,:QCS x + Q C S y
and
scheme
and
maps
(The S h e a f
= Spe~c A
Y' + X', Y' ~ Y
separated
scheme.
f*:QCSy
Criterion
Suppose
~ QCS X be
is n a t u r a l l y
map
for I s o m o r p h i s m ) of a l g e b r a i c
f is
faithfully
the u s u a l
equivalent
U.
L e t n : U ~ X be X is s e p a r a t e d
applying the
fact
successively
an e t a l e
flat.
Let
functors.
Suppose
identity
functor.
to the
covering
so ~ is an a f f i n e the
that Y is affine~
fact
that
we h a v e
a cartesian
diagram:
U
-->U
X
~)y f
of X by
map,
f*f,
we h a v e U = S ~ c
= X x ~Spec f,~. u~'" = X x Spec~ F ( Y , f . ~ . ~ U Y Y Thus
L e t X'
f is an i s o m o r p h i s m . Proof:
Then
of a l g e b r a i c
X'
a quasicompact
f * f , : Q C S x + QCS X Then
~
an a f f i n e
a map
diagram
Y'
5.3:
algebra,
o v e r X".
~x-algebra.
a cartesian
Proposition
n-space
f:Y + X be
spaces
f*A.
the p o l y n o m i a l
an a f f i n e
so U = ~
= i, P r o p o s i t i o n ~, U~
= Spec
~. ~
U"
5.2, f*f,~. ~
U
) = X × Spec~ F(U, uO-) = X × U. Y Y
II. 5
We
122
fill
out one m o r e
cartesian
square
~2 -->U
~x outer
with
base
Finally,
------>
give
theory
since
Proof. be
The
an a f f i n e
two m a p s
flat,
Let Then
axb
diagonal
f~ is e t a l e
an e t a l e
By
surjective.
surjective
map I
a faithfully
is l o c a l Z be
an a l g e b r a i c show
the p r o d u c t
Let W = Y
flat m a p
of
epimorphism.
on Y so w e
We must
x Z be
map.
surjective.
f is a c a t e g o r i c a l
Let
schemes
f.
af = bf.
~Z
f by
of a f f i n e
etale
schemes,
f:X ~ Y be
assertion
that
square
and ~i
of
so is
scheme.
such
Let Y v the u s u a l
5.4:
spaces.
~,
for a f f i n e
f~ is an i s o m o r p h i s m ,
algebraic
f
x
the p u l l b a c k
Corollary
f~
a cartesian
f~ f a i t h f u l l y
flat d e s c e n t
1
~
corners
the m a p s
>U
I
~i
u
label
1
U x U
The
and
can
space
take Y to and Y ~ - ~
Z
a = b.
map
"-~
and Z--~ Z X Z the Z:
(z x z)
W
~ Z
a X b Y-
Then
a = b if and o n l y
~_ff_i_;'~ ~ z x z
if 4'
--
is an i s o m o r p h i s m .
Note
in g e n e r a l
II. 5
123
A is q u a s i a f f i n e separated.
so A'
N o w put
in the map
w ....
f
is an i s o m o r p h i s m
flat and
so since
A' is faithfully phism,
a simple
is n a t u r a l l y
implies
= X × W Y
-~ Z x Z
Thus A"
flat,
Using
the
fact
diagram
chase
shows
A'
implies
that
that A" is an isomor-
that A ' *
to the i d e n t i t y
for isomorphism,
this
is faithfully
!
A ,:QCS W + QCS W
functor.
By the
is an isomorphism,
which
that a = b.
a Stein m a p ~X
af = bf.
f is f a i t h f u l l y
Construction
map
since
a xb
-
flat.
sheaf c r i t e r i o n
and
f z
?Y
equivalent
quasicompact
f:X ~ Y and let W'
-->w
X
A"
is in p a r t i c u l a r
~
Let spaces. X' = Spec
I
5.5:
A map
f:Y ~ X of a l g e b r a i c
if f is q u a s i c o m p a c t
f*~Y
and s e p a r a t e d
is
and
the natural
map
of a l g e b r a i c
is an isomorphism.
f:Y + X be any q u a s i c o m p a c t Then
spaces
f,~y
f, 0-y.
separated
is a q u a s i c o h e r e n t Then
f:Y ~ X
factors
affine m a p d:X' ~ X
Y ~
fl
X
--)X'
~x-algebra. through
Let
the n a t u r a l
II.5
124
f' is a S t e i n with qf'
h an
map.
a f f i n e map,
any
there
factorization
is a u n i q u e
of
map
f:Y ~ g ~
q:X'
Z
h-Tx,
~ Z with
= g and h q = d. This
factorization
a Stein map ization
followed
of f.
We
Proposition spaces.
Then
of s e p a r a t e d
that
of a q u a s i c o m p a c t
by
an a f f i n e
call X' 5.6:
show
is c a l l e d of
f into
the S t e i n
Factor-
f.
a Stein map
epimorphism
map
of a l g e b r a i c
in the c a t e g o r y
spaces. an e t a l e it w o u l d
m a p Y × X' + X'
that X is affine.
T a separated
separated
imaqe
f:Y ~ X b e
L e t X' ~ X be
induced
map
affine
f is a c a t e g o r i c a l
f is an e p i m o r p h i s m ,
assume
the
Let
algebraic
Proof.
the
Given
algebraic
covering be
of X.
sufficient
is an e p i m o r p h i s m .
Let
a and b b e
space,
and
show
to show Thus we
two m a p s
suppose
To
that can
X + T, w i t h
af = bf.
We must
a = b. Since
maps.
T is s e p a r a t e d
Hence
it is s u f f i c i e n t
a, ~X ~ b, ~X" Definition
5.7:
Let
spaces.
quasicoherent
sheaf
f,~y.
0 x ~ f, ~ y
exact
geometric
closure
is the
to find
an
a and b are
affine
isomorphism
~T-algebra
a
But a. 0 X = a , f , Oy ~ b , f , ~y = b, ~ .
of a l g e b r a i c
0 ~ I ~
and X is affine,
__qeometrically d e n s e
Then
f:Y ~ X be
a quasicompact
f is s e p a r a t e d Let
I be
the
so w e ideal
and
let X = S p e c
of
the s u b s p a c e
if Y = X
~X
can
immersion form
making
/I "
Y of X.
We
say
f is
the
II. 5
125
If f:Y ~ X is any i m m e r s i o n , dense
if for e v e r y
nonempty
open
w e say
f is t o p o l o g i c a l l y
s u b s p a c e U of X, U × Y X
is
nonempty. Proposition algebraic a)
5.8:
f:X ~ Y be an i m m e r s i o n
of s e p a r a t e d
spaces. If f:X ~ Y
epimorphism b)
Let
is g e o m e t r i c a l l y
in the c a t e g o r y
If f is
dense,
of s e p a r a t e d
g e o m e t r i c a l l y dense,
f is a c a t e g o r i c a l
algebraic
f must be
spaces.
topologically
dense. Proof. proof of closed
b)
5.2b,
is clear.
it is s u f f i c i e n t
note
that
to s h o w that
X
and W
This
assertion
are s c h e m e s
just
then W + Y
is local on Y and
and for schemes,
as in the
if W ~ Y is a
s u b s c h e m e w i t h W × X ~ X an i s o m o r p h i s m , Y
is an i s o m o r p h i s m . affine,
To see a),
for Y
this s t a t e m e n t
is E G A 1.9.5.6. Proposition
I 5.9:
algebraic
space.
Then
immersion
i:U + X such
L e t X be a q u a s i c o m p a c t there
is an a f f i n e
locally
separated
s c h e m e U and an open
that
a)
i is t o p o l o g i c a l l y
b)
the induced m a p U ~ U × X is a closed
c)
i is an a f f i n e m a p
dense immersion
II. 5
126
Proof. s c h e m e Y. open
Let n:Y ~ X be By 1.5.20
subspace U
finite
as w e l l
there
of X s u c h
an e t a l e
of X b y
is a t o p o l o g i c a l l y that
as etale.
cover
an a f f i n e
dense
affine
the m a p V = U × Y ~ U is X
T o see b),
consider
the c a r t e s i a n
diagrams
v
Since
X is l o c a l l y
immersions. sufficient But
To
is a c l o s e d finite, c) U ~U
U
>
UXX
X
-------------') X ×
X
U + U × X and V ~ U × Y
show U ~ U × X
is a c l o s e d
that V ~ U × Y
immersion,
so p r o p e r ,
is a p r o p e r
and
the
are
immersion,
factors V ~ V × Y + U x Y where
map the
it is
of s c h e m e s . first map
second map
is
so p r o p e r . is a s i m p l e
consequence
o f b)
since
i is
the c o m p o s i t e
×X~X.
I
Construction be
uxY
separated,
to s h o w
this map
~
the s h e a f
define
Xred,
5.10-
Let X be
of nilpotents the
associated
an a l g e b r a i c
of t h e s t r u c t u r e reduced
space
space.
sheaf
0 x.
Let Nx We
of X as Spp~ec
x/~ x The natural algebraic
m a p X r e d + X is u n i v e r s a l
spaces
to X,
for m a p s
so the t r a n s f o r m a t i o n
of r e d u c e d
X /x/~ Xred
is a
II. 5
127
left e x a c t
functor.
Proposition spaces.
Then
5.11:
Let
f:Y + X b e an i m m e r s i o n
f is t o p o l o g i c a l l y
dense
f r e d : Y r e d ~ X r e d is t o p o l o g i c a l l y Construction
5.12:
space
construction
of the r e d u c e d
construction. struction
closed
5.13:
finite c o l l e c t i o n X, w e can
The
of r e d u c e d
form the u n i o n
the c o l l e c t i o n ,
since
infinite
of open
unions
of r e d u c e d
closed
this
subspaces
subspaces
algebraic
space
5.14:
f:Y ~ X,
Y which
is the s u p p o r t
and ~(X)
is called
o f U is a local
subspace,
the c o n -
Given
of an a l g e b r a i c of the m e m b e r s
construction.
and i n f i n i t e
open
irreducible subspaces
Supports.
let ~(Y)
the
any
space of
Similarly,
intersections
exist.
and F a c o h e r e n t
ing m a p
Then
of subspaces.
is a local
of any two n o n e m p t y
L e t X be
of C is local.
or i n t e r s e c t i o n
subspaces
Construction
complement
lattice
H e n c e w e can d e f i n e X to be section
immersion.
if C ~ X is a closed
of the o p e n c o m p l e m e n t
Construction
I
of subspaces.
and U + X an o p e n
Also,
if
dense.
Complements
an a l g e b r a i c
if and o n l y
of a l g e b r a i c
of X is nonempty.
Let X be a noetherian
s h e a f on X.
be the r e d u c e d
of f*F.
if the i n t e r -
For
closed
subscheme
T h e n ~ is a local
the s u p p o r t of F, d e n o t e d
each c o v e r of
construction,
Supp(F).
II. 5
128
Proposition algebraic subspace Then
Supp(F)
there
is c o n t a i n e d
is an integer Let
this
In
5.17:
Proposition noetherian
the s i t u a t i o n
Hence Y
~
holds
5.18:
~-X/J"
of X, w i t h Y affine. = 0 for some
let C = Spj~ec ~ X / j m
T h e n F = i,i*F.
P of a l g e b r a i c space,
for every
and
closed
I
spaces
is
for any algesubspace
Y of X
holds.
(Noetherian
space
above,
Induction)
L e t X be
and P an i n d u c t i v e
a
property.
holds. L e t U + X be an etale
noetherian of closed
ascending
injection.
then P(X)
algebraic
Proof.
descending
~
I
A property
if P(Y)
w i t h Y r e d / Xred'
chain
subspace
(f*J)n(f*F)
if P is true of the empty
space X,
affine
the closed
that jnF = 0.
to show that
5.16:
Definition
T h e n P(X)
in the closed
n such
and i:C ~ X the c a n o n i c a l
braic
Suppose
is E G A 1.9.3.4.
C_orollary
inductive
sheaf on a n o e t h e r i a n
of ideals.
f:Y ~ X be a c o v e r i n g
it is s u f f i c i e n t
n and
L e t F be a c o h e r e n t
space X and J a sheaf
Proof. Then
5.15:
subspaces,
chain chain
if P(X) X where
scheme.
of closed of ideals
fails, P(Y)
covering
of X w i t h U an
T h e n X has no i n f i n i t e since
such w o u l d
subspaces
of U,
lift
to an i n f i n i t e
so an i n f i n i t e
in the n o e t h e r i a n
there m u s t be a m i n i m a l
fails.
descending
ring F ( U , ~ u ) . closed
We can assume Y = X.
Then
subspace for
I I. 6
129
every
closed
Hence
P (X) .
6.
Points
subspace
X'
of X, X ' r e d / Xred,
and the Z a r i s k i
6.1:
of a l g e b r a i c
spectrum
of a field k and
By abuse
of language,
k is the r e s i d u e k(p).
spaces,
we
field
say
A point
of X
i:p + X, w h e r e p is the
"p is in X"
monomorphism.
and w r i t e
of X a__~t p and w i l l
p 6 X.
sometimes
be
il:Pl ~ X and i2:P2 ~ X are e q u i v a -
is an i s o m o r p h i s m
by abuse
space.
i is a c a t e g o r i c a l
Two p o i n t s
len.___~tif there
Topology
Let X be an a l g e b r a i c
is a m a p
Again
holds. I
Definition
written
P(X')
of l a n g u a g e
e:Pl + P2 with
we c o n s i d e r
i2e = i I.
two e q u i v a l e n t
points
to be identical. A qeometric with
q the s p e c t r u m
a geometric
point
Proposition spaces w h e r e point
compact i.
open
j:q ~ X is any map of a l g e b r a i c
of a s e p a r a b l y
is u s u a l l y 6.2:
Let
q = Speck
p of X such Proof.
some
point
f factors X has
subspaces,
field.
(Note
that
a point.)
f:q + X be a map of a l g e b r a i c
for some
that
By 3.13,
not
closed
spaces
field k.
there
is a
q + p ~ X.
a covering
f:q + X m u s t
H e n c e we can assume
Then
{X. _____~x] 1 factor
by q u a s i -
q ~ X. + X for 1
that X is q u a s i c o m p a c t .
ii. 6
130
For
any covering
let ~(Y) This
be
the
is a l o c a l
quotient
m a p Y ~ X, w i t h Y a q u a s i c o m p a c t
(finite:)
underlying
construction.
o f an e t a l e
set of p o i n t s
of q X Y ~ Y. X S i n c e p is the
Let p = ~(X).
equivalence
scheme,
relation
~ ( Y × Y) ~ ~ ( Y ) , X
where
each
affine of
of ~ ( Y
× Y) and ~(Y) a r e f i n i t e d i s j o i n t u n i o n s of X of fields, p is a d i s j o i n t u n i o n of a f f i n e s p e c t r a
spectra
fields.
('See 1.5.15) .
is s u r j e c t i v e . Finally,
q × Y ~ ~(Y) is s u r j e c t i v e so q ~ p X p is the a f f i n e s p e c t r u m of a field.
Hence
~(Y)
= p × Y ~ Y is a m o n o m o r p h i s m X
so p ~ X is a
m o n o m o rph i sm •
M
Corollary
6.3:
Every
algebraic
s p a c e X / ~ has
at l e a s t
one point. Proof.
Let Y ~ X be a covering
and q + Y a p o i n t map
q ~ p such Theorem
a point
6.2,
Let X b e
Then
there
is an a f f i n e
that
x
(The p r o o f
uses
the notion
until
~X
IV.2.6,
it is s o m e t i m e s
of q u o t i e n t s
there
is a p o i n t
an a l g e b r a i c
such
be deferred but
By
of X by a s c h e m e Y p ~ X and
a
t h a t q ~ p ~ X = q ~ Y ~ X.
6.4:
of X.
map U-~)X
of Y.
map
useful
of a l g e b r a i c
factors
p.190.
M
space
scheme
U and
an e t a l e
x---~ U ~ ) X .
of s y m m e t r i c We have
- for e x a m p l e spaces by
and x - - ~ > X
product
no n e e d
and will
for t h i s
fact
in the c o n s t r u c t i o n
finite
group
actions
(IV.I.8).) W
11.6
131
Definition an a f f i n e
6.5:
A point
s c h e m e U and
p ~ X is s c h e m e - l i k e
an o p e n
immersion
if t h e r e
U ~ X such
is
that p ~ X
f a c t o r s p ~ U ~ X. Proposition is an o p e n
subspace
p ~ X is in U if and o n l y We
6.6: U
if and
if all
sometimes
L e t X be an a l g e b r a i c of X s u c h only
t h a t U is a s c h e m e ,
if p is s c h e m e - l i k e .
its p o i n t s say
space.
that,
are
Then and
there
a point
X is a s c h e m e
scheme-like.
"U is the o p e n
I
subspace
where
X is
a scheme. " Proposition a scheme
almost
like points
an o p e n
iff they
L e t X b e an
everywhere,
i.e.,
is t o p o l o g i c a l l y
Proof. ei
6.7:
By
3.13,
X has
immersion.
are d e n s e
The
algebraic the open
space.
T h e n X is
subspace
of s c h e m e -
dense. a covering
scheme-like
in e a c h X
•
Hence
{X
l
--~ ~ X }
points we
can
with
in X
each
are d e n s e
a s s u m e X is
1
quasicompact. quasicompact
L e t R ~ U be schemes with
dense
open
subspace
lence
relation
quotient
V o f U,
S = R
an e t a l e
~
such
equivalence
relation
of
X.
By 1 . 5 . 2 0 ,
there
is a
that
the i n d u c e d
etale
equiva-
(V × V)
V is e f f e c t i v e
in the
(u × u) category dense
of schemes.
open
is a s c h e m e
s u b ~ contains
L e t Y be t h e q u o t i e n t . of X. Y.
The
open
T h e n Y ~ X is a
s u b s p a c e U of X w h e r e
X I
II. 6
132
Corollary there
6.8:
L e t X be a n o e t h e r i a n
is a t o p o l o g i c a l l y
dense
open
algebraic
immersion
space.
Then
U + X w i t h U an affine
scheme. Proof: subscheme and then open
Using Proposition
of s c h e m e - l i k e such
this p r o c e d u r e
must
underlying of points
language
of X
the
IX[,
space
by taking
topology
on
6.10:
and a one-one
]X],
of
Ixl.
hence
affine,
Q c
and by
The ]x]
the zariski
between
space
associated
as the c o l l e c t i o n set
IxI
By abuse topoloqy and there
of X and open
reduced
is given
to be closed
Y of X.
subspaces
Also x~[x
The
is defined
subspace
open
correspondence
subsets
U = U 1 U...U U n
space.
is a topological
between
and affine.
I
a subset
]X] is called Ixl
open
X is n o e t h e r i a n , s o
n steps.
of points).
for some closed
correspondence
x and closed
of X,
structure ]Y]
pick U 1 C X
dense.)
equivale~ce
form
etc.
schemes,
open
assume X is a scheme,
p i c k U 2 c X-Ul,
say,
of affine
(modulo
Proposition one-one
after,
X by its d e n s e
(Specifically,
Let X be an algebraic
topological
a topological is of the
union
6.9:
we can
pick U 3 c X-UI-U2,
U is t o p o l o g i c a l l y
Definition
replace
constructed.
terminate
disjoint
construction
I.e.,
If U 1 is not dense,
If U 1 U U 2 is not defuse,
is a finite
points.
a U is easily
and affine.
6.7 we can
closed
I is a functor.
if Q
of
on x. is a subsets
subspaces
of of
II. 6
133
Proof: different
By
5.13iX
subspaces
nonempty.
is a t o p o l o g i c a l
of X:
say the c o m p l e m e n t
C is a n o n e m p t y
a 6 U 2 - U 1 so
IU21 ~
subspace
IUII.
so has
Similarly
If f:X ~ Y is a map of a l g e b r a i c b y 6.2 there
space.
is a p o i n t p + Y
L e t UI, U 2 be two
C of U 1 in U 2 is
a p o i n t q by 6.3 and
for r e d u c e d spaces
closed
subspaces.
and ~ ~ X a point,
and a map o ~ p such
that q + X ~ Y
=q~p~y.
I
6.11:
Note
definitions
that X is i r r e d u c i b l e
of t o p o l o g i c a l l y
o p e n map,
oper~ map
via
We
IXl.
associated every
then
and i m m e r s i o n
say that
map
IYI ~
algebraic
dense
iff
subspace,
surjective
are e a u i v a l e n t
a m a p of a l g e b r a i c IXI
Ixl is and our p r e v i o u s
is closed,
map,
universally
to the usual
definitions
spaces Y ~ X
and u n i v e r s a l l y
space X' ~ X m a p p i n g
to X,
is c l o s e d
closed
if the
if for
the induced map Iy × X' I + X
is closed. 6.12:
IXl can in fact be c o n s i d e r e d
taking
for an open s u b s e t
reader
should
separated Example
IU
note c a r e f u l l y
algebraic
spaces,
6.13:
~ixl)
space,
= F(U, ~ X ) •
The
this
functor,
restricted
to l o c a l l y
is n e i t h e r
faithful
nor
(See
Atomizations.
S p e c R w h e r e R is a c o m p l e t e
full.
local
An atom
ring w i t h
is an affine
Separably
scheme
closed
field.
L e t X be an a l g e b r a i c residue
that
IXl, F(IuI,
ringed
2 in the I n t r o d u c t i o n . )
Construction
residue
of
as a local
field k(p),
space.
For each p o i n t p of X, w i t h
let ~ be the affine
spectrum
of the s e p a r a b l e
IX.
II.6
134
closure
of k(p)
and
i:q + X the n a t u r a l
map.
Let ~
= Spec
~ ( a , i * ~X) .
P Then
~
is an atom,
the
atom
of X at ~.
L e t ~ be
the d i s j o i n t
union,
P = ~p,
over
all p o i n t s
~ X is c a l l e d
the
Proposition
a)
p of X.
atomization
6.14:
For
~,
of X.
L e t X be
any p o i n t
or m o r e
specifically,
X is a t o m i c
an a l g e b r a i c
p in X,
the m a p ~
the m a p
if X = ~.
space. ~ X is
flat.
P
b)
T h e m a p ~ ~ X is
c)
The m a p ~ is a c a t e g o r i c a l of a l g e b r a i c
faithfully
flat. epimorphism
in the c a t e g o r y
spaces.
Proof: a)
L e t ~ : Y + X be of a f f i n e
an e t a l e
schemes.
The map ~
+ X
covering
Let
factors
~ be ~
P Hence well b)
theorem.
assume
one
reduces
is w e l l
~ Y identifying
~
~(q)
with ~ P
X is an a f f i n e
scheme
= p. . q
where
this
is
apply
the
easily
of b)
techniaue
is a w e a k
to
the c a s e
of a f f i n e
X where
known.
is a c o r o l l a r y
This
of Y w i t h
union
known.
this
We now
a point
Y a disjoint
P
we c a n
Again,
c)
with
version
Deligne
Theorem
for s c h e m e s ,
is not
a trivial
conseauence
and
I
of a t o m i z a t i o n
to p r o v e
for a l g e b r a i c
mentioned of
5.4.
spaces
previously.
its v a l i d i t y
the
of
Note
for s c h e m e s .
following
the s t r o n g e r the r e s u l t
II.6
135
Theorem
6.15:
is q u a s i f i n i t e ,
Let
f:X ~ Y be a map of a l g e b r a i c
of finite p r e s e n t a t i o n
spaces
and separated.
Then
which f is
auasiaffine. Proof:
The
Y = Spec R. subrings
assertion
is local
R can be c o n s i d e r e d
RI and b e t a k i n g
on Y so we can assume Y is affine:
as a d i r e c t
a sufficiently
s c h e m e Y o = Spec R.l can be found w i t h separated
map
X = X~oY'-x
locally
Once we
of its n o e t h e r i a n
large Ri,
a noetherian
a m a p Y ~ Yo'
of finite p r e s e n t a t i o n
show X o > Y o
limit
X
o
~ Y
is ~ u a s i a f f i n e ,
and a a u a s i f i n i t e o
so that
it follows
that
X ~ Y is q u a s i a f f i n e . Also Stein
I.i.21,
and separated,
ft
X
X 1 ~ Y where
fl is ~ u a s i f i n i t e ,
are thus r e d u c e d Lemma
of
f is ~ u a s i c o m p a c t
factorization
Applying We
since
6.15a:
to showing Let
fl* 0 X
take its
= ~Xl and d is affine.
of finite p r e s e n t a t i o n a n d
the f o l l o w i n g
Y a noetherian
separated.
lemma:
f:X ~ Y be a a u a s i f i n i t e
finite p r e s e n t a t i o n w i t h
w e can
separated
affine
scheme.
Stein map Then
f is
an open immersion. Proof:
W e use
the t e c h n i q u e
flat and a ~ o n o m o r p h i s m .
F i r s t we deal w i t h
an atom and the image of the m a p L e t X' ~ X be a (quasifinite) s c h e m e X'. III.4.4.3)
By Z a r i s k i ' s X'
of a t o m i z a t i o n to show
f contains
is i s o m o r p h i c
to an open
f is
the c a s e w h e r e Y is the c l o s e d p o i n t
etale c o v e r i n g
Main Theorem
that
(for affine subspace
of Y.
of X b y an affine schemes--see
of an affine
EGA
scheme
II. 6
Y'
136
finite
over Y.
of atoms Since
Since Y is an atom,
(Hensel's
so X'
at least one p o i n t
of the c o m p o n e n t s Y'.
Lemma)
Thus X 1 m a p s
onto
a component
and finite
H e n c e X + Y is a Stein map
is nonempty.
Hence
f~
+ Y,
Consider
one
of
X 1 + X is so its
image
Since X + Y is Stein, m u s t be all of X.
and X 1 is affine, schemes,
of Y,
to a c o m p o n e n t
of X.
sum
sum of subspaces.
the closed p o i n t
this c o m p o n e n t
of affine
In the case of general
disjoint
But X 1 + X' ~ X is etale
(being an atom)
S i n c e X 1 ~ X is etale
f-l(p)
onto
so X 1 ~ X is finite.
is closed.
and Y is c o n n e c t e d
that
is a finite d i s j o i n t
of X' maps
is finite
so its image
is open.
is a finite
X 1 of X' m u s t be isomorphic
The m a p X 1 ~ Y
proper
Y'
so is X
(by 1 . 5 . ~ ) .
hence an isomorphism.
let p be a p o i n t
the c a r t e s i a n
in Y such
diagram:
xx z YjP
JP f
X
Since ~
~ Y is flat
.~Y
(6.14b),
the top m a p
is a stein map.
(we are
P here
also
separated,
applying
4.18).
X × ~ ~ ~ y P P
is also q u a s i f i n i t e
so the map m u s t be an i s o m o r p h i s m
in the special
case.
Now
by
let Y 1 be the d i s j o i n t
the above union
and
argument
of all P
for all p o i n t s cartesian
p in Y
for w h i c h
f-i (p) is nonempty.
We have
diagram f'
'~Y1 with
X
">
f' an i s o m o r p h i s m
a
I1.6
By
137
6.14,
~ is
faithfully
so f a i t h f u l l y
maps
flat.
flat,
Hence
and ~ is
f:X ~ Y is
Also
f is a m o n o m o r p h i s m .
with
fa = fb.
To
flat.
flat. a let Z ~ X b a diagram
see this,
L e t Z 1 = Z X YI" Y
f' is an i s o m o r p h i s m
We have
be
two
a !
a Z
%
X
where
~a'
monic. a~' by
= a~',
Hence
= b~'. 5.4,
9'
To
By
flat.
= b'f'
6.14,
our
Let
The~ Since
Hence
Since
Since implies
f' is an i s o m o r p h i s m , a' = b'
faithfully Hence
can
flat
of
implies so ~'
~a'
is
it is
= ~b'
faithfully
now
show
the
following
a flat m o n o m o r p h i s m
of
and of
finite
f is an o p e n
assume
presentation, subspace
and Y is the q u o t i e n t X × X = X, Y
the
f is s u r j e c t i v e ,
the e t a l e
of the e t a l e induced
map
lemma:
finite
equivalence equivalence
X ~ Y
(which
it is open.
of Y so w i t h o u t hence
X × X = X so f is n o n r a m i f i e d . Y monomorphism. of
flat,so
immersion.
f is m o n i c ,
surjective
implies
a = b.
we only need
flat
the i m a g e we
Y
f:X ~ Y be
f is
X is the a u o t i e n t
Since
->
f is an o p e n
of g e n e r a l i t y ,
an e t a l e
~ is
theorem
6.15b:
Proof: (By 3.5) .
= b~'
is an e p i m o r p h i s m .
presentation.
loss
a'f'
finish
Lemma
~b'
f
_
b
relation relation.
is our
faithfully Thus
f is
1 X ~ X
X × X----~ X. Y o r i g i n a l f) m u s t
I1.6
138
be the i d e n t i t y Corollary is locally Suppose
map. 6.16:
l Let
quasifinite,
locally
Y is a scheme.
Proof: quasicompact and such
By 3.13, algebraic
there
Corollary algebraic
6.17:
spaces.
is a c o v e r i n g
spaces X., 1
that X is a scheme
f is quasiaffine,
of finite p r e s e n t a t i o n
spaces w h i c h and separated.
Then X is a scheme.
assume X is q u a s i c o m p a c t 6.15,
f:X + Y be a m a p of a l g e b r a i c
with
{X. ~ l X) l
of X by
each ~i an open
iff each X. is a scheme. l
so f is q u a s i c o m p a c t .
immersion Hence
Applying
we can
the theorem
so by 3.8, X is a scheme, Let
f:X ~ Y be an etale
If Y is a scheme,
l
separated
then so also
is X.
map
of I
II. 7
7.
139
Proper
and P r o j e c t i v e
Definition
7.1:
if f is separated,
A map
morphisms
in the etale
The
first
of p r o p e r topology.
is proper
closed,
f is
~y-module. morphisms
and
A closed
finite
immersion
f:X + Y is proper.
two assertions
on Y so we can
spaces
and u n i v e r s a l l y
is a c o h e r e n t
The classes
A finite m o r p h i s m
Proof: is local
type,
and f* O X
7.2:
are stable
is finite.
f:X ~ Y of a l g e b r a i c
of finite
.finite if f is affi,e Proposition
Morphisms
assume
are s t r a i g h t f o r w a r d .
that Y,
and h e n c e X,
The
is affine.
last This
is then EGA II.6.1.10. 7.3: book--is
A
fundamental
the F i n i t e n e s s
noetherizn
algebraic
all the h i g h e r The m a j o r separably X is,
I
direct
Theorem:
spaces,
reduced
of X by taking
associated
with
coherent
sheaves
In one case
Rqf,F
are c o h e r e n t
H e r e Y = Spec k,
sheaves
morphism
of
then
~y-modules. over
f:X + Y is proper, the usual
of the v e c t o r
Rqf,F
of this
sheaf on X,
T h e n one d e f i n e s
the d i m e n s i o n s
the c o h e r e n t
theorem
is w h e n X is a "variety
and irreducible.
invariants
the m a i n
If f:X ~ Y is a p r o p e r
theorem
field k".
fact
and F is a c o h e r e n t
images
use of this
closed
say,
theorem--in
for v a r i o u s
a and numerical
spaces canonical
F on X. the
finiteness
morphism,
f* ~ X
is a c o h e r e n t
sheaves.
Since
f is affine,
theorem ~y-module
all
is obvious.
If f is a finite
so f, p r e s e r v e s
the R q f , F
coherent
= 0 for q > 0, so these
II. 7
140
are certainly coherent.
The general case of the theorem involves
considerably much more work and we will only be able to prove it after setting up two chapters of machinery. Proposition 7.4:
Let f:X ~ Y be a finite etale morphism with
X and Y noetherian. Then f, ~X is a locally free Y is irreducible.
Then the rank of f* ~ X
~y-module.
is constant
Suppose
(and is called
the degree of f). Proof:
For the local freeness,it is sufficient to assume Y
is the affine spectrum of a local ring. f* ~ X
is a finite and flat
~y-module,
Since f is finite and etale, hence free.
For the second assertion, note the Nakayama lemma implies that f, %
has constant rank in an open neighborhood of any point.
7.5:
I
We now define projective and quasi-projective morphisms.
There are several ways these can be defined and the detailed theory involves such notions as the Pro~ construction and ample sheaves. (See EGA If)
Our definition however, will entail that,
is a projective or quasi-projective morphism, then X is a scheme.
if f:X + Y
and Y is a scheme,
Thus any projective construction
in algebraic
spaces over a base scheme is the same as the scheme-theory case. We will give an indication of stability of projective morphisms in the etale topology but otherwise leave to the reader the task of transcribing the detailed theory of projective morphisms.
II.7
141
Definition
7.6:
Let ~n be p r o j e c t i v e Spec Z
affine s p e c t r u m of the ring of integers Z. is familiar with this object.) define
n ~ X as
map
n ZX +
invertible
Let X be any algebraic
sheaf
~(i)
on
n ~ S p e c Z to induce a c a n o n i c a l
w h i c h w e also d e n o t e A map
f is p r o j e c t i v e Note
X is a scheme,
n - s p a c e o v e r X.
There
~n and we use the Spec Z n i n v e r t i b l e sheaf on ~X'
if there
i:Y ~ ~Xn such that f is the c o m p o s i t e
if for some such i, i is a closed
(applying 3.8)
7.7:
m o r p h i s m of algebraic
if f:Y ~ X is q u a s i p r o j e c t i v e
The classes spaces
of p r o j e c t i v e
and
are closed
and q u a s i p r o j e c t i v e
subcategories.
(Note they are not stable in the etale topology. class of p r o j e c t i v e maps of schemes in the Z a r i s k i
Proposition finite type.
We
then so also is Y.
Proposition
of schemes
space.
spaces is q u a s i p r o j e c t i v e
is an integer n and an immersion
immersion.
the reader
0~(i).
f:Y + X of algebraic
Y ~ ~xn ~ X.
(We assume
n × Z) X, p r o j e c t i v e ~ S p e c Z (Spec
is a c a n o n i c a l
n - s p a c e over the
7.8:
N Indeed
the
is not stable in the c a t e g o r y
topology.)
A quasiprojective
A quasiprojective
map is separated
map is p r o j e c t i v e
and o f
if and only if it
is proper. Proof:
The only h a r d p a r t of this is p r o v i n g
n + X is p r o p e r w h i c h comes down to showing ~X is proper.
This is p r o v e d
in EGA II.5.5.3.
that the map
that the map ~n ~ Spec Z Spec Z n
II.7
142
Definition i
7.9:
be an i n v e r t i b l e
projective,
and
Let
sheaf on Y.
and
~wk
is isomorphic
We say ~
of algebraic is f-ample
spaces.
for some integer
k,
the k - f o l d
to i * ~ ( 1 )
where
Let
if f is q u a s i -
factorization, of f, y ~ ~Xn + X, with
for some
immersion, of ~
f:Y ~ X be a m a p
i an
tensor product,
(~(i)
is the c a n o n i c a l
sheaf
n on ~X" As in scheme and
theory,
if f:Y ~ X is a map
an i n v e r t i b l e
Oy-module
~
an a p p r o p r i a t e
imbedding
Y ~ ~Xn so
, the c o n s t r u c t i o n
is f-ample,
is local
~=
of algebraic
spaces,
one can r e c o n s t r u c t i*~(1).
on X so we can
(Since,
from given
take X, hence Y to be
a scheme. ) Proposition
7.10:
Let h
u__
-]
Y
If V be a c a r t e s i a n Suppose
diagram
•-) X
of a l g e b r a i c
g is q u a s i - p r o j e c t i v e
invertible
sheaf ~ on Y such
spaces
and suppose that h * ~
with n etale further
and surjective.
that there
is g-ample.
is an
Then ~ i s
f-ample,
so f is q u a s i - p r o j e c t i v e . Proof:
See EGA IV.2.7.2
Theorem
7.11:
(The Serre
f:X ~ Y be a p r o j e c t i v e F be a c o h e r e n t
| Finiteness
morphism
sheaf on X.
Theorem).
w i t h ~ an ample
We w r i t e
F(n)
~AC
she~
for F @ ~
en
XXIX~ on X.
Let Let
143
I I. 7
Then (i)
Rqf,(F)
(2)
There
is c o h e r e n t , is an i n t e g e r
R q f , (F(n)) (3)
There map
Proof: Let
E ~ Y be
N,
such
that
for all n > N and q > 0
N,
such
that
for
is an i n t e g e r ~ F(n)
assertions
an a f f i n e
the n a t u r a l
is s u r j e c t i v e .
are
etale
all n ~ N,
all
cover
local of Y.
on Y in the Then
there
following
sense.
are c a r t e s i a n
diagrams ~2
E ×x.
n
E ~(F(m))
~
~Y
.
= "2*(F) (m)
and,
of f:X ~ Y,
R ~ f ' (~*(F(m))) L
are r e d u c e d
to the c a s e
7.6,
X is a s c h e m e
and
>x
1
I~E~
with
,
= 0.
f*f,F(n) The
for all q > 0.
f' :E × X + E be the p u l l b a c k Y = ~*(Rqf(F(n))) (see II.4.17) . T h u s we
where the
letting
y
is affine.
assertion
Then
is p r e c i s e l y
by
the r e m a r k
EGA
in
III,2.2.1.I
I I. 8
8.
144
Integral
Algebraic
Definition algebraic
8.1:
space.
Spaces Let X be a ~uasicompact
%~e say X is i n t e g r a l
locally
separated
if X is also i r r e d u c i b l e
and
reduced. 8.2: an open U
L e t X b e an i n t e g r a l
immersion
is affine,
o t h e r hand,
with U quasicompact,
then U is the s p e c t r u m
x
o o
Then
is X. is X.)
point
there
(I.e. x
o
of X.
Clearly
the only r e d u c e d
is c a l l e d
By
8.3:
5.9,
spectrum
corresponding
o
On the
not be integral.
e X such that subspace
p o i n t of X. x
lie in U.
of an i n t e g r a l
(Y
algebraic
the c l o s u r e
of
of X c o n t a i n i n g o
is a s c h e m e - l i k e
affine
open
As m e n t i o n e d
domain.
Then
s u b s p a c e U of X. above,
U is
the p o i n t
of U
ideal of F(U, ~ X ) is the d e s i r e d
5.9 U can be c h o s e n
x
~ U ~ X is an affine map. 8.4:
o
closed
there is a d e n s e
to the zero
Definition
If
+ X is affine.
By o
domain.
L e t X be an i n t e g r a l
the g e n e r i c
any such p o i n t m u s t
the a f f i n e
then U is also integral.
of an i n t e g r a l
is a u n i q u e p o i n t x
The m a p x
Proof:
If i:U + X is
of course.)
Proposition-Definition
x
space.
if Y ~ X is an etaie map, Y need
m u s t be r e d u c e d
space.
algebraic
so that
A map
the m a p U ~ X is affine.
x o.
Then 1
f:Y + X of a l g e b r a i c
spaces
is b i r a t i o n a l
I1.8
145
if there is a ~ o p o l o g i c a l l y )
dense
open
subspace
U of X such
of f to Y × U + U is an isomorphism. X C o n s t r u c t i o n 8.5: ( D e c o m p o s i t i o n of a n o e t h e r i a n
that
the r e s t r i c t i o n
separated
algebraic
noetherian
locally
space
into
separated
topological
space
I.e.,
are no infinite
there
its components.)
algebraic
IXI of X i s t h e n
L e t X be a
space.
The
a noetherian
descending
chains
locally
associated
topological of closed
space.
subsets
of Ixi We claim irreducible point
of
closure
that
IXI
closed
is the union
subsets.
IXl is c o n t a i n e d of the point)
and every
irreducible
this).
Ixl is the union
cible be
closed
subsets
the r e d u c e d
integral
its generic
point
if the set
subset
x.. l
point
each
of X with
subset
(the
closed
subset
is c o n t a i n e d
the Zorn
IVil
the m a x i m a l i t y
[Ci]ie I is i n f i n i t e
of
IXl - C l Z
closure).
irredu-
Each V.~ is
Each
(Since
we can p i c k
closed
: IX[ a
for
let V. ~ X l
= C i.
and d i s t i n c t n e s s
an infinite
IXl
lemma
maximal
such C., l
A l s o x. 6 C. iff i = j. l 3
~ I and o b t a i n
(where--denotes
closed
x. + V. • C. is the c l o s u r e 1 l l
I' = [1,2,3,...] subspaces
any
(apply
For
of m a x i m a l
that
of the set of d i s t i n c t
subspace
a generic
C. c C contradicting i -- j Hence
closed
number
first n o t e
irreducible
{Ci)ie I of X.
algebraic
so admits
T o see this,
in an i r r e d u c i b l e
in a m a x i m a l Thus
of a finite
IXl
otherwise of the C.'s). l
a countable
descending -(C 1 C 2) a
inclusion
here
of
subset
chain o f ''"
is p r o p e r
since
I I. 8
146
X i e IXl
-C 1 ... Ci_l)
thenoetherian
but not in
hypothesis.
Ixl -(c I ... ci).
Hence
the index
This c o n t r a d i c t s
set I is finite.
Say
I = {l,2,...,n~. L e t U.x = IxI - (Cl u...u subspace ~i:Xi
of
IXl since
~ X be an open
and s e p a r a t e d
c i _ 1 u c i + 1 ...u Cn) •
the C. are closed. l subspace
so wc can
L e t W be the d i s j o i n t
with
Ixil
form W o = the l
union
Let now = ui
We
for each
i,
" ~i is q u a s i c o m p a c t
geometric
of the W.. l
u.x is an open
closure
then h a v e
of X. in X. l
the
following
proposition: Proposition algebraic finite
space.
disjoint
natural
Then union
L e t X be there
of i r r e d u c i b l e
W is a d i s j o i n t
Proof:
The only
the open
satisfies
a noetherian
is an algebraic
m a p W ~ X is birational,
reduced,
But
8,5:
union
thing
subset X which
closed
projective of integral
left
to check
is the union
locally
separated
space W w h i c h subspaces
is a
of X and the
and surjective. algebraic
If X is
spaces.
is the b i r a t i o n a l i t y . of X.l -
(Xl U ' ' X i - ~
X i + ~ Xn)
the requirement.
8.6:
W e now d e f i n e
the notion
AS x w i l l
denote
the c a t e g o r y
space X,
and QCS x the c a t e g o r y
is a m a p of algebraic associated
inverse
spaces
of stalks
of a b e l i a n
of sheaves.
sheaves
of q u a s i c o h e r e n t
and Q 6 ASy,
image of Q in AS x.
on an a l g e b r a i c sheaves.
we w r i t e
Also
As usual,
fabQ
if Q e QCSy,
If f:X + Y for the w e use
II .8
147
fabQ to d e n o t e convention 8.12.
will hold
(See 4.4 m
and
the a b e l i a n
f
sheaf
between
This
through paragraph
the two functors
fab:ASy ~ AS x
:QCSy "~ QCSx. )
on X at a p o i n t
definition
i:x ~ X is F
F(x,-) :AS schemes
+ Ab.
x
this m a k e s
to c h e c k
at x)
, taking
at all p o i n t s
F(x,-)
an exact
x
of an a b e l i a n
The
functor
functors
spaces,
functor.
at
AS x and
or the c a s e of
Since One
i
ab
. is always
then shows
s h e a f F to the d i r e c t
of X is not o n l y exact but o f sheaves
sheaf F
(Stalk
iabAsx
is exact.
any a b e l i a n
that a s e q u e n c e
is exact,
that sum
faithful.
it is s u f f i c i e n t
at the stalks".
In the e t a l e F(x,-) :AS
of the two
topology,
(Stalk
the functor AS x + A b
"to c h e c k
= F(x,iabF).
In the case of t o p o l o g i c a l
in the Z a r i s k i
of its stalks
o f the stalk F
x
x) :AS x + A b is the c o m p o s i t e
Thus
image of Q in AS x.
throughout this discussion,
for the d i s t i n c t i o n
The classical
exact,
inverse
topology
of s c h e m e s
+ A b is no longer
x
separably
closed
different
kinds
Definition
field.
This
or a l g e b r a i c
exact unless leads
spaces,
x is the s p e c t r u m
to the d e f i n i t i o n s
the functor of a
of two
of stalks. 8.7:
L e t X be an a l g e b r a i c
space
and F an a b e l i a n
s h e a f on X. (i)
Let
i:x ~ X b e
a geometric
point
of X.
Then
the g e o m e t r i c
I1.8
148
stalk (2)
of F at x is the a b e l i a n
L e t i:x + X be a p o i n t is the a b e l i a n
(3) either
o f X.
group Then
is its s t a l k
quasicoherent
Proposition (Geometric
as an a b e l i a n
s h e a f F at a
is the k (x) - m o d u l e
(geometric
F(x,iabF
s h e a f on X,
sheaf.
The
or ordinary)
its stalk
point
i:x + X
~
Stalk) :AS x ~ A b w h i c h
takes
at all the g e o m e t r i c
each
abelian
points
Then sheaf
the functor into
of X is exact
I
Definition generic point b y K = ~ k(Xo) . 8.10:
8.9:
i:x
o
The
With
L e t X be an i n t e g r a l
~ X.
The
function
function field
A l s o K is a flat QCS x ~ QCS
map
x
~X-mOdule
where
since
is not a c o n s t a n t
and Y has i r r e d u c i b l e
components
6 U,
and thus c h e c k
of i
ab
Indeed,
and separated.
image
:AS x + AS
definition
sheaf.
o
it is clear.
the inverse
the c o r r e s p o n d i n g
i,
x o .ab K = F(Xo,1 G X ) --a fact w h i c h
i is q u a s i c o m p a c t
is just the r e s t r i c t i o n
o that unlike ~
since
space w i t h
field K of X is d e f i n e d
s u b s p a c e U of X w i t h x
in the c a s e w h e r e X is affine, K is q u a s i c o h e r e n t
algebraic
s h e a f ~ on X is the s h e a f
the above notation,
one can c h e c k on any open
schemes,
the
and
faithful.
Note
in
fiber of a
~ x ) = F(x,imF) . X L e t X be an a l g e b r a i c space.
8.8:
sum of its stalks
the stalk o f F at x
s h e a f iabF on x.
If F is in fact a q u a s i c o h e r e n t
sense
F(x,iabF).
in t h e
functor x
which
is exact.
o theory
of
if Y + X is an etale
YI,...,Yn
with
generic points
II .8
149
yl,...,yn, of F (X,~)
then F(Y,~)
note that since
generic point,
~
8.11:
~*X
divisors
0 ~
sheaves.
~ ~*
~ ~
on X.
~X'
and
~ X is the
o
~ X + ~" integral
algebraic
an injection
be the q u o t i e n t
0 is exact. ~
A Cartier divisor
space
fields.
Let
the s h e a f of units of ~.
~ X + ~ induces
Let~ +
i:x
sheaf and K its sheaf of function
be the sheaf of units of
Then the injection
0 ~
where
Let X be a separated
its structure
of abelian
= i,iab~x,
there is a natural inclusion
Definition ~X
field extension
= K.
Finally,
and
= ~ k ( y i) - a finite separable
0 ~ ~
(abelian)
is called
~ ~*
sheaf so
the sheaf of C a r t i e r
on x is an element of F ( X , ~ ).
The class of p r i n c i p a l C a r t i e r d i v i s o r s
is the image of F ( X , K * ) in
r (x,,5). Theorem
8.12:
H 1 (X,~*)
= 0.
Thus there is an exact sequence
of a b e l i a n groups
o -~ r ( x , O x ) -~ r ( x , ~ * ) Proof: easy--~*
= i,~*
is a c o n s t a n t sheaf.
x
o .
-~
0
t o p o l o g y this is
But we h a v e s o m e t h i n g
to prove.)
~ X b e the inclusion of the generic p o i n t in X, so T h e n there is a spectral
sequence
o ~*
As always,
-~ H i ( x , O x )
(In the case of schemes in the Z a r i s k i
Let i:x ~*
-~ r ( x , / ~ )
this gives
m o r p h i s m H 1 (X,~*)
Xo
an injection
H 1 (Xo, ~ : ) . o
o i0 ~ H 1 so there is a m o n o 0 ~ E2 (Another w a y to see this is to note
II.8
that
150
the C e c h
complexes
defining
H e n c e w e only need
H'(X,~
) and H ' ( X o , ~
x ") are identical.) o )= 0. But this fact is
show that H l ( x o , ~ x o
the H i l b e r t
Theorem
T h u s we h a v e classifies principal
both
8.13: sheaves fixed
locally
Group
the function
a free
8.13:
integral
We w r i t e
point
field
fact that H I ( X , ~ x
divisors
)
modulo
we w r i t e H I ( X , ~ x *)
algebraic
we d i s c u s s
= P i c X,
coherent
space X which w i l l
b x for its s t r u c t u r e
and K for the
sheaf,
function
sheaf,
field.
remain
i:x ° . x
Recall
that ~,
is q u a s i c o h e r e n t .
L e t G be a c o h e r e n t
subspace
V of X such
that
sheaf on X.
Then
the r e s t r i c t i o n
there
is a
of G to V is
~V-mOdule.
Proof:
By
can c l e a r l y simple
As usual,
Cartier
In the final p a r t of this section,
for the g e n e r i c
open
spaces X the usual
of X.
throughout.
dense
|~,3.3).
free s h e a v e s , a n d
divisors.
on a n o e t h e r i a n
Lemma
(See S G A A
for a l g e b r a i c
Cartier
the P i c a r d
90:
6.8, X has
assume X = U,
corollary
m a p F ~ F ® ~. X
8.14:
affine
open
subspace
that X is affine.
free m o d u l e
be c a l l e d
Definition
i.e.,
of the N a k a y a m a
The rank of this of U and will
a dense
This
U and we is then a
lemma.
1
is c l e a r l y
the @eneric r a n k
of the c h o i c e
of G.
L e t F be a c o h e r e n t
L e t F 1 + F be the kernel.
independant
~X-mOdule.
Consider
Then F 1 is c o h e r e n t
the
and
11.8
]51
Supp F 1 is a p r o p e r sheaf
closed
subspace
of X.
We say that F is a torsion
if F = F 1 and that F is t o r s i o n - f r e e Proposition a)
b)
For
zero
sheaf F is torsion
if and only if F has generic
if and o n l y if Supp F ~ X.
any c o h e r e n t
~:0 c)
8.15:
A coherent rank
if F 1 = 0.
sheaf F,
there
is an exact
~ F 1 ~ F + F 2 ~ 0 w i t h F 1 torsion
If F is a torsion
sheaf
sequence
and F 2 torsion-free.
and G is a t o r s i o n - f r e e
sheaf,
HOmQcsx(F,G) = 0. Proof:
a)
is clear.
b)
Given
F, d e f i n e
F 1 as in 8.14
sheaf of F 1 ~ F. e
~ is a flat ~ X - m O d u l e
K is exact.
we have
labeled 0
Consider
some maps 9 F1
~>
a~ 0 ~ F
1 ® ~
F 1 is torsion,
Applying c)
the snake
the
so the induced
following
diagram
sequence
w h e r e we
for c o n v e n i e n c e : F -
> F2
I
0
bj
-->F e ~
0x Since
and let F 2 be the q u o t i e n t
>F 2 ® ~
°x
2 Q
Ox
F 1 ~ ~ = 0, so a is the zero map.
lemma,~er
L e t ~0:F ~ G be any map.
b = 0.
L e t H be the image
Supp H ~ Supp F so H is torsion.
of ~.
~ is the zero m a p
iff the
I1.8
i52
inclusion H ~ G is the zero map. is an injection. F
aI
~
~
Hence we can assume
Consider the commutative diagram: ~G
b~
~ h e r e ~' is injective since ~ is a flat b are both injective so a is injective. free as well as torsion.
~X-mOdule~
~ and
Hence F is torsion-
Hence F = 0 so ~ = 0.
CHAPTER THREE
QUASICOHERENT LOCALLY
i.
SHEAVES
SEPARATED
ALGEBRAIC
SPACES
i.
The C o m p l e t e n e s s / E x t e n s i o n
2.
The Serre C r i t e r i o n . . . . . . . . . . . . . . . . . . . . . . . . . . .
159
3.
Schemehood
.....................
165
4o
chevalley's
Theorem ...........................
169
5,
Devissage .....................................
and N i l p o t e n t s
The C o m p l e t e n e s s / E x t e n s i o n Let
X
be a noetherian
The C o m p l e t e n e s s / E x t e n s i o n categories
quasicoherent Phrased
X
in the t e r m i n o l o g y
the c a t e g o r y noetherian
algebraic
..............
separated
is a b a s i c
and c o h e r e n t
algebraic
tool
sheaves
Completeness
of G a b r i e l
173
says
sheaves
is that,
and every q u a s i c o h e r e n t
space.
on
X.
the
There
that every subsheaves.
XIV on a l o c a l l y
space is a "locally
assertion
155
for r e l a t i n g
is the u n i o n of its c o h e r e n t
of q u a s i c o h e r e n t
The Extension : U ~ X,
locally
involved.
sheaf on
Lemma
Lemma
Lemma
of quasicoherent
are two a s s e r t i o n s
i
ON N O E T H E R I A N
noetherian
for e v e r y
sheaf
F
on
separated category."
open immersion X,
and
for every
III. 1
154
coherent there
subsheaf
GU
is a c o h e r e n t These
in the
are
sense
the p r o o f
subsheaf
equivalent
that
each
of e i t h e r can be
seen
coherent
sheaf
G
The We
first
Hence
X
by
Now
affine
F
X
with
on
F
easily
from
in the
is m o r e
assertion:
canonically
determined
of
schemes
the o t h e r
involved.
extension
U,
i*G = G U-
can be proved scratch
to
The
but
difficulty
the r e q u i r e d by
Gu
but
ad hoc.
that
Extension X.
of
of
category
of the p r o o f
observe
the
schemes of
idea
G
i*F
in the
is not
constructed
restriction
assertions
from
involved
must be
of the
in the
case
completeness
lemma
holds be
of s c h e m e s
is t r i v i a l
for o p e n
let
X
any
scheme
open
subschemes
U
for
affine
subschemes
and
.
is as follows:
U
of a f f i n e
a covering
{Ui}iE I
First
schemes.
assume
Z =
{1,2],
1
so let
X = U 1 U U 2. G1
be
Let
F
any c o h e r e n t
be
a quasicoherent
subsheaf
of
F1
sheaf
on
Then
Gli IU I N U 2
IU 1 a coherent
subsheaf
of
Fi
so b y
the
Extension
IU I N U 2 U 1 N U 2 c U2,
there
lemma
and is
for
i
is a c o h e r e n t
sheaf
G2
of
FI
with U2
! G2 IUIQU 2
X,
= GII .UIQU 2 .
G1
Hence
and
glue
G2
together
to g i v e
I
a coherent
subsheaf
G
of
F
on
X
with
GI
= G1•
This
shows
U1 that
if
~
is the u n i o n
of the c o h e r e n t
subsheaves
of
F,
then
m
IUl = F IU 1
Similarly
~ IU2
=
F IU2"
Hence
F = F.
Thus
any
III.l
155
scheme which
is c o v e r e d b y at m o s t
completeness,
hence
to the g e n e r a l affines
U. m a p p i n g l
such a t e c h n i q u e
on an a l g e b r a i c
etale
the e x t e n s i o n
X:)
to X,
false
Criterion
the c o m p l e t e n e s s
reason
of D e v i s s a g e
assertion
for p r o v i n g
(5.2)
of
to us since schemes
injectively.
for an a r b i t r a r y
(And e t a l e map
u n i o n of two c o p i e s
assertion
in a d i f f e r e n t
of way
as a c o r o l l a r y .
this
lemma is its use in the
and in the p r o o f
of the S e r r e
(2.5).
1.1
(The C o m p l e t e n e s s
s h e a f on a n o e t h e r i a n limit b e i n g
T[ : U ~ X ,
and n o t n e c e s s a r i l y
the e x t e n s i o n
Our main
Proof:
is not a v a i l a b l e
- just take U to be the d i s j o i n t
and d e d u c e
Theorem
One proceeds
space is given b y a f f i n e
l e m m a is o b v i o u s l y
Hence we prove
proof
lemma.
satisfies
by induction.
the t o p o l o g y
9X
the e x t e n s i o n
schemes
case w h e r e X is c o v e r e d by a finite n u m b e r
Unfortunately
U.
satisfies
two affine
taken
and separated.
algebraic
L e t F be a q u a s i c o h e r e n t
space X.
over all c o h e r e n t
(P. Deligne) with U
Lemma.)
T h e n F = L i ~ F i , the
subsheaves
F ! of F.
By II.2.9,
w e can
find an e t a l e
a noetherian
affine
scheme,
The c o m p l e t e n e s s
assertion
covering
of X,
and 11" q u a s i c o m p a c t
is c e r t a i n l y
true
for
III. 1
156
affine
algebraic
sub - ~ U - m O d u l e s T~
spaces•
unions
GT II.5.4).
exact
(unless X is separated)
(in fact all
Hence
W e now d e f i n e following
of c o h e r e n t
~. 1
is not right
preserve
so TTWF is a union
~T["F
filtered
= L~I~
direct
b u t it does
limits
- see,
e.g.,
i .
F. as the p u l l b a c k 1
(intersection)
in the
diagram: m
i
c~
"~,-7, ~ i
/
,l
F
~-
"2 n,, 1"~ F
By the A B 5 p r o p e r t y Finally,
(II.4.3)
•
F = Lim F . ~ i
for each i, there
is a c o m m u t a t i v e
diagram
of
•U-mOdules: 1
1
Since F. is a s u b s h e a f l is injective. is c o h e r e n t
so
of F,
and
is flat,
1
the map ] ~ F . ~ > D ' F l
Hence
the top m a p
~VF. ~ ' ~ ~. 1 1
~F.
is coherent,
i-~is a c o v e r i n g
1
is injective.
~. 1
so F. is coherent, m 1
III. i
157
Corollary subspace Let
F
be
a quasicoherent
on
X
sheaf
of
with
We define
Lemma).
separated
sub- ~ - m o d u l e
G
Proof:
(The E x t e n s i o n
of a l o c a l l y
coherent sheaf
1.2
on
i*F.
Apply
G1
by
Then
i*i.Q = Q injective.
there
space
G ~ - i*F
>
which
so
is exact,
for all q u a s i c o h e r e n t
a
is a c o h e r e n t
diagram
of
~-modulesx
i.i*F
G1
is a s u b s h e a f
to this diagram. Q
on
U
since
of
F.
Note U - X
Thus
L G~ i*G 1 = % .
X.
> F
i*G 1
so
algebraic
and
the c a r t e s i a n
is i n j e c t i v e
i*,
X,
b e an o p e n
G ~ = i*G.
i.G u
map
U - X
noetherian
G1
The b o t t o m
Let
> i*F
1 "> i*F
is cartesian,
is
III.l
158
Hence by
i*
there
to
is s o m e
G~.
Since
subsheaf G
of
F
which
is c o h e r e n t ,
and
pulls back G1 =
Lim
G. 1
Gi~G 1 G. c o h e r e n t 1
by the
1.3.
lenlna,
there
It s h o u l d b e
Theorem
is m u c h
spaces
X.
map with
Then
for
Now we copy
sheaves the
there
Corollary
1.4.
Let
separated
algebraic
maps open are
~
Then
subspace
U
f:
f
of
o f the C o m p l e t e n e s s
separated
noetherian
U - X
be
is a f f i n e .
form
and proof
is t h a t
X
be
of
X,
the m a p
Let ~
~
of
an e t a l e One
first
sheaf
F
G
l e m m a D,
be
: H -- F
remarks
F
on
X,
to p r o v e on
noting
U. the
is a f f i n e .
a coherent
sheaf
sheaf
such that of
there
that
locally
the restrictions
the
the
noetherian
and a coherent
algebraic
surjective
quasicoherent
U - X
an i n t e g r a l
space.
and
f,G,
i * G = G~. I
the p r o o f
so it is s u f f i c i e n t
of the
is an i n t e g e r
: H -- ~ mX
that
with
for a n y q u a s i c o h e r e n t
statement
used
there
let
F ~ f,f*F
fact
Then
here
G ~ G1
in the c a s e
case,
affine.
is an i m m e r $ £ o n theorem
remarked
In t h i s U
a coherent
easier
l e m m a B above.
only
is
H
for ~
on
on
X. X
and
some dense and
~
to
U
isomorphisms.
Proof:
B y [.8.~3
there
is a d e n s e
open
subspace
U
of
X
such
III.2
159
that the r e s t r i c t i o n there HU c
(~X
X
~ F) I U lemma,
with
H ~ ~m
2.
F
is an i s o m o r p h i s m
extension on
of
The
Serre
The separated
Serre
there
H ~ F.
Criterion
noetherian
of a l g e b r a i c
In p r o d u c i n g
which
we label
Criterion only
~U-mOdules.
Let
subsheaf maps
H
By the
of
~X
t h e n give
m
@ F
two maps
is an i s o m o r p h i s m
on
U.I
algebraic and d e s c e n t
space
to b e
of s e c t i o n
criterion a scheme.
for a AS in the
i, the p r o o f
not g e n e r a l i z e
directly
for to the
spaces. for this
two a s s e r t i o n s
the W e a k
(e.g.,
Lemma
a new proof
are a p p a r e n t l y
full
Thus
isomorphism.
is a c o h o m o l o g i c a l
in EGA/does
there
The
of this
The p r o j e c t i o n
algebraic
category
just this w e a k
of
~module.
criterion
as given
criterion
: ~m U " F I U
Each of these maps
schemes,
category
is a free
is a c o h e r e n t
case of the C o m p l e t e n e s s
separated
U
b e the graph
H I U = HU"
and
X
~
to
Serre
spaces
theorem,
involved.
Criterion,
holds
and is a p u r e l y
theory
fact.
the results
Several
in s e c t i o n
3)
we found The
that
first,
for q u a s i c o m p a c t
formal
abelian
applications
of the
are c o n s e q u e n c e s
of
statement. Serre
Criterion
and the C o m p l e t e n e s s
in the case of n o e t h e r i a n
follows
from the w e a k
Lemma.
Thus we h a v e
separated
algebraic
Serre this
theorem
spaces.
I II. 2
160
(This is s l i g h t l y which holds
stating
of the theory
F : ~ exact
-
Let
M
of a b e l i a n
~
if the f u n c t o r
a functor
if
P
F
A n object Hom~(P,-)
2.2.
Hom : ~
A
be
(P,M) -
categories
~ in
~ 0. that
some
~
F
(If
F
is faithful is exact M, N
and
(Ab)
the c a t e g o r y
arbitrary
P
in
~
is f a i t h f u l l y
: ~
~
(Ab)
the ring
sums
is exact, HomdL(P,P ) .
is an A - m o d u l e F(M)
With F
the
is injective.)
has
projective,
short
, the s e q u e n c e
(~-
Mitchell)
and
if for every
for every
category
(A-modules),
(Gabriel,
is f a i t h f u l l y
categories.
in
(F(M),F(N))
W e assume
Let
schemes.)
we r e v i e w
is exact
is exact
b e an abelian
sums.
M 6 ~,
theorem,
F
to the a s s e r t i o n
groups.
for schemes
separated
be abelian
functor.
~L,
-- H o m ~
(= coproducts).
preserves
~
M ~ 0, F(M)
Hom~(M,N)
Theorem
and
in
map
criterion
categories.
- 0
is e q u i v a l e n t
let
this
-- F(M")
this
Now
quasicompact
0 - M' - M - M" - 0
- F(M)
if for every
every
~
an a d d i t i v e
sequence
the Serre
and p r o v i n g
of a b e l i a n
2.1.
0 - F(M')
than
for a r b i t r a r y
Before
Notation
weaker
so
Hom~
projective
faithful, Then
and
for
(P,-)
gives
= Hom~(P,M).
the a s s u m p t i o n s
is an e q u i v a l e n c e
of 2.1, of
III. 2
161
Proof:
See Bass,
Algebraic
W e are n o w ready denotes space
the c a t e g o r y and
X,
Theorem
~x _ -
K-Theory,
to p r o v e
p. 54.
our theorem.
of q u a s i c o h e r e n t the s t r u c t u r e
As usual,
sheaves
sheaf of
on an a l g e b r a i c
X.
2.3.
(The W e a k
Serre Criterion)
Let
quasicompact
separated
algebraic
Suppose
section Then
functor
X
F ( X , - ) : QCS x ~
is an affine
Proof:
Since
F(X,-)
commutes
X
is quasicompact, w i t h direct
the h y p o t h e s e s
object
in
QCSx,
(F(X, ~ x ) - m o d u l e s ) Let A-modules
(Ab)
X
be
is exact
a
the global and
faithful.
scheme.
For any Thus
space.
QCS x
sums.
has
all
sums
(See II.4.17,
and
and [GT]).
F 6 QCS x, F(X,F)
say that
so b y
QCS x
~
= HOm~CS (~x,F)U X is a f a i t h f u l l y p r o j e c t i v e
X
the t h e o r e m
2.2,
is an e q u i v a l e n c e
F(X,-)
: QCS x -
of categories.
A = F(X, ~ X ) = H o m CS ( ~X' ~ X )" The c a t e g o r y of Q X is e q u i v a l e n t to the c a t e g o r y Q C S s p e c A' so we have
a functor
F
: QCS X - Q C S s p e c
A
which
is an e q u i v a l e n c e
of
categories. We n o w c l a i m spaces
y
separated.
that
: X ~ Spec A Applying
there
is a n a t u r a l
such that
II.4.6,
Y*
y
m a p of a l g e b r a i c
is q u a s i c o m p a c t
: QCSx ~ QCSspec
A
and and we
III. 2
162
c l a i m y, = F. To see this, of X. call
let Spec B ~ X be an a f f i n e
etale covering
S i n c e X is separated, this
scheme
Spec B × Spec B is a f f i n e - X Then the e x a c t s e q u e n c e of rings
Spec C.
A ~ B ~ C induces
a m a p of a l g e b r a i c
COkAlg. S p a c e s ( S p e c
spaces
C + Spec B) ~ Spec A.
quasicompact
and separated,
quasicompact
and separated.
~, is an e q u i v a l e n c e
S i n c e X is
and Spec A is affine, It is
then clear
of c a t e g o r i e s ,
~ is
that ~, = F.
and its i n v e r s e
-I e q u i v a l e n c e y,
: QCS S p e c A + Q C S x m u s t be a left a d j o i n t
of y,.
the usual
But ~*,
inverse
sheaves,
is a left a d j o i n t of ~,.
adjoint,
y* is n a t u r a l l y
point
two c o n s e q u e n c e s :
naturally is exact
equivalent and
image of q u a s i c o h e r e n t By uniqueness
equivalent first,
to ~
y*y,
so the map y
From
this w e
: QCS x + QCS x is
to the i d e n t i t y
faithful,
-i
of the
functor,
and second,
~*
: X ~ S p e c A is f a i t h f u l l y
flat. W e now apply which
says that Lemma
a coherent Suppose
2.4:
the S h e a f C r i t e r i o n ~
m u s t b e an i s o m o r p h i s m ,
L e t X be a n o e t h e r i a n
s h e a f on X.
IF = F.
for i s o m o r p h i s m
so X is affine, l
algebraic
space
L e t I be a sheaf of ideals
T h e n Supp(F)
N V(I)
= ~.
(II.5.3)
and F be
on X.
Ill • 2
163
Proof:
The conditions
is a f f i n e . finite
Hence
type,
representing
and
we have
local
a ring R,
IM = M.
a point
are
Let p be
x
on X so w e can a s s u m e
ideal
I, m o d u l e
any p r i m e
e X = S p e c R,
such
associated
p M = M.
sheaf
of
ideal that x
P T h e n p ~ I so
M
X
E V(I). P
Hence
is z e r o
by
the N a k a y a m a
lemma,
in a n e i g h b o r h o o d
of
x
the , so
P x
~ S u p p M. P Theorem
noetherian F(X,-) :
2.5:
(The S e r r e C r i t e r i o n )
algebraic (Coherent
space.
Sheaves
Suppose on X) ~
Let
X
the g l o b a l (Ab)
be a s e p a r a t e d section
is exact.
functor
Then
X
is affine. Proof: all
coherent For
Lemma
says
HI(x,F) using
that
F
on
F = Lim F
subsheaves. = 0
by
this,
closed
the h y p o t h e s i s
sheaves
in the
sheaf
i
just
F
the
that
we use noetherian
subspace
on
HI(x,F)
HI(x,F)
= 0
for
the C o m p l e t e n e s s
we
direct
F
on
can c o n l u d e is a l s o
induction Xlred
and
limit
= L~m HI(F,F
sheaves
F(X,-)
X 1 of X, w i t h
X,
(filtered)
By I I . 4 . 1 7 ,
Serre Criterion, showing
form:
X.
for all q u a s i c o h e r e n t
the W e a k
affine
use
any q u a s i c o h e r e n t
coherent
show
We
X.
l
so
Thus,
t h a t X is
faithful. assume
/ Xred,
)
of all
that
is a f f i n e .
To every
III. 2
164
The sheaves sheaf
induction on
on
a proper
X
such
i
done;
ideals sheaf
X
such on
that
subset
such
0.
be
an
is
X
Spec
with
the
that
F =
V(X,F/IF)
proper
closed
the
induction
Supp
subset
closed
subset
Now F(X,F)
=
let 0.
of
X,
Since
F
be
any
By
the
coherent
subsheaves,
so
V(X, Fi)
each
of
so
there
is
for
some
and
of
F
is
a proper F1
since
C
If
closed
on is
C.
affine
U = X
we
I
be
a sheaf
= X - U.
Let
F
be
a coherent
Consider
the
exact
of
sequence
the (U
support is
Hence C
is F =
functor
~ 0 - ~(X,F/I F ) ~ 0
= F.
nonempty,
is
a
F/I F = 0
Using F
lemma
by 2.4,
is
a proper
X
such
the
union
0.
U F I•
Hence
F/I F
Supp
quasicoherent
F =
of
nonempty), IF
Completeness
= 0.
a coherent
support
subspace.
section
Since
X
0
be
IF ~ F ~ F/I F ~ 0
hypothesis.
F n C = ~.
the
coherent
Let
= 0.
global
= 0.
F
i,F 1 =
to
not.
0 - F(X, IF)
Hence
and
open
~X/I
Let
II.5.14
Suppose
F(X,F)
(exact)
By
affine
0 ~ Applying
0
= F(C, FI)
affine.
that
=
X.
applied
way.
~(X,F) of
F =
U -- X
can be
following
= F(X,i,Fl)
Hence
Let
the
: C - X
F(X,F)
F 1 = 0.
are
in
closed
subspace Then
X
hypothesis
sheaf
Lemma, Then
each
F
F
on is
0 = ~(X,F)
l
=
0
so
=
F =
that of
its
U F ( X , F i) 0.
I
III. 3
3.
165
Schemehood
Lemma
3.1.
spaces.
Let
Then
associated
f
3.2.
Then
X
Proof: since
algebraic
scheme,
Let
X
is local
on
in w h i c h
case
of schemes,
if and only
spaces
first
the second
fred
if the
: Xred " Yred
Xre d
that
so we can take X
this
if
is clear,
Xre d
(X x X ) r e d =
~
to
i
algebraic
space.
is separated.
(Xre d × Xred) red
the u n i v e r s a l
In the c o m m u t a t i v e
i
Y
is also a scheme
separated
if and only
satisfies
(X × X) red - X × X.
Y
b e a locally
is s e p a r a t e d
Note
immersion
of a l g e b r a i c
immersion.
in the c a t e g o r y
Lemma
b e an i m m e r s i o n
is a c l o s e d
The q u e s t i o n
b e an a f f i n e and
f : X - Y
m a p of r e d u c e d
is a closed
Proof:
and N i l p o t e n t s
property
for
triangle
(X × X ) r e d =
(Xred× X r e d ) r e d
Xre d × Xre d j
is a closed
immersion,
so separated.
Applying
I.i.21,
III. 3
166
L
A
is a c l o s e d
immersion
if and o n l y
immersion,
which by Lemma
X ~ X × X
is a c l o s e d
Theorem
3.3.
Let
space.
Then
X
an a f f i n e
By L e m m a
affine,
then
so
be
scheme
3.2,
locally
if
separated
if and o n l y
we can assume the c l o s e d
~X/J"
is an i n t e g e r there
let
J
Since n
if
algebraic
Xre d
immersion
with
is s e p a r a t e d . Xre d
sheaf
of
be
the
X
is n o e t h e r i a n ,
with
is a s e q u e n c e
X
subspace
j n = 0.
of c l o s e d
X r e d = X 1 - X 2 - ... ~ X n = X. closed
if and o n l y
I
a noetherian
is an a f f i n e
clearly
X r e d = Sp~ec
so t h a t
is a c l o s e d
immersion.
the c o n v e r s e ,
so t h e r e
happens
i
If
X
of n i l p o t e n t s
Let
J
satisfying
Hence
Each map
we
of
Let
separated if
Y
f : Y ~ Z
algebraic
is affine,
be
spaces
so is
Z.
a closed with
X,
is c o h e r e n t
X i = S~
~X/ji
X i ~ Xi+ 1 with
is a
I = ji/
i+l
are r e d u c e d
immersion
Y = Spec
an
J to the
following
lemma:
Lemma.
is
immersions
X i = Sp~ec ~
12 = 0.
X
is a f f i n e .
Xi+i/I ideal
is
scheme.
Proof:
For
X
3.1,
if
of n o e t h e r i a n
~Z/I'
12 = 0.
Then
III. 3
167
Proof
of Lemma.
Consider
the
exact
are b o t h
F/IF
modules.
Let
F
annihilated
Apply
the
H 1 (Z, IF)
and
IF
and
vanish.
Z
Proposition
~ ~(Z,F)
- H 1 (Z,F)
H
1
(Z,F)
functor
Z.
0.
IF
~Z/I
and
=
~Y-
V(Z,-) :
- H I(Z,F/I F )
~y-mOdules,
= 0
are
on
~ ~(Z,F/I F )
Y
= H I(Y,F/I F ) .
Hence
Criterion,
I ~ so t h e y
section
are
sheaf
0 ~ IF - F ~ F/I F by
global
F/I F
H I(Z,F/IF)
a quasicoherent
sequence
0 ~ F(Z, IF)
Since
be
HI(z, IF)
is a f f i n e
so
F(Z,-)
= HI(y, IF)
so b o t h
is exact.
of
these
By the
Serre
|
is affine.
3.4.
algebraic
space.
separated
etale
Let
X
be
a noetherian
locally
Then
there
is a o n e - o n e
correspondence
maps
Y ~ X
and
separated
etale
separated
maps
between Y'
- Xre d
given by Y ~ X
This
~
y'
correspondence
Proof:
The
assume
that
Y - X, Y schemes,
preserves
one-one X
a scheme
the o n e - o n e
(Xre d) ~ X r e d.
and r e f l e c t s
correspondence
is affine.
must be
= Y ~
Hence
is local for
(by II.6.5).
correspondence
affine
any
on
X
separated In the
is 1.4.20.
schemes.
so w e can etale
category
map of
III. 3
168
Note only
if
that Yred
Corollary space
and
Y × (Xred) X is affine.
3.5.
Let
X
= Yred"
By 3.3,
Y
is affine
I
be a noetherian
locally
the c a n o n i c a l
inclusion.
f : Xre d ~ X
if and
separated Then
algebraic
the
two functors f,
: AS X
- AS red
are inverses of a b e l i a n
f*
: AS x - AS X
X
to each other.
(AS
red
as usual
h e r e means
the c a t e g o r y
|
sheaves.)
Corollary
3.6.
algebraic
space.
Let
X
Then
be
a noetherian
X
is a scheme
locally
separated
if and only
if
is
Xre d
a scheme.
Proof:
A n open c o v e r i n g
an o p e n c o v e r i n g
of
X
of by
Xre d affine
by
affine
schemes,
schemes
lifts
to l
III.4
4.
169
Chevalley's
This
Theorem
theorem
separated
gives
algebraic
space
us w i l l be in the p r o o f Theorem and
Y
4.1:
f
:
is an affine
criterion
for a n o e t h e r i a n
to be a scheme.
Its a p p l i c a t i o n
of C h o w ' s
Lemma
Theorem)
Let
separated
algebraic
space.
X ~ Y which
is finite
for
(See IV.3.1).
(Chevalley's
a noetherian
is a m a p
another
X
be an a f f i n e Suppose
and surjective.
there
Then
Y
scheme.
morphism.
Y red' also a finite s u r j e c t i v e X × Yred Y By C o r o l l a r y 3.3 or the Serre Criterion, it is
sufficient
to show that Y r e d
Proof:
finite
Consider
and surjective.
is affine.
X r e d ~ X × Y r e d is Y Thus w e can assume that X and Y
are reduced. We prove assuming
the
that
for every
Y l r e d / Yred' on
Y, w i t h
theorem b y n o e t h e r i a n closed
Y1 is affine.
subspace
Hence
Supp F ~ Y, HI(y,F)
induction Y1
of
for every
on Y,
Y, w i t h
coherent
sheaf
F
= 0. I
Suppose component
Y
is not
of Y.
F' = j,j F and
irreducible.
For any c o h e r e n t p
L e t Y' ~ Y be a
~y-module,
: F + F' b e the n a t u r a l
map.
F,
let
Let
G = Ker p
scheme
III.4
and G
170
K = Imp. and
K
have
HI(y,K)
= 0
sheaves
F
Y
p
is an i s o m o r p h i s m
support
not
H 1 (Y,F)
sO on
Y
equal
= 0.
so b y
the
on
to
This Serre
Y'
Y.
-
(Y N Y')
Hence
holds
for
criterion
so
HI(y,G)
all
=
coherent
2.5,
we are done;
is affine. On the o t h e r hand,
We now need
Lemma: map
Y
is i r r e d u c i b l e ,
so
integral.
a lemma:
In the
situation
of s h e a v e s
subspace
suppose
U
there
: _ _ ym . f * ~ x
u
of
above,
Y,
is an i n t e g e r
such
the r e s t r i c t i o n
m,
t h a t on a d e n s e
of
u
is f i n i t e
so
to
U
and a open
affine
is an
isomorphism.
Proof
of
lemma:
~y-module. su30space
of
Y
integer
f
TI', 2,15
Applying U
for s o m e
The map
, there
and an i s o m o r p h i s m m.
Let
X
with
the
is an a f f i n e restriction
g's 1 ..... g ' S m F (X, ~ X ) .
scheme g'
These
u IU : ~ ym I - f. ~ X U
{hi} I U
u I : ~y
m
so t h e r e
is the m a p
u
affine
open
~ f.
U Let
is an e l e m e n t a unit,
to s e c t i o n s
define
]
= U × X. Y b e the e l e m e n t s
6 F(U', 4~X)
extendable
is a c o h e r e n t
is a d e n s e
U' = f-l(u)
s 1 ..... Sm E ~(U', ~ X ) = F ( U ) f . ~ x ) u I.
f* ~ X
m : ~ y ~ f. ~ X taking
the
defining g 6 F(X, ~ X ) ,
and w i t h
bl,...,bm
I U
of and
element
III.4
171
(0 ..... 1 ..... 0) But w
since
with is
g'
1
in t h e
a unit
on
f. ~ X I
: f. ~ X t U
composite
i
U',
taking
th
position
there any
is
to
b i I = g ' s .l U isomorphism
an
x ~x/g'
The
U
w'u I U
is
the
isomorphism
u I.
the
theorem.
Our
Hence
is
u IU
an isomorphism.
We that By
can
now prove
every
torsion
sheaf
F
on
Y
satisfies
every
sheaf
F
on
Y
fits
II.8.15,
0 - F'
-- F - F"
Hence F,
it is
H I(Y, F) Let
F
" f* ~ X
torsion T
is
be
Cok an
v
H
1
and
~WX-mOdule.
(Y, G)
=
H1
an
F
that
HI(y,F)
into
and
for
an
F"
every
(X, GI)
on
is
= F
T
is
says
= 0.
exact
sequence
torsion-free. torsion-free
sheaf
The
G -- H o m ( f . defines
exact
cokernel,
a map
sequence
necessarily
0 - Hom(T,F) so b y
~x,F).
a
v m - G -- F
II.8.15,
injective. sequence
so
HI(y,
Cok
v)
0 ~ G v Fm . Cok
= 0.
G = Hom(f.
is
an affine
map.
X
with
f . G I.
so
and
above
.
sequence
exact
= 0,
m
the
the
on
Y
constructed
exact
v
X - Y G1
sheaf
is t o r s i o n - f r e e
Hence
torsion
sheaf
where
gives
consider
coherent
show
m -Hom(~y,F)
~x,F)
= 0.
is
to
torsion
a torsion-free
" T -- 0,
torsion
Now
F'
m : ~y - f. ~X
u
sheaf,
Hom(T,F)
with
assumption
= 0.
: G = Hom(f.
~Y
- 0
sufficient
The% t h e m a p v
induction
H
1
G =
(Y,F)
= 0.
Hence
v ~ 0.
~x,F)
there
is
is a
Hence This
holds
for
all
III. 4
172
torsion-free Y.
sheaves
on
Y, h e n c e
By the S e r r e criterion,
Corollary
4.2.
Y
for all c o h e r e n t
sheaves
on
is affine.
I
Let f
X
> Y
S
be a commutative h
an a f f i n e map,
surjective
Proof: S
diagram g
separated
and finite.
The assertion
is an affine
Chevalley
scheme
Theorem.
of n o e t h e r i a n
Then
algebraic
and of finite g
type,
and
f
is affine.
is local on in w h i c h
spaces w i t h
S
case
so we can a s s u m e this
is e x a c t l y
that
the I
III.
5
173
5.
Devissaqe
Devissage coherent space. of t h e
is an i n d u c t i v e
sheaves We will
in E G A
on a n o e t h e r i a n use
finiteness
The proof
this
here
Definition
but
5.1.
o f the o b j e c t s
(We n o t e
0 E K'
2)
For
Let
Theorem
If
5.2.
separated
K
for e v e r y G 6 K'
are
space K'
of
algebraic
the e q u i v a l e n t
the definition does
an a b e l i a n
not
in
of exact
assume
in 5 . 1
the d i r e c t
in the
theorem.)
A subset
category.
0 ~ A'
K',
then
A1
and
Let
X
be
and
K
the
an e x a c t
closed
subspace Then
~ A -- A '~ -- 0
so is the
be
S u p p G = Y.
assertion
K'
if
sequence
(Devissage)
integral
with
be
A 1 • A 2 6 K',
Let
that
is e x a c t
any exact
algebraic
~X-mOdules.
of
IV for t h e p r o o f
it as an e x t r a h y p o t h e s i s
K
if t w o t e r m s 3)
category
separated
in C h a p t e r
- Grothendieck
uses
of
i)
technique
is a t r a n s l a t i o n
altered
sum c o n d i t i o n
locally
for the
theorem.
(III.2.1.2).
is s l i g h t l y
technique
A2
category
subcategory
every
K,
third. are
a noetherian
Y - X,
in
K'.
locally
o f all c o h e r e n t of
there
coherent
in
K
such
that
is a s h e a f
sheaf belongs
III. 5
to
174
K'
(Recall
integral
Supp G = Y
algebraic
space
for a c o h e r e n t
Y
entails
that
sheaf
the
G
stalk
and G
of Y
G
at t h e g e n e r i c
Proof:
Consider
subscheme
Y
contained
in
remains
P(Y')
then
P(Y) Let
F 6 K'. ( Then
the
of Y
y
X:
Every
that
holds
Say
have
Y
of a c l o s e d
D-module X
with
By n o e t h e r i a n
closed
support
induction
subscheme
subscheme
are
of c o h e r e n t
exact
Thus
of
Y'
X
of
it such
Y,
- F/
where
we can a s s u m e
Y
Y.
ideal n
We will
I
such
of
prove
~X"
that
By
InF = 0.
j
Since to s h o w
K'
is exact,
that
each
IF = 0,
i.e.
is the
is r e d u c e d .
is r e d u c i b l e .
- F/ n-i - 0 I F
InF
can assume
F = j,(j*(F))
an
in
sequences
induction we
by
is an i n t e g e r
~-modules. X by
contained
is d e f i n e d
InF
Y
P(Y)
is a c l o s e d
support
) there
there
a)
property
K'.
for e v e r y
Yred
~.5°8
K'.
is n o n z e r o . )
coherent
to
if
0 ~ In-iF/
in
Y
is true. F 6 K
sufficient
of
following
belongs
to s h o w
that
point
Say
it is
FK = I
k-i
F/k I F
is
that
injection
Yred
We now distinguish y = y, L~y,,
with
~ X.
Thus
two cases: Y'
and
III. 5
Y"
175
two reduced
Say
Y',
Y"
F'
The a
closed
are d e f i n e d b y
= F ~Y® ( ~ y / j , )
canonical
maps
: F - F' D F".
Ker a
and
Hence
F E K'
F"
are
Cok
each
unequal
to
b)
Y
coherent
in Y,
we
F E K'
O ' ym e
We
Since
on
Y.
F - F"
using iff
F'
of
~y.
define on
the F'
and
Y -
Put
a map (Y' n Y")
induction
@ F" F"
E K'
so
hypothesis. iff
each have
so i n t e g r a l .
Applying
~y -module
of
Ker
9,
H
1.4,
Let
there
and m a p s
is an
F'
and
support
~
~
are i s o m o r p h i s m s .
~,
all h a v e
and C o k
assumption,
Im ~ 6 K'
iff
they
H 6 K'
subspace
support
are all iff
be
any
integer
: H ~ ~m y
dense
and
open
~
F
on some
this
first
which
s u p p G = Y.
This
Hence
~ y n E K'
F E K'.
J"
Y.
in
m
and
U
of
Y,
Thus not
equal
K'.
to
Hence
Im ~ E K'
iff
coherent
sheaf
. apply
argument
J',
from
are done.
induction
iff K'
K',
Im a E K'
such that
By the
distinct
isomorphism
is i r r e d u c i b l e ,
C o k ~,
Y
F" = F ~Y® (~y/j,,) .
and
a r e in
K'.
the restrictions
Y.
and
is an
iff
sheaf
: H - F
a
of
ideals
F ~ F'
a
and a c o h e r e n t
K e r ~,
subspaces
for
for
any
any c o h e r e n t
to the p o s t u l a t e d shows
that
integer sheaf
~y
m
E
n k 0. F
on
K'
for s o m e
Now Y,
applying
~y
E K'
G
m ~ 0. the so I
for
CHAPTER
FOUR
THE F I N I T E N E S S
i.
Actions
of a F i n i t e
2.
Symmetric
3.
Chow's
4.
The F i n i t e n e s s
Powers
THEOREM
Group
of P r o j e c t i v e
Theorem
in the title,
is the F i n i t e n e s s
as that
in the case of schemes
one applies
an i n d u c t i v e
using C h o w ' s
Lemma
generalization
Chapter
of Devissage;
is the same
spaces
specifically step.
where Devissage,
The p r o b l e m
lies not in the
in the
final
in the g e n e r a l i z a t i o n s
III can b e viewed
this c h a p t e r
of
as e s s e n t i a l l y
as e s s e n t i a l l y
a proof
Lemma.
The p r o o f
of C h o w ' s
powers
of p r o j e c t i v e
groups
acting
in S e c t i o n
in this
(cf. E G A III.3.2.1)
procedure,
b u t rather
202
theorem
The p r o o f
for the initial
of F i n i t e n e s s
of C h o w ' s
Theorem.
to a l g e b r a i c
the ingredients.
the m a j o r
185 192
....................
chapter
a proof
Spaces ......
Lemma ..............................
As i n d i c a t e d
proof
177
.................
Lemma
spaces,
on a l g e b r a i c
uses
and a few notions
spaces.
1 the bare m i n i m u m
the notion
of
of s y m m e t r i c about
For the latter, foundations
finite we give
necessary
for
IV. 1
177
the later proof. foundations spaces
The reader h o p e f u l
of the
should
theory
i.
all a l g e b r a i c
of a F i n i t e
Definition an a l g e b r a i c
1.1:
~
fixed
are a s s u m e d
map
a group homomorphism
f:X ~ Y s a t i s f i e s
is trivial
G-invariant
Oy
spaces
L e t G be a f i n i t e group.
to b e
the s u p e r s c r i p t
elements
the a c t i o n of G.)
stable
e t a l e map)
(resp.
an etale map)
A stable open
and w h i c h q
satisfies,
~-~E
X - - - - - - . ~ - ~ q '~ X
is cartesian.
space.
"G" d e n o t e s
f:E ~ X is a G - m a p w h i c h
E
as the i d e n t i t y map. A
as a
there is a m a p of s h e a v e s
(f, ~ X )G w h e r e
L e t E and X be G - s p a c e s .
is
G ~ Aut(X).
f:X + Y w h e r e Y is taken
(Note in this case
under
A G-space
for all q in G.
and Y be any a l g e b r a i c
is a G - m a p
G-space.
fq = q f
if e v e r y q in G acts
Let X be a G-space
trivial
on a l g e b r a i c
Group
space X w i t h
of G - s p a c e s
A G-space
actions
and separated.
Actions
A G-map
group
the g e n e r a l
see 1.8.
In this chapter, Noetherian
of finite
of finding
as usual
subspace
the
(resp.
is an open i m m e r s i o n for all q in G,
IV. 1
178
L e t X b e a G-space. is a g ~ o m e t r i c
Note
quotient
A map of algebraic of X b_y_GG if
i)
n is G - i n v a r i a n t
2)
The map
that,
and
finite
of s h e a v e s O y ~
as an i m m e d i a t e
Also;by Chevalley's
spaces ~:X + Y
(~,OX)
consequence,
Theorem/X
is a f f i n e
G
is an i s o m o r p h i s m .
~ is affine
and proper.
if and o n l y
if Y
is
affine. Example group
acting
point
locus
1.2:
L e t X b e an a l g e b r a i c
on X.
G acts f r e e l y
of ~ is empty.
I.e.,
space
and G a finite
if for any ~ e G, the
fixed
the
is
following
diagram
cartesian: emp t y space
~
X
We write G × X for each ~ e G,
GXX
A
~X
for the d i s j o i n t
and use
the p o i n t x in the a
X
th
the n o t a t i o n c o p y of X.
--~
xX
sum of c o p i e s (c,x)
There
of X,
one
E G X X to m e a n
are two m a p s
X
~2 defined etale
by ~ l ( S , x ) = x, ~2(o,x)
equivalence
geometric
relation
quotient.
= ~(x),
on X, w h o s e
and this d e f i n e s quotient
an
~ : X ~ Y is a
IV. 1
179
Proposition geometric
quotient.
the n a t u r a l quotient. Y' ~ Y
action
then
Proof:
and ~ : X ~ Y
an e t a l e m a p .
under
× X, Y' × X ~ Y' is a g e o m e t r i c Y Y if G a c t s on X, and X ~ Y is G - i n v a r i a n t , and Y'
× X ~ Y° Y
is a g e o m e t r i c
is X ~ Y.
The definition
L e t X'
1.4:
of geometric
quotient
is c l e a r l y
× X, Y action.
X' ~ Y'
affine
n-space
T h e n X' ~ Y'
is c l e a r l y
property,
to the a f f i n e
and ~ : X ~ Y
= Spec ~y[Zl,...,Zn]
= Y'
the q u o t i e n t
to r e d u c e
Let X be a G-space
L e t Y'
the i n d u c e d Proof:
we
Thus
case.
o v e r X, b e
and
A [ Z I , . ..,Zn] <
the a s s e r t i o n
,
is c l e a r ,
.... A G
is t h a t
AG[Zl,...,Zn]
=
To
if X = S p e c A and Y = S p e c A G,
A
. .. ,Z n] ~
finite.
quotient.
the l a s t p r o p o s i t i o n
J
AG[Zl
n-space
a G-space
is a g e o m e t r i c
G-invariant can use
a geo-
be affine
we have
which
Then
of G on Y'
so a l s o
quotient.
o v e r Y.
and
a
on Y.
metric
check
a G-space
L e t Y' ~ Y b e
surjective,
Proposition
under
Let X be
Conversely,
is e t a l e
quotient,
local
lo3:
(A[Zl,...,Zn]) G
i
IV. 1
180
Proposition algebraic acts
1.5:
spaces.
Let X
Suppose
~
:X
)Y which
is also q u a s i p r o j e c t i v e a stable
open
quotient
of U,
of
S-spaces, Proof:
to b e of Y
subspace
The
the o r d e r of G). (EGA I I . ~ . ~
>X
V
~Y is
an is
case
~V
open
m a p of ~
G
S - s p a c e Y,
of X b y G.
if U - - - ~ X
Y
is
is a g e o m e t r i c
diagram
immersion.
local
on
all of this
S,
so
we can
except
take
S
the q u a s i p r o j e c t i v i t y
V.
of q u a s i p r o j e c t i v i t y . of Y,
and c o n s i d e r
These define
Let
m f:Y--'~ PS
the c o m p o s i t e s
a map y__~(pm)n
(where n is
Imbed(pm) n into P (m+l)n-I b y the S e g r e ).
We then have
the g r o u p G acts
of Y in P (m+l)n-I
~l:U
U
imbedding
quotient
Furthermore,
and
each
is a u n i q u e
is a c a r t e s i a n
the p r o o f
~-~-a2P~ , ~ e G .
is an o p e n
of X,
from SGA 6 0 - 6 1
be a projective
such that
o v e r S.
proposition
indicate
there
is a g e o m e t r i c
V,.--~Y
in w h i c h
is q u o t e d
Y--~Y
Then
then there
where
affine,
We
map
X is a G - s p a c e w h e r e
as an S - a u t o m o r p h i s m .
and m a p
) S be a quasiprojective
a projective
linearly
on Y.
The a c t i o n of S e x t e n d s
immersion.
imbedding
Let Y b e the c l o s u r e to Y and Y / G - - - ~ Y / G
H e n c e w e can a s s u m e Y = Y.
Y = Proj A, w i t h A a g r a d e d
S-algebra
of Y
on w h i c h
Thus
G acts
linearly.
IV.I
181
The geometric a map
quotient
Proj A - - ~ P r o j
essential.)
Spec A - ~
AG
(here the
The only difficulty
is p r o j e c t i v e .
A G is o f ~ i n i t e
generaters.
Unfortunately
the d e g r e e
i p a r t of A G = ~ A
EGA I I . 5 , 5 , 1
k.
and h e n c e
the g e n e r a t e r s
We g~t
around
G
gives by restriction
linearity
of the a c t i o n
n o w is to s h o w that
S i n c e A is of finite
and G is finite,
Instead
Spec A
type over S,
type,
so has
these generaters G n
to a s s e r t
that
Proj A
lie in d e g r e e s b e t w e e n
this d i f f i c u l t y
Proj A G. property
A bit that
it is g e n e r a t e d 1.6:
quotient.
) Proj
shows
this
affine
and of
finite type over Y.
as a c l o s e d
subspace
Proposition
1.4,
of a f f i n e
n-space
w e are r e d u c e d
of a f f i n e
affine
of in
is p ~ o j e c t i v e ) .
, (k~)
=
tAG ) a l g e b r a has
and
n-space
o v e r X, u n d e r
of d e g r e e
7[:X----~Y b e
to the case that Y ' ~
a
type,
over Y for some n.
induced
i.I
to
can be considered
o v e r Y is a g e o m e t r i c the
the
so we can t a k e Y'
H e n c e Y'
n-space
integer
(AG) (k:) =
for e v e r y Y'----~Y of finite
~ ' : X ~ q Y'---gY' is an o p e n m o r p h i s m . Y Proof: T h e a s s e r t i o n is local on ¥' be
G
over A~ b y the t e r m s
Let X b e a G - s p a c e Then
number
not all b e
by considering
EGA I I . 2 V, 7
of c o m b i n a t o r i c s
proposition geometric
~y
S is n o e t h e r i a n ,
1 and some
.
for w h i c h
G
(as t h e y n e e d b e to a p p l y
G
~r=0Ark~
Proj A
a finite will
is
action
Y is a c l o s e d
By
quotient
of G.
Hence
immersion.
IV. 1
182
The proposition in w h i c h
c a s e ~ is o p e n
of n to a c l o s e d We will of the
following
quotient.
on Y so we can
(SGA 60-61V).
subspace
in its d o m a i n
take Y affine, the r e s t r i c t i o n
is an o p e n m a p p i n g . E in the p r e c i s e
form
corollary:
1.7:
L e t X be a G - s p a c e
L e t Y' ~ Y be a finite
of Y'
Hence
later use this p r o p o s i t i o n
Corollary
space
is local
Then,
defining
X'
U'
>X'
and ~ : X ~ Y a g e o m e t r i c
type m o r p h i s m
and U'
to m a k e
and U a subthe
following
cartesian:
U
>X
- - > Y '
->Y
the n a t u r a l m a p
(Closure of U'
is an i s o m o r p h i s m (Closure h e r e Proof: to be affine.
(Closure
of the u n d e r l y i n g
is taken
topological
is local
It is also c l e a r l y hence
of U in Y')
× X' y,
spaces.
in the sense of II.5.7)
The proposition
is a l g e b r a i c a l l y , have
in X') ~
on Y'
sufficient
topologically,
dense
so we can
take Y'
to a s s u m e in Y'.
that U
Then we
IV. 1
183
U'
-
-p
!
J
~j
U
a cartesian dense open
pY'
diagram
of s c h e m e s w i t h U --~ Y'
immersion
necessary
In this
algebraic exist
is t o p o l o g i c a l l y
section,
Deligne
spaces,
- that
has
quotients
is, q u o t i e n t
geometric
and c a t e g o r i c a l
geometric
or c a t e g o r i c a l
We briefly on w h i c h write
indicate
a finite group
Y ~ and X ~ for the
X ~C
f-l(y~
).
We
this
is an e q u a l i t y
quotients
depends
algebraic
s p a c e X,
a s c h e m e U, G on U,
say
shown under
maps
that
which
the b a r e m i n i m u m
actions
of s e p a r a t e d always
are b o t h universal
quotients). the proof: G acts.
Let X and Y b e
Given
fixed-point
any G - m a p
locus
~£ G.
of any
~
union
The existence i)
Given
surjective
schemes,
an etale F P R G - m a p of s e p a r a t e d
G-spaces
then
(FPR)
if
of g e o m e t r i c
there
is of
is FPR.
If f : X - - - > Y
and g e o m e t r i c
f/G is etale.
if w e
in G,
an a c t i o n
G-map U---~X which 2)
spaces
a separated
a finite g r o u p G acts, of a f f i n e
algebraic
f:X---->Y,
f is f i x e d - p o i n t - r e f l e c t i n g
on w h i c h
f/G exist,
l
(although not always
II.6.4).
Y / G and
in X' .
only
finite group
(The p r o o f of this uses T h e o r e m
X/G,
1.6).
in the c a t e g o r y
X--~X/G
on two lemmas:
and an e t a l e
~y
lemma.
quotients
for each
a disjoint
dense
we have proved
for the p r o o f of the C h o w
In general,
a topologically
and X'---> Y' an o p e n m a p p i n g
It is t h e n c l e a r t h a t U' 1.8:
X'
is
quotients
G i v e n these,
we
IV. 1
184
start w i t h U~X,
a G-space
compatible
X, use
with
I) to find an affine
the G-action,
covering
form R = U ~
U and X
note
that
each m a p R L ~ U
R, and U are d i s j o i n t f/G all exist Its q u o t i e n t Thus under
group
example
notion
shows.
We abstain
Chap.
U/G
and
relation.
of time
in XXVI,
Varieties
finite
is c l o s e d
group
the c a t e g o r y
of all schemes
separated
schemes
base
actions.
And,
is so closed.
is not so closed,
of a l g e b r a i c "right"
spaces
as the
justification space
(rather
generalization
But
for the that the
of the n o t i o n
scheme.
outline
5, S e r r e s ' s
Abelian
R/G,
category,
category
is a n o t h e r
(for the m o m e n t
both
this
noetherian
under
This
is the
a general
in p r i n t
(i.e,,the
schemes
that the n o t i o n
(for reasons
Since
equivalence
algebraic
Inside
of affine
of s e p a r a t e d
of q u a s i p r o j e c t i v e
exists
schemss
is c l o s e d
of scheme)
giving
quotients.
the c a t e g o r y
on p . 1 4
assertion
schemes,
is an etale
of s e p a r a t e d
over a given
space)
the c a t e g o r y
covering.
is X/G.
quasiprojective
of course,
of affine
and R/G c-~ U / G
of q u a s i p r o j e c t i v e
algebraic
unions
the c a t e g o r y
finite
is a FPR etale
Groupes (Oxford
at least)
of the t h e o r y and length). XXXV,
from the task
of finite g r o u p The g e n e r a l
S G A V, Bourbaki:
Algebriques, 1970).
Alg.
and M u m f o r d ' s
of actions
theory Comm.
IV. 2
The
185
fact that q u a s i p r o j e c t i v e
quotients known" the
under
but
finite g r o u p
indication
to some e x t e n t indications
of a l g e b r a i c problems
2.
of U 1 u n d e r
permutations P we defing immersion
etale
above
symmetric
is to take
an a c t i o n b y the
V
,~
e t a l e m a p U---) v, map UI--~ U
symmetric
induces
P
snp
reduced
XXXI.)
group S
for a g i v e n p r o j e c t i v e
n ~
can b e
) V
) V is a g e o m e t r i c
Pn---)snp
U --9 P of the U a b o v e
quotients
Spaces
and such that U 1
uI
groups
a particular
Then
But t h e r e
in finding
(See S e s h a d r i
a finite
it b y
powers
and h e n c e
can b e g e n e r a l i z e d
groups.
algebraic
of P r o j e c t i v e
of n letters.
"generally
reference,
involved
finite groups.
and d o m i n a t e
is finite
quotient
and d e f i n i t i o n s
spaces b y g e n e r a l
to b e
above.
that the p r o b l e m s
The object here
which
seems
to the case of a l g e b r a i c
involving
n,
admit q u a s i p r o j e c t i v e
a specific
of p r o o f of this
Symmetric~owers
of d e g r e e
actions
it is h a r d to g i v e
All the t h e o r e m s
are
varieties
n
of space
of P and s h o w e v e r y a cartesian
diagram
to
186
IV. 2
Construction affine
algebraic
2.1:
L e t n : U ~ V be a finite e t a l e m a p of
spaces.
T h e n ~ is a f f i n e
w h e r e ~,0 U is a c o h e r e n t is of d e g r e e
n, i.e.,
locally
so U = Sp~ec ~,0 U
free O v - a l g e b r a .
that ~,O u is l o c a l l y
We assume
free of d i m e n s i o n
n over % . R e m a r k 2.2: the d i s j o i n t general,
The simplest
sum of n c o p i e s
assertions
about
can a l w a y s b e r e d u c e d
local ring,
that U
is a d i s j o i n t
of V
this
situation
case,
which
case,
V b y the a f f i n e
in w h i c h
is to t a k e U to b e
(the t r i v i a l
to this t r i v i a l
a r g u m e n t w e can r e p l a c e hensel
example here
case.).
are local
since
spectrum
of a s t r i c t implies
of V.
Consider symmetric each ~
the n - f o l d p r o d u c t ~ = U X U x ... X U. V V V group S acts on ~ by permutation of factors. n
e Sn,
the
fixed p o i n t
which we denote U
locus
, is d e f i n e d b y the c a r t e s i a n
by the c a r t e s i a n
diagram
~
l X ~
V
The For
of the a c t i o n of o on ~
~->UxU or e q u i v a l e n t l y
on V
in any local
the h e n s e l p r o p e r t y
sum of n c o p i e s
B u t in
diagram
IV. 2
187
S i n c e ~ ~ V is e t a l e Hence ~
is a c o m p o n e n t
Notation:
U1 = ~
and U 1 ~ V is etale, the
action
stable)
in
remark
2.1
Intuitively, u i £ U,
fact
quotient
separated,
-
~ ~6S
U
.
a component
of ~,
n
freely
of U 1 b y Sn.
n(ui)
U 1 is t h e n
U 1 is s t a b l e
to r e d u c e
u i ~ uj,
and c l o s e d .
of d e g r e e
Sn a c t s
U 1 is the
A is o p e n
of ~.
finite
o f Sn on ~,
and
geometric use
and
and
(since on U I.
To
to the
n!
In
its c o m p l e m e n t
is
Indeed,
U 1 ~ V is a
see all of this,
trivial
case,
set o f all n - t u p l e s = ~(uj),
surjective.
where
where
we
can
it is easy.
( U l , U 2 , . . . , u n) ,
~ denotes
the o r i g i n a l
m a p U ~ V. Finally
we note
that
there
is a c a r t e s i a n
diagram
U U 1 ~-----------~U n
rT1 U1 ~ _ ~ ' ~
with
~'~i
of U 1 w i t h map which
the p r o j e c t i o n
maps~
~ the d i s j o i n t on
the i
th
V
copy
U 1 a disjoint n sum of the i d e n t i t y
(Again,
using
2.2,
this
maps,
and ~ the
of U 1 is the c o m p o s i t e :
UIC--J2 ~ = U × U × U V
sum of n c o p i e s
... × U
V is c l e a r
th ~.-
P r ° 3~e c t i ° n ~
V in the
trivial
case.)
U
188
IV. 2
2.3:
Symmetric
Powers
Definition: (over S D e C The
Z) .
symmetric
Let
group
exists,
n-fold
P be
L e t pn d e n o t e S
n
also quasi-pro3ective tient
of P r o j e c t i v e
which
acts
call of P,
F o r ~ e Sn,
let J
+ P
_ J
pn
its o p e n
power
by p e r m u t i n g 1.5,
~ : p n ~ snp.
power
= pn
the n - f o l d
on
symmetric
and K
a quasi-projective
so b y P r o p o s i t i o n we
Spaces
n
and
snp
is a g a i n
be the
clearly
stable
of K u n d e r
S
n
n
u n d e r t h e a c t i o n o f Sn.) T h e n by p r o p o s i t i o n
.
1.5,
P
is c a l l e d
n
is
quo-
the
quasiprojective.
Let K =
i s an open s u b s p a c e o f pn on which S
... × P.
a geometric
~ K
P × P × factors.
fixed point
complement.
scheme
acts
locus ~ a ES
of ~,
K
.
Then
n
freely.
(K i s
L e t L be t h e q u o t i e n t there
is a c a r t e s i a n
diagram K
c~-------~> pn
~
~n
L ~__--------~ S p
Since
S
n
acts
freely
2.4~Combinin@ map
of d e g r e e
above
so ~ i : U l
group
Sn.
on K, K + L is etale.
2.1 and
n of a f f i n e
2.3):
algebraic
~ V is a g e o m e t r i c
Sn acts
freely
quotient
of the e t a l e
(Example
1.2) .
L e t ~ : U ~ V be spaces
quotient
on U 1 so V can be
equivalence
relation
a finite
and U 1 b e
as
of U 1 u n d e r identified Sn × U 1 ~3
etale
the as the U I"
IV. 2
189
Let Then
U ~ P be
there Let
is a n
the
action
the
etale
above of
S
n
immersion
immersion
K ~ pn be
(as d e f i n e d
the
.
S
U1
Hence
the
0
pn
defines
n
--~-?p
n
is c a r t e s i a n ,
UI ~
freely S
n
immersion
on w h i c h the
on K × K
S
~ - >
K
and
~ )
K
so L
is
i is
an
i
;, P
n
V
K ~__---~---) L
>V
~ - - ) L
square
i
K
immersion,
acts
freely
of K u n d e r
the quotient
UI
b
P.
U 1 + pn.
quotient
U 1 .----~ K ...
each
n
space
of
7-~_~ L.
UI
Since
L X
an
of pn
factors
o)
J : V + L.
Sn × U I
S
so
a projective
diagram
X K
a map
into
and L be
relation
Sn × U 1
S
~ ~ pn,
B)
acts
n
of U
subspace
in p a r t
equivalence
Clearly,
sion.
an
the
square
with
i
an
immer-
IV. 2
790
is c a r t e s i a n
and j is an i m m e r s i o n .
subspaces.)
Combining
this
K --
v r,
is cartesian,
we
.
result with ~
.
.
.
a cartesian
i
V
~ snp
j an i m m e r s i o n . 2.5:
Arbitrary Finally, redone
taking
above,
to A l g e b r a i c
not that this all spaces
entire
to b e
separated
to m e a n
affine
quasiprojective
over ~,
etc.
o v e r S,
application
of a q u a s i p r o j e c t i v e page space
129.
Recall
and x ~ X
in p a r t i c u l a r ,
Spaces
section
algebraic
Another
and,
Over
an
Base
over a noetherian base
2,6:
diagram
P
as i d e n t i f i e d
Generalization
fact that
n .>
with
for
>snp
u I ......
all the m a p s
the
theory
pn
finally have
with
(By d e s c e n t
variety
this
and of finite
space S,
a p o i n t of X.
interpreting
to m e a n
of the n o t i o n
is the p r o o f
theorem
2 can b e
states:
affine
quasiprojective
of s y m m e t r i c
of T h e o r e m
Let X b e
Then there
type
power
II.6.4,
an a l g e b r a i c
is an a f f i n e
scheme U
IV. 2
and
191
an e t a l e m a p Proof:
~
Given
and W--~W' W~-~X
U
such that
an e t a l e
an a f f i n e
open
is q u a s i a f f i n e
1.5.12)
and
etale
covering
y
finite
etale
extension
separable W''
assume
J
k(y)
exists
the n-fold
since W/X
a map y ~ y y
5- ~ W n,
is a l s o
symmetric
- - ~ W and w e
y is a G - s p a c e it f a c t o r s
through y
L/k(y),
normal
of d e g r e e there
n,
say.
is an a f f i n e
diagram
SGA
I).
over k(x).
Let
group. product
is q u a s i a f f i n e .
~ -_ <~,j
applying
diagram
(applying
--:W
G = [~,j .... , ~,] b e t h e G a l o i s Form
Then
L - --~ W''
t
can
is,
f~[eld e x t e n s i o n ,
extension
y we
y.
X
of W a n d a c a r t e s i a n Spec
Hence
containing
J
k ( y ) L - - k (x) a s e p a r a b l e any
let y =W;l(x)
) W
x ~ . ~
Given
of W'
x---~ U - - ~ X .
of X,
each map W XW-~W X is a commutative
i with
factors
W' ~ ' J X
subscheme
(since
and there
x--'~X
of W/X,
S y m m xn (W), w h i c h
For e a c h m a p
let y - - ~
Symm
(W) b e
~£ in G,
there
the m a p g i v e n
is
by
. ,~-.>. S y m m xn( W ) is G - i n v a r i a n t ,
and t h e m a p y - - j the q u o t i e n t ? x
S y m m xn( W ) c o n t a i n s
k(x')
= k(x).
:
~ S y m m n (w)
x
Hence
y/G=x
)
a point
X
x' w i t h
~(x')
= x,
and
so
IV. 3
192
Now we claim is l o c a l hence
W
S y m m ~)( W n ) ~_
on X, w e c a n
assume
is a s u m of n c o p i e s
a s u m of n of f a c t o r s of c o p i e s
n
S y m m xn( W )
3.
we
Theorem
Thus
at x'
X is a s t r i c t of Xo
on w h i c h
freely.
Since
the construction
Hensel
local
ring
and
Then
W ~ W~ . ~W is k X X t h e g r o u p of p e r m u t a t i o n s
the q u o t i e n t
Symm~(W)
is a sum
o v e r X.
t a k e U to b e
containing
Chow's
morphism
of X,
of X so e t a l e
Finally,
exists
X is e t a l e
the p o i n t
an a f f i n e
open
subspace
x'
of 1
Lemma 3.1:
Let
of s e p a r a t e d
an a l g e b r a i c
f:E ---~Y b e
noetherian
a separated
algebraic
finite
spaces.
s p a c e ~ and a m a p g : ~ E
such
Then that
type there
IV. 3
and
193
i)
G : V ~ E is p r o j e c t i v e
2)
f g : V + E ~ Y is q u a s i p r o j e c t i v e .
Proof:
The proof
reductions. there
are
has been from
Then
done
II.8.5,
is a b i t
facts
there
II.
of the Chow
Reduction:
We
lemma can
appropriate map
V
l
If
~ E. s a t i s f y i n g l
V. ~ E w i l l l Second
Reduction:
the convention projective
satisfy
that
means
choose
an a f f i n e
X ~ P,
all
covering
mention
Construction: an a f f i n e dense
etale.
open
can
projective
noetherian subspace
in E G A
of t h e
E. w e l
etc.
can
find then
affine
(Indeed,
a projective
is l o c a l
on Y,
an e t a l e
s p a c e X.
base
an the
By
take V
immersion so w e
are
just
scheme.) of E by
1.5.20, finite
affine.
o v e r Y,
once we
covering
V of E with U = V × E ~ V X can
map
W e do t h i s b y m a k i n g
Let X ~ E be
so w e
By
for E.
noetherian
E is n o e t h e r i a n
is a d a p t e d
irreducible
for E., l
of an a f f i n e
algebraic
facts
and b i r a t i o n a l
affine means
X + E and
easy
finally
irreducible.
union
i g n o r e Y.
o v e r Y,
and
two
II.5.6.
the t h e o r e m
the p r o o f
involves
of the s e c o n d
irreducible
theorem
surjective)
first of these
surjective,
from n o w on,
the r e s t o f
suppressing
the
We
The
disjoint
for e a c h
(hence
of c o n s t r u c t i o n
t a k e E to b e
is a p r o j e c t i v e ,
of E.
first
The proof
W + E with W = [JE. a finite l components
lemma
to p r o v e .
in C h a p t e r
the p r o o f First
of Chow's
there
two h a r d
and b i r a t i o n a l
there
is a
and
E is i r r e -
IV. 3
194
ducible
so U + V h a s
an o p e n
immersion
immersion
The
a projective
Section
is
an i n d u c e d space,
n.
L e t X ~ P be
s p a c e P. immersion
is the n - f o l d
Then
the
V ~ snp. symmetric
power
2.)
immersion
V ~ E × snp
degree
of X in a p r o j e c t i v e
U + X + P gives
(Where snp, of P - - s e e
a well-defined
V + snp
factors
the p r o d u c t
and
the m a p E X snp + snp
the
geometric closure
of the
V ~ E X snp + snp w h e r e immersions,
is the
so an i m m e r s i o n ,
second projection.
of V in E × snp.
L e t V be
Then we have
a string
of m a p s V ~
We
now
take
V ----~ E × s n p
the d e s i r e d
projective closed
since
projective snp
E
birational
is p r o j e c t i v e
the b i r a t i o n a l i t y ,
V
~--~
v.~
(?)
~
~
s i n c e V × snp + E × snp
the c l o s u r e since
E x snp ~ >
cover
of E.
and V ~ E x snp
It is is a
immersion.
To c h e c k
where,
(*)
the c o m p o s i t e
~>
to be
~snp
of V
in V × snp.
consider
the c a r t e s i a n
n . . . . . -~2 V x S P
--
-> E x S P - - ,
is e t a l e
But V is c l o s e d
~ V
-~
surjective,
squares
E
(?)
in V × snp
is
IV. 3
195
V
-~ V × snp
snp
is cartesian
A
~
snp
snp
and snp is separated.
Hence V = V × V so V + E
is birational. The hard part of the proof now comes in showing is quasi-projective.
We do this by showing
that
that the map
~ snp is quasiaffine. By II.6.15., it is sufficient separated,
of finite presentation,
is clearly
separated,
and quasifinite.
of finite p r e s e n t a t i o n
so it is sufficient
to show that
topological
image
inverse
to show that V ~ snp is
and quasicompact,
for any point q E snp,
]ql = IF ~ q] is a discrete snp
To show this, we go back to the string of maps the pullback
U1 ~ )
of everything
U1 ~
~ ~
i V
The map
by the geometric
pn
>
E ×
the space.
(*) and take
quotient pn + snp:
pn
> P
n
sn P .
----7 V ~
2 E x snp ~
"7 S nP
where we have used 2C to identify U 1 =
the algebraic × pn snp
_
~E
closure ×
pn
pn. U1 is here V ~ snp pn. of the immersion U 1 ~ E × Since
is a closed
subspace
containing
UI,
it
IV. 3
196
contains UI"
U1 m a y be identical with V
~ pn --this is not snp
c l e a r - - b u t b y 1.7 it is c l e a r that the map U1 ~ ~ an i s o m o r p h i s m
of the u n d e r l y i n g
topological
spaces.
W e wish to show that the map V ~ snp has fibers.
Since pn ~ snp is finite, × pn
~ pn is snp
finite d i s c r e t e
and
-~ pn
snp
•t .....
is cartesian,
it is s u f f i c i e n t
finite d i s c r e t e
fibers.
_>
l
snp
to show that V
Since U1 + ~ ~ pn gives an i s o m o r snp
p h i s m of the u n d e r l y i n g
topological
spaces,
to show that U1 + pn has finite d i s c r e t e done once we have p r o v e d Lemma:
~ pn ~ pn has snp
the following
it is s u f f i c i e n t
fibers.
H e n c e w e are
lemma:
U1 + pn is an immersion.
Notation:
We have the open immersion X ~ P.
the p r o d u c t P × P ×
Let P
... × X × ... × P, with all P's,
for an X in the i th slot. and the d i s j o i n t u n i o n
~
. be 1
except
Then pn i ~ pn is an open immersion Pn
i
n
l
+ P n is a union of open subsets
of pn. Claim:
~] p n i
The image of U1 + pn lies inside the image of
~ pn. 1 F i r s t let's see why the truth of this claim will p r o v e
the lemma.
IV. 3
197
The image o f ~ it
is
P n, ~ pn is open, and if the claim is true, l i to check, for each i = 1,2,...,n, letting Uli
sufficient
be the product U1 p>~n P n i '
-
that
n
Uli
~ P i
is
a subspace.
Thus
we can take the string of maps
U1 ~ U1 ~ E × pn ~ pn
and replace
it by
Ul ~ Uli ~ E × pn
+ pn
l
1
where Uli is the closure of U 1 in E × pn .l
(Note the image of
U1 ~ pn lies inside pn. since U 1 lies inside X n = I ~ P n .) l i l Consider
now the map pn . . . .~i l
~ E which
tion followed by the covering X ~ E. graph pn. -----~ E × pn. is a closed l l U 1 ~ E × pn. lies inside l is a commutative
this closed
Hence
~ Q
Since E is separated, immersion. subspace.
the
The subspace I.e.,
there
diagram:
P
U1
is the i th projec-
Uli~?
n
E × pn
l
-~
pn
l
the closure Uli of U 1 in E X Pni factors Uli- ~ Pni ~ E × ~ni
IV. 3
198
so the c o m p o s i t e U-l i ~ p n i is a subspace. Hence,
r e t r a c i n g our steps, U1 ~ pn is a subspace
so
~ snp is quasiaffine. P r o o f of Claim:
We have a diagram
LiP n l i
n 9 E × P-
Ul ---->Ul-
Consider
pn
7T~
also the d i a g r a m
E
UI ----) U1 -----) E x pn
X
W e now combine
these two and form lots of c a r t e s i a n
squares:
/llo pn
•
[3 U 1 i
~
~)
~
UI
i
,I
UI •
____-~)
T
/.-J u 1 i
~-j E × pn. l i
~>
•
~I
01
--9 ~J Ul i
~'
(
_~
~
.i T
E×P
i
pn
n /
~2
rt 1
S E
X X pn
X
IV. 3
199
In i d e n t i f y i n g
the v a r i o u s
pullbacks
Y + Y ~ Z to d e n o t e
the c l o s u r e
the r i g h t
are etale,
Finally, l-J U I, i
identification
maps
The
pn,
the
the d i s j o i n t
Several
image
hand maps
are
claim
of ~: ~ i
by open
of UI'
labeled
closure
is that
the
pn • ~ pn. l
image
to p r o v e
the
of ~ 2 ~ : U I
e is a c o v e r i n g
so ~ is an o p e n the c l a i m
with
left h a n d
of U 1 is
facilitate
the n o t a t i o n
Y ~ Z.
commutes
of the l o w e r
to
used
of an i m m e r s i o n
sum of n c o p i e s
subspaces,
Hence,
we have
from
pullback.
corner
as
section
2.
following
discussion.
~ pn
inside
lies
of s o m e
covering
we need
Since
only
subspace
the of
of s o m e p a r t show
that ~ is
surjective. N o w ~ : X ~ E is an e t a l e ing, be
hence
surjective.
sufficient
Unfortunately, slight
to
To p r o v e
find
such
modification
covering,
a map
this
t h a t ~ is s u r j e c t i v e ,
~: ~ U i
a ~ seems
so ~ is an e t a l e
1 ~
hJ U 1 such i
impossible
to find.
coverit w o u l d
that ~
= 11,.
But with
a
idea works.
th ~8 : lJ U 1 ~ X × pn is an i m m e r s i o n and on the i i s u m m a n d e S i : U 1 ~ X × pn is the p r o d u c t of th pn UlC-> U × U × ... x U i and the i m m e r s i o n U I -> The map
•
= pn Let U 1 ~ X X denote Then
there
Pr°3ecti°~X~~
the c l o s u r e
is a c o m m u t a t i v e
diagram
of this
i th i m m e r s i o n
~8 i.
IV. 3
200
L/ U 1 i
6
D
LJUI
¢
_~XXp
n
i
w h e r e ~ is surjective. Now let's rewrite out big d i a g r a m
inserting
the map 0:
] 7 2 / 3 ~i Pni
~ u I ------) LJ~ l l
l
l
-------~ hJ E >< p n l i P
"~..5 ~iUl
-) ~ u1 I --
n
E
~D X
Now we can use our idea--to $~
(which is surjective)
showing that,
i
show ~ is s u r j e c t i v e w e show that
factors through ~.
for each i = 1,2,...,n,
~. maps to the i th copy of U1 in i
the i
~ UI" i
We do this b y th
copy of U 1 in
Fix i then,
and
IV. 3
201
write Uli and Uli
for these i th copies.
_
Since Uli = (UI) ~n sufficient
to give a map Uli ~ Uli,
pn
l" such that -Uli
1 n ----~ P i
l~1
rt2"Y -2~ pn
Since ~. is an open immersion, l
is to show that the image of ~ 2 Y ~ i : U l i
~
what we need to do pn
To see this we will prove that the following
--Uli
~2~i
lies inside diagram
X
--
and P is separated.
. l
pn ----~
injection
of separated Thus to show
(**)
~ p
Note now that the injection U 1 ~ Uli is a categorical phism in the category
pn
commutes
projection
to show
it is
to take the given map @~i:Uli ~ U1 and to find a
new map X:Uli ~
commutes.
(Pni)'
algebraic
spaces
(**) commutes,
epimor(II.5.8).
it is sufficient
IV. 4
202
Ul
injection
1
1
p r o jection
pro j ection \ X
is a c o m m u t a t i v e hence
4.
diagram,
-~ p
which
is clear.
spaces
4.1:
(The F i n i t e n e s s
and F a c o h e r e n t
Theorem.
Cf.
of n o e t h e r i a n
sheaf
on X.
Oy-modules.
Proof:
the set of all c o h e r e n t
Clearly
for all q "~ 0, is exact
it is s u f f i c i e n t
that there Rqf,(F)
is a c o h e r e n t
coherent
integral,
By Chow's
w i t h ~ surjective,
~
0.
of devissage.
integral
W + X,
Thus we can assume X is
is to find a c o h e r e n t
and Rqf,(F)
Lemma,
alge-
F, w i t h Rqf,(F)
in the sense
for every
separated
sheaf F on X, with W = Supp F, and
for all q
and the p r o b l e m
Supp F = X,
position
to show,
[EGA 3.2.1])
Then Rqf,(F),
0, are c o h e r e n t
coherent Hence
the claim,
Theorem
f:X ~ Y be a proper m o r p h i s m
q ~
Hence
l
The F i n i t e n e s s
braic
with
injection
the theorem,
Theorem Lef
---> pn
there
coherent
for all q
is an a l g e b r a i c
projective
and b i r a t i o n a l
h.-X' ~ X ~ Y projective.
sheaf F on X, 0.
space X ' and w i t h
~
X
the com-
Since X is integral,
we
IV.4
203
can assume X' is integral. large n (where~x,(1)
is an ample sheaf relative to ~:X' + X).
Since ~ is projective, n * ~ . ( ~ X , (n)) ~
Let F = ~. ~X, (n) for some very
F is coherent.
Also
~X' (n) is surjective since n is large.
Since
the generic point of X' is mapped onto the generic point x e X. Fx ~ 0.
Also h:X' ~ Y is projective
so R q h . ( ~ x, (n)) is coherent
for q ~
0. (We are here applying II.7.11.)
There is a spectral sequence
E2Pq = RPf.(Rqn.( O X, ( n ) ) ) ~
RP+qh.(O~x(n))
Since n is large. Rq~.( ~X' (n)) = 0 for q > 0 so
R P f * ( ~ * ( ~ X ' (n))) = R P h . ( ~ X , (n))
or RPf. (F) = RPh* ( ~ X ' (n)) . all p
~ 0.
Hence RPf. (F) are coherent for
CHAPTER
FIVE
FORMAL ALGEBRAIC
i.
Affine Formal
2.
Formal Algebraic
3.
The Theorem
4.
Applications
5.
Completions
6.
The Grothendieck
1.
Affine Formal
Schemes ..................................
204
Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
215
Functions ...................
224
to P r o p e r M o r p h i s m s . . . . . . . . . . . . . . . . . . . . . . .
255
of H o l o m o r p h i c
of M o d u l e s
Throughout
SPACES
of H o m o m o r p h i s m s
Existence
................
241
Theorem .....................
245
Schemes
this chapter,
all rings w i l l
be assumed
to be
Noetherian. Definition ideal
I, wuch
is s e p a r a t e d where
i.I:
that
Any
ideal
the d i s c r e t e open
I is c a l l e d
two ideals
ring R is c a l l e d
on R is the I - a d i c
in this
I n is b o t h
such ideal
I 1 and 12 are
the t o p o l o g y
and c o m p l e t e
each R / i n has
that each
A topological
topology. topology.
(I.e.,
adic
if R has
topology,
and R
Lim R = ~-~-- R / i n ,
In p a r t i c u l a r
this m e a n s
and closed~ an ideal of d e f i n i t i o n
of d e f i n i t i o n
of R,
an
of R.
If
there m u s t be i n t e g e r s
V. I
205
n 1 and n 2 w i t h (R,I)
Ilnl c 12 and I2n2 c__ I I.
to specify
is t r i v i a l l y
a particular
adic
(of discrete)
Since R is a n o ~ t h e r i a n of d e f i n i t i o n radical
I.
of J.)
f: (R,I) f-l(j)
~
(S,J)
of definition.
adic ring,
R has
largest
is a c o n t i n u o u s
is c o n t i n u o u s
write
An adic
a unique
of d e f i n i t i o n
the n th truncation
I is the u n i q u e
A map of adic rings
sometimes
ring
if I is nilpotent.
(For any ideal W e define
ring R/in , w h e r e
ideal
We will
iff there
largest
ideal
J of R, I is the
of R to be the d i s c r e t e
ideal
of d e f i n i t i o n
ring h o m o m o r p h i s m . is an integer
of R. (Note
n with
D_ I n .)
Let
(R,I)
be an adic ring
if M =~i___m M / i n ~. n P r o p o s i t i o n 1.2: (Krull)
and M an R-module.
M is a c o n t i n u o u s
R-module
as always)
and M
Proof: A map for every)
of finite
type.
(noetherian
T h e n M is continuous.
of d e f i n i t i o n
is an adic map
if for some
I of R, the S-ideal
f(I).S
(hence
is an ideal
of S.
Proposition
1.3:
Let
f:R ~ S be an adic map.
Let M be any S-module.
M is also c o n t i n u o u s
Then
S is a c o n t i n u o u s
Then M can be c o n s i d e r e d
(by r.m = f(r) .m, r £ R, m £ M)
an S-module,
be an adic ring
I
f:R ~ S of adic rings
of d e f i n i t i o n
R-module
(R,I)
See EGA 0 1 . 7 . 3 . 3
ideal
R-module.
an R - m o d u l e
Let
as an
and if M is c o n t i n u o u s
as an R-module.
as l
V. 1
206
Let
(R,I)
be an adic ring and M an R-module.
of M is d e f i n i e d Proposition
by ~ = L i m M/InM"
1.4:
With
the above
(i)
~ is a c o n t i n u o u s
(2)
M = ~ iff M is a c o n t i n u o u s
(3)
A is a right R-modules,
Proof: The
exact
functor
left adjoint
(i) and
(continuous of
consequences
1.5:
of the definitions.
for all R - m o d u l e s
N and all
= HornR(~,M)
is clear.
By EGA O i . 7 . 3 . 3 , A is left exact
The c o m p l e t e
functor
M
equivalence
Definition
to c o n t i n u o u s
A is left exact on m o d u l e s
is s p e c i f i c a l l y :
R-modules
natural
from R - m o d u l e s
to the inclusion
(2) are simple
HornR(N,M) This
R-module.
type.
last a s s e r t i o n
continuous
notation:
R-module
R-modules)£--~ (R-modules). finite
The completion
Let
(R,I)
tensor p r o d u c t
Hence
the right
on m o d u l e s
exactness
of finite
be an adic ring
of M and N, d e n o t e d
of A
type.
I
and M and N R-modules. M ~ N is the R
completion
of the usual
tensor p r o d u c t :
M ~ N = M ~ N. R
The c o m p l e t e the usual
module
hom-set:
O f c~urse,
of h o m o m o r p h i s m s
Hom~(M,N)
R
is the a o m p l e t i o n
of
= H O m R (M,N) •
if M and N are R - m o d u l e s
M ® N and H o m R ( M , N ) so these m o d u l e s
of finite
are a l r e a d y
type,
so are
complete.
R
Definition
1.6:
Let
f:R ~ S be an adic map.
We
say
f is o f
V. 1
207
finite
type if for some
the a s s o c i a t e d finite
(hence
for every)
ideal
m a p R/I ~ S/I S i d e n t i f i e s
of d e f i n i t i o n
S/I S as an algebra
I of R, of
type over R/I.
Proposition f is of finite
1.7:
Let
type.
f:R ~ S and g:R ~ T be adic maps.
Then S ~ T is a n o e t h e r i a n
Suppose
adic r i n g and
the
R
map T ~ S ~
is adic of finite
type.
R
Proof:
This
Definition (resp.
is a c o r o l l a r y
1.8:
faithfully
of E G A 01.7.5.5.
L e t F : R + S be a map of adic
flat)
rings,
f is flat
if the functor
e S: (R-modules)
~
(S-modules)
R
is exact
(resp.
As usual,
exact
and
to check
faithful).
that
f is flat it is s u f f i c i e n t
to look
at the r e s t r i c t i o n ® S:(R-modules
of finite
type)
~
(S-modules
of finite
type).
R
Thus
it is s u f f i c i e n t
to show
A e S : (continuous
that
R-modules)
~
(continuous
S-modules)
R
is exact
(or exact
Proposition adic rings,
then
and
1.9:
faithful). If R ~ S is an adic
for every R - m o d u l e M ~ S ~ M
~
R
is exact
and we h a v e
(cf. 1.3.2 Proof:
the usual
faithfully
M of finite
type,
flat m a p o f the s e q u e n c e
S ~ S ~ M R
R
descent
theory
for such modules.
) SGA VIII,
I
V. 1
208
Definition etale
(resp.
I of R,
i. I0:
a formal
the induced
in the usual
For
f:R + S of adic rings
if there
exists
given
correspondence
etale
surjective)
is formally
of d e f i n i t i o n
etale
between
formally
there
of R.
(2)
(discrete)
rings
etale
then
and I is the m a x i m a l
f(I)'S
is the m a x i m a l
ideal
ideal
of
are c o r o l l a r i e s
of S G A 1.8.3.
(3) is a I
1.12:
f:R ~ S be a formally
Then
f is flat.
also
faithfully
Let
is
etale m a p s R ~ S
of 1.4.9(2).
Let
that
of S.
(i) and
Proposition
if and
I of R such
I of an adic ring R,
and etale m a p s R/I ~ T of
If f:R ~ S is formally
definition
(2)
of d e f i n i t i o n
by T = S/I s .
of d e f i n i t i o n
(i)
an ideal
of d e f i n i t i o n
of adic rings,
corollary
(resp.
ideal
is etale.
any ideal
a one-one
Proof:
if for every
is formally
i.ii:
R/I ~ S/IS
(3)
covering)
map R/I + S/I S is etale
An adic map only
(2)
etale
f:R ~ S of adic rings
sense.
Proposition (I)
An adic map
etale map of adic rings.
f is formally
etale
surjective
iff
f is
of adic rings.
Let
flat.
f:R ~ S and g:R + T be adic m a p s
h : R ~ S × T be the induced
m a p of R into
the c a r t e s i a n
V. 1
209
product etale
(3)
of S and T.
iff both
A formally
and h is formally
f and g are.
etale
universally
Then h is adic,
surjective
effective
map
of adic rings
epimorphism
is a
in the c a t e g o r y
of adic
rings. (4)
Let
R
s
T
be a c o m m u t a t i v e etale.
diagram
of adic rings with
If if is formally
formally
etale.
etale
surjective,
If g is formally
etale,
h formally then
g is
so is f.
Proof: (i)
For
an R - m o d u l e
M of finite
type, M ~ S is c o m p l e t e
so
R M ® S = Lim((M
R
n
R
Completion (2)
This
® R/in
and
This proof
~, S/in S are b o t h R/in
exact.
is the axiom S 1 of I.i.19
is a n a l o g o u s of the
relation (4)
This
~
S/InS ) .
R/in
and is clear
from
the truth
(I.4.5).
to axiom S 2 of I.i.19
fact for etale maps
(I.4.5)
and
follows
from the
and the a d j o i n t n e s s
1.4 above.
is axiom S 3 of I.i.19,
statement Definition
S/InS ) = L i m ( M
n
of S 1 for etale maps (3)
~
R/in
for etale maps
1.13:
Given
again
following
from the similar
1.4.5.
arJy ring R and ideal
l I of R, we can
form
V.l
210
A
the c o m p l e t i o n ----°f R along I, ~I = ~nnLim R / I n R , which is then an adic ring w i t h ideal of d e f i n i t i o n I.
If R is noetherian,
so is R I.
(EGA 01.7.2.6) The c o n s t r u c t i o n property: rings
of c o m p l e t i o n s
has the following
functorial
there is a map i:R ~ ~I of rings such that for any map of
j:R + S w h e r e S is adic w i t h
ideal of d e f i n i t i o n J and w h e r e
for some integer n, j-l(j) D__ I n , there is a unique c o n t i n u o u s map f:~I + S with
fi = j.
of R and J = f(I)'S,
If f:R ~ S is any map of r i n g s , I an ideal then there is a canonical
induced map of
c o m p l e t i o n s w h i c h w e d e n o t e ~:~I ~ ~J" Proposition
1.14:
Let f:R ~ S be an etale map of n o e t h e r i a n
rings and I an ideal of R.
Let J = f(I)"S.
~I ~ ~" is formally etale. 3
A l s o ~j = S ~ ~I" R
Suppose
further
Then b y 1.3.1a,
Then the induced map
that f:R ~ S is etale and faithfully
flat.
there is an exact sequence R ~ S - ~ S ® S. But now R
we also have an exact sequence R I ~ S j ~
Sj
~ Sj.
(I.e.,
"completion
I along an ideal is local in the etale topology".) Proof:
The first a s s e r t i o n
is clear.
The second
the fact that R I ~ Sj is formally etale s u r j e c t i v e plus L e t R be a n o e t h e r i a n of R.
M then induces Proposition
1.15:
ring and M an R-module.
follows
1.12(3).I
Let I be an ideal
an R i - m o d u l e , ~ ~
the notation
above,
if M is an R - m o d u l e
of finite type the canonical map M e ~I ~ ~ is an isomorphism. R
Furthermore,
from
~I is a flat R-module.
V. 1
211
Proof:
EGA 01.7.3.3
Definition dual
1.16:
of the c a t e g o r y
write
l
The c a t e g o r y of adic rings.
S p f R for the a s s o c i a t e d
is an affine write
~n
affine
formal
scheme
= Spf Rn,
the t h
formal
the first
Since are dual, finite
schemes,
truncation
(or carrier.)
of ~
of affine Given
affine
and R
n
now on, d e f i n i t i o n s
carriers
.
If ~ = Spf R of R, we
In the c a t e g o r y
will
be e x t e n d e d
resp.
formal an open
is an i m b e d d i n g
and affine
formal
if the a s s o c i a t e d
An adic m a p of affine
of their
of {
of adic rings
in this w a y w i t h o u t
imbedding,
scheme.
of
.
or flat,
a closed
formal
truncation
is the
an adic ring R, w e
is the n th t r u n c a t i o n
w e can say a m a p of affine
two c a t e g o r i e s
schemes
~ = L--> i m ~ n" G i v e n an affine formal scheme n ~ i w i l l be called the a s s o c i a t e d closed s u b s c h e m e
the c a t e g o r i e s
type,
formal
schemes
map
back
schemes
of rings
and
further
schemes
is etale,
or of
is such.
From
forth b e t w e e n
explicit
these
mention.
is an i m b e d d i n g
imbedding) (resp.
formal
(resp.
if the a s s o c i a t e d
closed
imbedding,
map
resp.
an open imbedding). The Global
etale
is the G r o t h e n d i e c k etale maps.
topology
("Associated"
The Local the r e s t r i c t i o n formally
topology
etale
topology
over
{ .
associated
with
of affine
formal
the s u b c a t e g o r y
schemes
of formal
in the sense of I.i.16)
of the global
etale
on the c a t e g o r y
on a given etale
affine
topology
formal
to affine
scheme ~ is formal
schemes
V.I
212
A sheaf
(of sets,
a contravariant ) + make
(Sets)
an extra
functor axiom
case
functor
satisfying
scheme
~
Proposition
(2)
Every
schemes)
is the u s u a l - -
say)
of continuity:
etale o v e r
sheaves,
we
is a c o n t r a v a r i a n t
satisfying
) = Lira F ( ~ n ) . A m a p n t r a n s f o r m a t i o n of functors.
the sheaf
For every
of sheaves
affine
in either
1.17:
formal
schemes
Let
be an affine
sheaf.
(Sets)
~
formally
For global
(of sets,
~
an asiom
representable
~
axiom.
A sheaf
scheme
schemes
F( ~
is a natural
(1)
formal
formal
formal
the sheaf
restriction:
F:(Affine
on an affine
F:(affine
satisfying
and also
formal
say)
Let
truncations
functor
is a global
of 5 •
in the c a t e g o r y
scheme
sheaves
Then
of affine
sheaf.
formal
~" be the n
on the c a t e g o r y
and
~'the
associated
~=
of sheaves
associated
to the n
th
Lim ( ~ n ) n of sets.
- the direct
limit
Continuity
is i m m e d i a t e
from
Proof: (i)
The sheaf
axiom
is 1.12(3).
the definitions. (2)
is p r e c i s e l y on global imply ~ndeed
sheaves.
that, take
to assert
the c o n s e q u e n c e Note
for an affine ~
that
= ~
.
of the c o n t i n u i t y
that
the second
formal
In fact,
scheme
to assert
the set of sections
requirement
assertion
does not
,
) =-->Lim ~ n ( n this would be
of a direct
limit
of
).
V. 1
213
sheaves is the same as the direct limit of the sets of sections. 1.18:
Let
~=
In general this is not true. Spf R be an affine formal scheme.
map i: ~ i + ~
induces
topology on ~
and the etale topology on
i*:
l
(by 1.11(2))
(Abelian sheaves on ~ )
The natural
an isomorphism of the local etale ~i"
Thus the functor
~ (Abelian sheaves on 21)
is an
equivalence of categories. Note that the local topology on each affine formal scheme is noetherian in the sense of GT. The structure sheaf
~of
adic rings on the category of affine
formal schemes is the functor F( ~, ~ ) a sheaf by 1.14.
We write
~
~c
~=
Spf R.
This is
for the restriction of this sheaf to
a particular affine formal scheme a global sheaf of ideals
= R if
~
.
Applying 1.11(3)• there is
~ , assigning to each ~ =
maximal ideal of definition of R.
Spf R, the
For an affine formal scheme
we have the obvious definitions of coherent and quasicoherent ~M ~modules. each
~ ~ ~
A quasicoherent
~-module
F is continuous if for
formally etale, ~ = Spf S, F ( ~ ,F) is a continuous
S-module. Proposition 1.19:
Let ~
category of quasicoherent enough injectives. Hn(~,F)
be an affine ~ormal scheme.
~-modules
is an abelian category with
If F is a quasicoherent
= 0, n ~ 0, where H n ( ~ , F )
The
~-module,
we have
is the cohomology of F as
V. 1
214
abelian
sheaf
Proof:
in the local Exactly
etale
topology.
as in the case of affine
N o t e w e do n o t r e q u i r e
that the i n j e c t i v e s
p a r t l y b e c a u s e w e do not need b e c a u s e w e do not k n o w w h e t h e r
schemes
be c o n t i n u o u s
this e x t r a r e q u i r e m e n t enough
such
(I.4.15) .
injectives
modules--
but mainly can be
found.l
•V . 2
2.
215
Formal A l g e b r a i c
Definition
2.1:
Spaces
A
(separated noetherian)
space is a c o n t r a v a r i a n t
functor F : ( A f f i n e
formal
algebraic
formal schemes)
+
(sets)
such that
(1)
F is a sheaf in the global
(2)
There is an affine ~°
formal etale topology
formal scheme
~
~ F such that for any global
an affine
formal scheme,
of global
sheaves
and a map of sheaves
sheaf G, r e p r e s e n t e d
by
and map G ~ F, the p r o d u c t G ×
is r e p r e s e n t e d b y an aff~ne
F formal scheme,
and the map G ×F ~ ~ G is r e p r e s e n t e d by a formal etale covering. (3)
Given ~
F as above,
the map of affine schemes
~IXF~I
~ Jl × ~i
is a closed immersion. A morphism
of formal
algebraic
spaces
f: ~
~
~2
is a map of
sheaves. 2.2:
At this point,
p r o v e d in C h a p t e r
II for algebraic
" n o e t h e r i a n ....separated" algebraic
we have the analogs of a number of theorems
formal algebraic
spaces are "separated"
of these analogs
spaces.
are somewhat
Since we have only defined spaces,
all maps of formal
and "quasicompact"
the v e r i f i c a t i o n s
easier than in the p r e v i o u s
case where
w e tried to b e m o r e general. In particular, affine
the formal etale t o p o l o g y on the c a t e g o r y of
formal schemes extends
to a formal etale t o p o l o g y on the
V.2
216
category of formal algebraic
spaces and we have the usual definitions
and sorites on quasicoherent
sheaves.
The category of formal
algebraic
spaces is closed under the formation of quotients
separated
formal etale equivalence
separated
if ~ i
-~ ~i × ~ i
relations
(where ~ - - - ~
is a closed immersion)
formation of fiber products
~ ×9
of is
and under the
whenever one of the maps ~-~
OC or
~+~
is of finite type.
(in the sense of EGA 1.10)
flat and finite type morphism
in the obvious way to formal algebraic
notion of an affine map generalizes in the obvious way and if ~ continuous
quasicoherent
= S pf ~ ever
~
formal scheme
"is" clearly a formal algebraic
The notions of etale, generalize
A separated noetherian
for the associated
and immersions
spaces.
The
to the notion of formal affine map
is a formal algebraic
~-algebra
space.
space,
and ~
a
of finite type, we write
formal algebraic
space formally affine
.
The construction
of the completion
along a closed subspace extends algebraic
spaces,
algebraic
space
the completion ~
Proposition
to a local construction
affine scheme in formal
,~(or ~AC ) of a noetherian
separated
along a closed subspace C.
2.3:
Given a noetherian
X and a closed subspace C = ~ adic formal algebraic The maps Spec
of a noetherian
space.
~X/I'
separated
algebraic
space
we can consider X as a trivially
Let ~ be the completion
of X along C.
~ ~ X induce a map i:X/&~ X which is adic if and ~/i n
V. 2
2~7
only if I is nilpotent. (i)
This map has the following properties:
If Y ~ X is an etale map and 9 denotes the completion of Y along the closed subspace Y ~<XC, then the map ~ ~ is formally etale and the map iy:~ ~ Y is the fiber product of the map i (in the category of formal algebraic spaces).
(2)
i,:QCS~ ~ OCS x is exact and faithful
(where as usual
QCS T is the category of quasicoherent sheaves on T). (3)
i i,:QCS~ ~ QCS~ is naturally equivalent to the identity functor,
i* is an exact functor on the subcategory of
coherent
~r'X-mOdul es
Proof:
(1) follows from the truth of the statement in the affine
case, which is 1.14.
Hence
(2) and (3) are local in the formal
etale topology and we can assume X is affine,
in which case both
are clear from 1.15 and 1.4. Definition 2.4:
I
The canonical sheaf of ideals of definition I
on the category of affine formal schemes extends to the sheaf of ideals of definition, algebraic spaces.
again denoted I, on the category of formal
The restriction,
I)£
, of I to the local formal
etale topology on a fixed formal algebraic space ~ , ideals of definition of ~ .
The n th truncation of ~
the trivially adic formal algebraic space Spf ~ / i
n.
is the sheaf of is defined as Equivalently,
if ~
is defined by the formal etale equivalence relation ~
then
~niS
the auotient of the etale equivalence relation
~-~,
V. 2
~n~
218
~n
~
~n .
Hence
Theorem 2.5:
Let ~
~
= Li~ ~ n . n be a formal algebraic
space.
Then ~
an affine formal scheme if and only if the first truncation is affine. ~ Proof: Conversely,
is a formal scheme if and only if ~ i If ~
is affine,
by III.3.3
so are each of the Definition
and III.3.6,
~n'
2.6:
or a scheme, if
A map f: ~ ~
1
is a scheme.
clearly so is
~ 1 is affine,
andhence by EGA I.i0.6,
is
so is
E 1.
or a scheme, ~
.
of formal algebraic
I
spaces is
proper if f is of finite type and the associated map of carriers fl: ~ i + ~ i
is a proper map of algebraic
Proposition
2.7:
spaces.
Let f:X ~ Y be a map of noetherian
algebraic spaces which is of finite type. subspace
and D = C XyX = f-l(c).
is proper.
Suppose
Let C + Y be a closed the associated map D + C
Then f: ~D ~ ~C is proper.
The following material
I
on graded rings and modules
Leffler condition is given here in preparationfor Functions
Theorem to be proved
minimal necessary definitions details, Chap.
separated
such as proofs,
Definition
2.8:
identity as always)
the Holomorphic
in the next section.
We give just the
and facts for our purposes.
the reader may consult Bourbaki
III, and EGA II.2 and
and Mittag-
For more Alg. Comm.,
Oiii.13.
A @raded rinq R is a ring R (commutative with given with a collection
of abelian groups such that
{~,
k = 0,1,2,...]
V.2
219
i)
R O is a r i n g
ii)
Each
iii)
For
R i is an R 0 - m o d u l e each
i,j ~ 0,
there
aij:R i ® R R0 J iv)
R =
.~ R with •~ 0 i
is an R 0 - h o m o m o r p h i s m
Ri+j
the m u l t i p l i c a t i o n
on R d e r i v e d
from
the
a.. above. ~3 Proposition associated defined
2.9:
graded
L e t R be
rin@,
gr(R),
by R. = R / i i + l R. 1
Proof:
Bourbaki
Definition M is an R - m o d u l e R0-submodules
b i j : R i ® M. + Mi+j,
Chap.
for e a c h
such
derived
from
a graded
R-module.
Then
of R.
The
= ~ R. l i so is gr(R) .
2.10,
gr(R)
Cor.
5.1
ring.
A graded
[Mk,k = 0 , 1 , 2 , . . }
R-modUle
of
an R 0 - h o m o m o r p h i s m with
the s t r u c t u r e
of M
as
b
b... 13
Suppose
integer
mn
2.12:
an R - m o d u l e
a graded
M is of
m ~ 0,
:R m ® M + M m+n R0 n S e e B o u r b a k i Alg. C o m m .
Definition L e t H be
ring
a graded
i,j ~ 0,
L e t R = ® R k be
R-module.
for a n y
the m a p
Proof:
be
that M = ® Mk,
the m a p s
2.11:
M = ~ Mk
that
I an i d e a l
k
Proposition
such
III,
a collection
R0 3 R-module
the g r a d e d
Let R = ~
and
and
If R is n o e t h e r i a n ,
M given with
of M
is
AI~ Comm
2.10:
any r i n g
there
noetherian
finite
type
exists
an i n t e g e r
is s u r j e c t i v e III.3.1,
L e t R be a n o e t h e r i a n
ring
and H = K 0 D__ K 1 D__ K 2 D__ .
as an nO
for all n > n O .
Prop. ring
and
3.
1
and I an ideal.
. be
a filtration
of
V. 2
220
H by sub-R-modules.
The filtration
[Ki} is called I-@ood
(1)
I'K
(2)
T h e r e is an integer no, with I'K n = K n+l'
c K for all j ~ 0 3 -- j+l
Proposition H an R - m o d u l e
if
2.13:
Let R be a n o e t h e r i a n
of finite type.
Let
Let K = ~i Kl, w h i c h
over the ring S ~ ~ I i. l
Then K. is an I-good l
type over S.
I an ideal and
[Ki, i ~ 0} be a filtration of H
with I'K i c K i + l , i _> 0.
if K is of finite
ring,
for all n > n o •
is a graded filtration
algebra if and only
In this case there is an i s o m o r p h i s m
~'l ~ ,Limj H / I n H ~ <----LimH / K i . Proof: Prop.
See B o u r b a k i Alg. Comm.
III.3.1,
Thm.
1 and III.3.2,
4.
l
2.14: algebraic
Let R be a n o e t h e r i a n
ring and I an ideal.
space proper over Spec R.
Let F be a c o h e r e n t
Let X be an sheaf on X.
W e w r i t e IkF for the sheaf which is the k e r n a l of the map of ~X-mOdules
F ~ F e R/ik. R
IkF is a c o h e r e n t
n ~__ 0, we can form Hn(x,IkF), (by IV.4.1, Let S +
which
the F i n i t e n e s s Theorem).
® I k where by d e f i n i t i o n k>0
(by I k. I j ~ I k+j ia 6 H n(x,Ik+jF)
~-module X
is an R - m o d u l e Consider
I 0 = R.
and for each
of finite type
the sum E =
Then S is a graded ring
and E is a graded S - m o d u l e
(by, for i e i k ,a e H n (X,I j) ,
is the image of a in the map of c o h o m o l o g y
Hn (X, IJF) + H n (x, IJ+kF)
~ Hn(x,IkF). bo
induced by the map I j
I j+k which
groups is
V. 2
221
multiplication
by i).
Proposition graded
ring
2.15:
the n o t a t i o n
and E is an S - m o d u l e
Proof:
Applying
S is noetherian. there
With
Bourbaki,
The second
for X a scheme p r o p e r
case of a l g e b r a i c
spaces,
as above,
of finite AI~.
type.
Comm.
assertion
Chap.
III,
2.10,
is E G A III.3.3.2
over Y carries
given
S is a n o e t h e r i a n
Cor.
whose proof
over w o r d - f o r - w o r d
our p r o o f of the
finiteness
to the theorem
IV.4.1.
I
Definition abelian
2.16:
category
projective each
is said
all n 6 N,
for all k 0> . k_
ML
is one
A
to satisfy
n
of objects
In the
in C,
i < j < k
the M i t t a q - L e f f l e r
set and C an
following
we c o n s i d e r
where ~ij:A. + A. for 3 i 6 N.
Such
condition,
a projective ML,
if for
is a k 0 > n E N such
that I m a g e ( ~ n k ( A k ) ) = I m a g e ( ~ n k 0 ( ~ 0 ) ~
The
of a system
for which
case we say objects
[Ai,~ij}
be an index
limits.
and ~ij ~ j k = ~ik'
there
An
L e t N = [0,1,2,...}
with projective
systems
i < j e N,
system
[Ai,~ij] involved
simplest
example
all the m a p s ~ij is strict. satisfy
[Ai,~ij]
are epimorphisms.
Another
example
the d e s c e n d i n g
chain
satisfying
In this
is ~vhen all condition
the
for
subobjects. u. Proposition sequence Suppose
5,
2.17:
of p r o j e c t i v e IAi}
satisfies
Let 0 systems ML.
+ A
l
1
v. B. i C. + 0 be an exact l 1
of abelian
Then
groups w i t h
the s e q u e n c e
index
set N.
V. 2
222
0 ~ Lim A. ~ Lira B. ~ L i m C. + 0 is exact. Proof:
E G A 0iii.13.2.2.
Proposition
2.18:
of abelian groups, of degree i.
Let
I [Ki}i£ N be a p r o j e c t i v e
K 1 = (Kn) n6N,
in w h i c h
system of c o m p l e x e s
the d e r i v a t i o n
F o r each n, there is a canonical
Ki) ~ .-Lira Hn(KI). If for each n, the system l l satisfies ML, then each h is bijective. n
Let X be a n o e t h e r i a n
and [Fk]k6 N a p r o j e c t i v e F = Lim F k. 4----
[Kn]ieN
This is a special case of EGA 0iii.13.2.3.
T h e o r e m 2.19:
Suppose
~hk are e p i m o r p h i s m s
separated
system of c o h e r e n t
the system
is
h0momorphism
h n:H n ~
Proof:
operator
I
algebraic
~X-mOdules
and
[Fk,~hk ] is strict--i.e.,
in the c a t e g o r y of <~X-mOdules.
space
Then
the maps for all
i > 0, the canonical maps h. :H i(x,F) 1
-~ Lim H i(x,Fk)
are bijective. Proof: Y.
Let Y ~ X b e an etale c o v e r i n g of X b y an affine scheme
For each c o h e r e n t
sheaf G on X, w e h a v e the Cech c o m p l e x
F(X,G) ~ r(Y,G)__--~ F(Y X Y , G ) ~
.
X K" with K n = (Y × Y × ... × Y,G), X X X
giving a complex of F(X, ~ v ) - m o d u l e s
the p r o d u c t being taken n+l times.
Since each of the Y × Y × .. × Y are affine, the c o h o m o l o g y of G X X X v a n i s h e s on them and thus the c o h o m o l o g y of G can be computed as the Cech c o h o m o l o g y
of this complex.
G i v e n now the p r o j e c t i v e
H e n c e Hn(X,G)
system
= H n ( K ") .
~Fil of sheaves,
we have a
V. 2
223
projective
system of complexes
hypothesis
that all of the maps F k ~ F h are epimorphisms
the p r o j e c t i v e
systems
the last p r o p o s i t i o n hn:Hn(Lim~ Ki) ~ i m 1
Finally
[Ki} constructed
n {Ki}ie N are strict,
we have canonical
as above.
The implies
hence satisfy ML.
that
Applying
isomorphisms
Hn(X,Fk ) . 1
it remains
to show that Hn~Lim~ K~)
= Hn(X,F).
For this
1
it
is
sufficient
is the inverse
to
show
that
the
Cech
complex
limit of the Cech complexes
K"
associated
K~ associated
to
F
to the
1
F.'s. l
Specifically
we need to show that for each n > 0, if i is -n
the map Y x Y × . . >< Y + X X X X But i
n
(n+l factors) , then Lim(i *F~) ~n K
is flat so i * is left exact and left exact n
Corollary
2.20:
Let X be a noetherian
space and C + X a closed
subspace.
defining C.
Let F be a coherent
F ~
Let ~ = ~C'
~X/~k"
Then H n(2,~) Proof:
i:~ ~ X
Let ~ c
= i *~im F k. n k
functors
separated
commute.I
algebraic
~ X be the sheaf of ideals
~X-mOdule
and write F k for
the canonical
map and ~ = i*F =
= ~Lim k ~ Hn(X,Fk ) , n > 0. By 2.3,
i,~ = Lim F k and,
since
affine, lqn(~,~) ~--Hn(X,i,@).
The system
satisfies ML.
follows
The conclusion
i:~ + X is formally
{Fk} is strict
from P r o p o s i t i o n
and hence 2.19.
I
i~L F k.
V. 3
3.
224
The Theorem
Let algebraic
of H o l o m o r p h i c
Functions
f:X ~ Y be a proper morphism spaces.
of neotherian
Let Y' ~ Y be a closed
subspace
separated and X' = Y' × X. Y
Say Y' = Spec 0y /I'
and X' = Spec 0X/j, where J' = f*I'$0X,
A can then form Y = -->LimnSpec 0y/I'n consider
the proper morphism
A and X = Limn ~ Spec 0X/j.n
we and
A ~:X ~ Y A
Let F be a coherent Consider. (i)
(Rnf. (F)) A
(2)
L i m Rnf. (Fk) ~k nA A R f. (F)
(3) There
for each n >__ 0.
0X-mOdule , Fk = F/jkF
the following
and
F = Lim Fk.
0~-modules.
are natural maps A A Pn:Rnf.(F) A ~ Rnf.(F) nA ~n :Rnf*(F) A ~ Lim R f.(Fk)
~n:R
These
are constructed
x
W algebraic
spaces with
nA A f.(F) ~ Lira Rnf.(Fk ) 4k" as follows:
_
J
Let
>Y
~?
Z
be a commutative
diagram
of
V. 3
225
i,:(0W-mOdules ) ~
exact.
(0Z-mOdules)
Then there is a natural map i* (Rnf, (F)) ~ R n g, (j* (F)) .
where ~ is the image of the canonical
injection F ~ j,j*F under
the transformation
HOrny(F. j,j*F) ~ Hom z(Rnf.F,Rnf.(j.j*F))
= H o m z (Rnf,F.R n (fj) ,j'F)
= HornZ (Rnf,F.R n (ig) ,j'F)
= HornZ (Rnf,F.i,Rng, (j'F))
= HornW(i*Rnf.F,Rng. (j'F)) . Now for each k. n xk
is cartesian where Yk = Sp~cec O Y / i k .
>x
Hence there are maps
%0n.k:Rnf,(F) k -~ Rnfk *(Fk)
V. 3
226
w h i c h by n a t u r a l i t y
are c o m p a t i b l e
n
~0n:R f.(F)
A
for d i f f e r e n t k~ giving a map
n ~ L i m R f.(F k)
~k
Similarly
the sguares
Xk
A X
~
A
J
X
)X
A Y
)Y
and
Yk
--9 Y
are c o m m u t a t i v e
giving
k, the c o m m u t a t i v e
~
Consider
now,
for each
squares
A
_------3 X
~
-
A Y
Yk ~ By the n a t u r a l i t y
the maps Pn and %n"
9X
~Y
of the construction,
Rnf,(F) A
Pn
the d i a g r a m
nA A -~ R f.(F)
Rn f k * (Fk)
(considered
as 0Q-modules)
commutes
for each ][.
H e n c e we have
V. 3
227
1]
R f, (F) A
Pn -~ R n Af, (F) A
\
5
~n
Lira Rnf, (Fk)
commuting. N o t e here that the domain and range of ~n are both t o p o l o g i z e d as inverse limite of d i s c r e t e modules. T h e o r e m 3.1: notation
above,
(The H o l o m o r p h i c
Rnf,(F)
Functions
is a c o h e r e n t
Theorem)
O~-module
for each n >_. 0
and each of the maps ~n' Pn' @n are isomorphisms. ~n is a t o p o l o g i c a l Proof: is affine. suppose Y'
With the
In particular,
isomorphism.
The entire assertion
is local on Y so we can assume Y
Say Y = Spec A, w h e r e A is a n o e t h e r i a n is defioed by an ideal I of A.
ring,
and
We w r i t e F k = F/ik+l F.
The assertion can then be stated in terms of the usual c o h o m o l o g y A-modules: In the d i a g r a m Hn(X,F) A
A A ~ Pn --7 H n (X,F)
~n
/On
Lira H n ( ~ , F k )
we have (i)
H n ( X , F k ) = H n ( X k , F k)
k > 0 (since ~
~ X is an affine map)
V. 3
228
and the projective
system {Hn(X,Fk) }k ~ 0 satisfies ML.
(2)
~n is an isomorphism
(3)
The kernels of the maps Hn(X,F)
+ Hn(X,Fk ) , k ~ 0, give
an I-good filtration on H n (X,F) (4)
~n is a topological
isomorphism.
To prove this assertion, we start with the usual cohomology theory on X and consider the exact sequence: Hn(x,ik+iF)
where
~ Hn(X,F)
~
Hn(X,Fk ) ~ Hn+l(x,ik+iF)
~n =~Lim ~n~k To simplify notation, we write H = Hn(X,F)
~
(for n fixed) :
= Hn(X,Fk ) Qk = Image ~nlk
= Ker ~nlk Thus there are exact sequences 0 ~ ~
~ H + Hk ~ Qk ~ 0
Let x be an element of Im
(m ~ 0) •
k ~ 0
The multiplication
by x in
IkF is a homomorphism IkF ~ Ik+mF and gives a homomorphism ~x,m:Hn(x,IkF)
~ Hn(x,ik+mF)
Letting S be the graded A-algebra
~
Ik, the multiplications
k>0 define on E =
~
Hn(x,IkF)
the structure of a graded module of
kz0 finite type over the graded ring S which is noetherian.
(Prop. 2.15)
V. 3
229
Claim:
The submodules
P r o o f of Claim: multiplication
~
of H define
on H ~ I - g o o d
First of all we show that I m ~
in H by an element x e I m being
filtration.
c ~+m'
the
the map ~x,0"
For any x e im the diagram
Ik+iF ....X
the horizontal
vertical
arrows
is commutative
which,
of Hn(x,Ik+iF)
~ Hn(X,F)
S-module R =
~
given
~ ~x,0
Hence
> H n(X,F)
the interpretation
shows I m ~
~ ~ k>0
~ Hn(x,Ik+IF) of E. k>O equivalent to the claim. Consider
is commutative.
~x,m___~ H n (X, Ik+m+iF)
H n(x,F)
M =
injections)
by x, and the
diagram
H n (X, Ik+iF)
graded
~F
arrows being multiplication
the canonical
the corresponding
2
X
F -(with
> Ik+m+iF
c ~+m
is a quotient
now the graded
in the same way as E above.
as the image
and also shows that the of the sub-S-module
Thus R is an S-module (Prop.
of ~
of finite type which
2.13)
S-module N =
I (Claim) ~ Hn+I(x,Ik+IF) k>0
It is an S-module
of finite
defined type and one
is
V. 3
230
has Qk c N k for all k and as in the claim above SmQ k = ImQk c Qk+m" Thus Q =
~ Qk is a graded sub-S-module of N and hence is of finite
k>_0 type. Let e Sm ~ So.
I k+l.
m
be the canonical
injection I m + A, which can be written
Since Ik+iFk = 0, the A-module Hn(X,Fk ) is annihilated by
S i n c e Qk i s t h e image o f t h e A-homomorphism Hn(X,Fk ) ~
Hn÷l(x'Ik+iF)'
Qk' as an A-module,
is also annihilated by I k+l.
Thus in the S-module Q, we have
°k+l(Sk+l)
Qk = 0.
Since Q is an S-module of finite type, there are integers k and h such that Qk+h = ShQk for k _> ko.
o
This statement and the
above imply that there is an integer r > 0 such that ~r (Sr) Q = 0 We now note that the canonical injection Ik+mF ~ IkF give an A-homomorphism :Hn+l(x,Ik+mF)
+ Hn+l(x,ikF)
m and for any x ¢ I m, one has the factorization
~x,0 :H
n+l. ikF) ~x,m Hn+l(x,ik+mF) (X,
v
Hn+I(x,IkF )
w h e n c e we conclude that for every sub-A-module P of Hn+I(x,IkF), we have, in the S-module N Vm(SmP)
= ~m(Sm) P
V. 3
231
Claim:
There is an integer m > 0 such that Vm(Qk+ m) = 0 for k _> ko.
Proof:
We can take m ~ r to be a multiple
of h.
Since
Qk+m = SmQ k for k ~ ko, we have ~m(Qk+m) Consider
derived
= ~m(Sm)Qk c ~r(Sr)Qk
now the commutative
Hn(X,F)
~
Hn(X,Fk )
Hn(X,F)
~ Hn(X,Fk+m)
from the commutative
0 ~
Ik+lF
~
i (Claim)
= 0
diagram
Hn+I(x,Ik+IF)
+ Hn+I(x,F)
+ Hn+l(x,ik+m+iF)
~ Hn+I(x,F)
diagram
+ F ~
Fk
~ 0
7 T 0 ~ Ik+m+iF + F ~ Fk+ m ~ 0. From this one deduces
the commutative
~H~
k > ko,
Hence ~,
they are equal.
for k' ~ k+m.
Qk
+ 0
As the final vertical
the image Hk+ m in ~
and also it contains
~
H -~ Hk+ m -~ Qk+m ~ 0
0 ~ ~+m which has exact rows.
~
diagram
is contained
Im(H ~ ~ )
arrow is zero for
in K e r ( ~ ~ Ok)
by commutativity
of the diagram.
This is true for all of the images
Hence
the condition
ML holds
= Im(H ~ ~ ) ,
of H k in
for the projective
V. 3
232
system
( ~ ) k>._O"
W e can then apply the C o r o l l a r y 2.20 and the canonical map n A A H (X,F)+ Lira H n ( X , F k ) is b i j e c t i v e
for all n > 0.
~-~-
AS
the p r o j e c t i v e
system
(H/p)
is strict,
one can pass
to
k>0 the p r o j e c t i v e
limit in the exact sequences
0
Since Vm(Qk+m)
Qk ~ 0.
= 0, we have
i s o m o r p h i s m L k m ( H / ~ ) ~Lim~ is I-good,
it defines
3.2:
o r d i n a r y algebraic support
~.
spaces)
and F be a c o h e r e n t
Then
space is Supp(F)
injection.
(Proposition
Rn
A
(f°u),(G),
(of
subspace of Y, and
nA A for all n > 0 R f,(F) ~
A. (Rnf,(F))
is a
closed subspace of X,
and for w h i c h u:Z ~ X is the II.~,l&)
w e have G k = U*Fk, R n f , ( F k ) = R n (f.u),(Gk)
F)~'~~
I
0X-mOdule , w h o s e
Let Y' ~ Y be a closed
c o h e r e n t 0 X - m O d u l e , and Z is an a p p r o p r i a t e
nA SO R f,(
H e n c e i~im(H/_ ) is
We can assume that F = u,(G) w h e r e G = u*(F)
whose underlying
of H
L e t f:X + Y be a m o r p h i s m of finite type
is p r o p e r over Y.
canonical
(~)
of H for the I-adic topology.
A A A define Y, X, f and F as above. Proof-
But as the filtration
on H the I-adic topology.
the s e p a r a b l e c o m p l e t i o n Corollary
i~L Qk = 0 so there is a t o p o l o g i c a l
Putting Gk
= G Y~=OY/Ik+l"
and R n f,( F ) = Rn(f'u),(G)
and now the above theorem applies.
I
V. 4
233
4.
Applications
Theorem
to P r o p e r
4.1:
(The C o n n e c t e d n e s s
f:X + Y b e a p r o p e r spaces.
Let
morphism
X
g
g
-i
(p)
the p r o p e r t y
theorem
its S t e i n
separated
factorization
and g is a p r o p e r
that
Z + Spec
IV.4.1,
Applying
fiber
g, ~ X
for all p o i n t s
course
Z 1 + Z is
still
Thus we may
ideal
defining L e t F be 3.1,
a point of Z,
flat
Let
algebraic
(II.5.3)
Stein
p ~ Z,
Let
f is p r o p e r .
morphism,
g
the
fiber
the
finiteness
so h is
finite.
as a S t e i n map.
g is p r o p e r ,
g must
be s u r j e c t i v e ,
so
p in Z is n o n e m p t y . and Z 1 the a t o m of Z at p so if w e w r i t e
g l : Z l + X 1 is a g a i n
p is a c l o s e d
in p a r t i c u l a r
us w r i t e
By
~y-module
as w e l l
(II.6.15)
pullback
assume
of Z.
and
is a c o h e r e n t
J/~Z" and
proper,
point
theorem
=
a point
associated
f, ~ X
g is p r o p e r ,
of g o v e r
L e t p be The map
f* ~ X
I.i.21,
Since
the
of n o e t h e r i a n
of Z a r i s k i . )
= p X X is c o n n e c t e d . Z
Proof:
the
be
h is a f i n i t e m o r p h i s m
also has
Theorem
) Z
Y Then
Morphisms
point
X 1 = Z 1 ~ X,
Stein
(II.4.17),
o f Z 1 and gl
t h a t Z is a f f i n e
Z = S p e c A and
(II.6.13) .
-i
then and of
(P) =
g-i
(P)"
and p is a c l o s e d
let m c A be
the m a x i m a l
p. any c o h e r e n t
there
sheaf
on X.
is an i s o m o r p h i s m
By
the h o l o m o r p h i c
functions
V. 4
234
~n:(Rnf,(F)) Ap + ~Lim Hn(f-l(p), F~yZ ~ ~Z/mk) " We apply this in the case F = ~X' n = O. Then (Rnf,(F))p = F(X,f,~Yx))A = F ( Z , ~ )A = A which is P Z p P' a local ring. The kth stage of the range of ~n above is F(f-l(p), ~X ~z~Z/mk) " If we write ~
= f-l(spec OZ/mk) , this kth stage is F ( ~ , ~Xk) •
Suppose X 1 is not connected.
Then we can write X 1 = UIJ' VI,
a disjoint union, and F(UI, O~UI) = F(VI, ~kVl) . ~
has
has the same underlying topological space as X 1 so there exists a decomposition ~ and V 1 = V k ~
= U k ~ Vk, a disjoint union, with U 1 = U k X I.
Thus the induced decomposition
~kXl
r(~, ~)
= F(U k, ~Uk) x F(Vk, O~,Vk) is compatible with the map F(~, ~ )
~ F(XI, ~Xl). Lira F ( ~ , ~ )
Hence = Lira F(Uk, ~U ) × Lim F~k, ~V )
(since inverse limits commute). Neither of Lira F(Uk, O U ) and ~im F(Vk, ~Vk) is trivial (each k contains at least the limits of the unit sections), so~Lim F ( ~ , d~ ) is a product of rings, hence not a local ring. contradiction.
This is a I
Another result proved in this context in EGA is Zariski's Main
V.4
235
Theorem.
As i n d i c a t e d
category
of a l g e b r a i c
Theorem locally map.
T h e n there
i)
'x ii)
algebraic exist open
such
f l
:X' ~ Y'
to the
is trivial. M a i n Theorem)
space
and
subspaces
L e t Y be
a noetherian
f:X ~ Y a q u a s i p r o j e c t i v e X'
of X and Y'
of Y, w i t h
is finite
in its
6 X, x is in X' fiber
The a s s e r t i o n
Then X m u s t
equivalent
theorem
|
F o r each p o i n t x
Proof:
of this
that
isolated
affine.
spaces
the e x t e n s i o n
(Zariski's
separated
f(X') c__ Y',
and
4.2:
below,
assertion
if and only if x is
f-l(f(x)) .
is local on Y
so we can
be a s c h e m e
(II.7.6)
for schemes
E G A III.4.4.3.
In the rest of this
section~
we
so this
apply these
assume follows
that Y is from the I
two r e s u l t s
in
the c a s e of v a r i e t i e s . Definition throughout
4.3:
the d i s c u s s i o n .
space X g i v e n w i t h separated
and of
quasiprojective; proper
L e t k b e an a l g e b r a i c a l l y
a map
A prevariety f:X ~ S p e c k
finite type. affine
closed
field
X is a r e d u c e d
algebraic
(the s t r u c t u r e map)
X is q u a s i p r o j e c t i v e
if f is affine.
fixed
which
is
if f is
X is a y a r i e t y
if f is
and X is also i r r e d u c i b l e .
If X is a p r e v a r i e t y , is an e t a l e c o v e r i n g
it m u s t be n o e t h e r i a n
(II.3.7)
n : Y ~ X w i t h Y an a f f i n e p r e v a r i e t y ,
so there the s p e c t r u m
V. 4
236
of a noetherian Given dimension stale and
a point
and n q u a s i f i n i t e .
p in a p r e v a r i e t y
of the c o m p l e t e
covering
this
ring,
~ : Y ~ X,
local
ring
the point
is the d i m e n s i o n
in
X,
the d i m e n s i o n
(the atom)
sense
is the
of X at p.
q in Y w i t h ~(q)
the u s u a l
of p
(EGA
= p,
For
any
dim p = dim q
OIV.;~, I )
and
is an
integer. The dimension dimension
of X is the m a x i m u m ,
i, X is c a l l e d
d i m ~ = m a x d i m p. If X h a s p6X if d i m e n s i o n 2, a s u r f a c e .
a curve;
I f d i m X = n and X is a v a r i e t y , is the u n i q u e point
p
integer
point
£ X with
d i m p = m,
we
n.
say
the
In t h i s
generic point
case,
given
the c o d i m e n s i o n
of X
another
o_~f p in X is the
n - m.
In I I . 6 . 8 , subspace point
of d i m e n s i o n
then
we
showed
X 1 c X which
x 0 of X must Theorem
codimension
any point
X has
In p a r t i c u l a r ,
in t h i s d e n s e
subset,
a normal
a dense the
open
generic
so x 0 is s c h e m e l i k e .
variety,
and x
£ X a point
of
T h e n x is s c h e m e l i k e .
giving 4. 5:
a prevariety
is a s c h e m e .
L e t X be
one.
Before Lemma
4.4:
be
that
the p r o o f ,
Let
we need
f:Y ~ X b e
~4ith q = f(p) .
the
a map
following
lemmas.
of p r e v a r i e t i e s
and p
c Y be
we can
assume
T h e n d i m q < d i m p. m
both
Proof:
Since
X and Y
are
Lemma
4.6:
the n o t i o n
of d i m e n s i o n
affine prevarieties. Let Y
is local,
This
and X be v a r i e t i e s
is t h e n E G A
and
f:Y ~ X b e
0iV.16.1.5.1 a birational
V.4
237
surjective
projective
morphism.
any p o i n t w i t h q = f(p) Proof:
its image
An i m m e d i a t e
Lemma
4.7:
Let
The
assertion
Since
schemes,
this is p r o v e d
Lemma subset
corollary
Then
affine.
Zariski Main
in X.
of L e m m a
4.5.
I m a p of v a r i e t i e s .
is local on X, Y is also
so w e can a s s u m e
affine.
in E G A I I I . 4 . 4 . 9
In the c a s e of
as a c o r o l l a r y
Let X be a p r o j e c t i v e such that all p o i n t s
variety
and A c X
topology.
of just one point,
or the c l o s u r e
of A is all of X.
The c l o s u r e
of A, A,
of c o m p o n e n t s .
of these c o m p o n e n t s in A,
X is separated)
P r o o f of t h e o r e m a projective over k ( s o
4.4:
Applying
Lemma
birational
in p a r t i c u l a r
a scheme).
B u t s i n c e Y ~ X is b i r a t i o n a l , subset
of Y.
map
(and l
IV.3.1,
there
is
f:Y ~ X w i t h Y p r o j e c t i v e
A priori,
Y m a y n o t be reduced.
and X is reduced,
Hence
one
and c o n n e c t e d
or none.
a
the g e n e r i c
of c o d i m e n s i o n
just one point,
surjective
open reduced
and h e n c e has only
If A is finite
Chow's
one
either A c o n s i s t s
all of X,
the o n l y p o i n t s
so A is finite. A must have
Then
is n o e t h e r i a n
I f A is not
are
a nonempty
of A are of c o d i m e n s i o n
in the i n d u c e d
contained
of the I
and A is c o n n e c t e d
points
that X is
Theorem.
of p o i n t s
finite n u m b e r
e Y be
f is an i s o m o r p h i s m .
f is affine,
4.8:
Proof:
Let p
Then c o d i m a ~ c o d i m p.
f:Y ~ X be a finite b i r a t i o n a l
S u p p o s e X is normal. Proof:
T h e n d i m X = d i m Y.
there
is a d e n s e
the m a p Y r e d + Y is b i r a t i o n a l .
This
V.4
238
m a p is also c l e a r l y p r o j e c t i v e
and s u r j e c t i v e
at the first considered Yred"
H e n c e we can assume Y is reduced.
so we m a y as well have
Y can also be chosen to be irreducible. with Y1 and Y2 closed
subsets of Y, then since Y + X is surjective,
there m u s t be a p o i n t p in Y1 p o i n t o f X.
For if Y + Y1 U Y2'
Then Y1 + X
(or in Y2 ) w h o s e image is the generic
(or Y2 ~ X)
is surjective,
projective
and
birational. H e n c e Y is a p r o j e c t i v e Clearly
variety.
By l e m m a 4.6, dim Y = dim X.
the generic p o i n t Y0 of Y is the unique p o i n t of Y m a p p i n g
to the generic p o i n t x 0 of X. one and y £ Y a p o i n t with
f(y)
N o w let x 6 X be a p o i n t of c o d i m e n s i o n = x.
By lemma 4.6, y m u s t be the
generic p o i n t of Y or a p o i n t of c o d i m e n s i o n Y / Y0'
so y is of c o d i m e n s i o n
c l o s u r e of f-l(x) a closed
( which
Also,
is contained
since
consists
is i s o l a t e d
that f-l(x)
is connected.
of a single p o i n t y.
But since x / x0J f is surjective,
in f-l(the closure of x))
subspace of Y but cannot be all of Y.
T h e o r e m 4.1 implies f-l(x)
one.
one.
the is
The C o n n e c t e d n e s s H e n c e by lemma 4.8,
In p a r t i c u l a r
this p o i n t y
in its fiber.
W e now apply the Zariski M a i n Theorem.
T h e r e is an open
subspace X 1 c X and an open s u b s p a c e Y1 c Y such that if f(Yl ) c Xl, the map
fl:Yl + X 1 is finite and Y1 c o n t a i n s
their fibers under
f.
Our map
all points
isolated
in
f is a b i r a t i o n a l map of i r r e d u c i b l e
V. 4
239
varieties,
so
lemma
fl m u s t
4.7, ~f x
be
fl is b i r a t i o n a l .
Y
Y1
projective
so Y1
is n o r m a l .
Applying
be an i s o m o r p h i s m .
E X is of c o d i m e n s i o n
in X I.
is n o r m a l
is a scheme,
variety,
one,
since
so Y 1
= Xl
then
as a r g u e d
it is an o p e n
is an o p e n
above,
subspace
neighborhood
x must
of a of x w h i c h
is a s c h e m e .
I
Theorem field k.
4.9:
L e t X be
~:Y ~ X which
quasifinite
can
By Chow's
and
apply
4.4,
one.
is a curve, well-known
it m u s t
closed
This
any o p e n
criterion that
e n
One
-i
Criterion
finite.
EGA
that Y
it by,
in
n-l(u)
and
~ is
a scheme,
III.4.4.2
the
irreducible
statement
for
is l o c a l and we
schemes.
+ U is an
the r e s t r i c t i o n s But
since be
for e x a m p l e ,
for p r o j e c t i v e I ) X - n(y)
the
scheme Y
affine.
- y is q u a s i p r o j e c t i v e
true
(~[~.
finite.
be
and m a p
As
Since
(This
Y - y of Y m u s t
can p r o v e
reduced
quasifinite.
and c o n s i d e r
subspace
to s h o w
Y is a l s o
of X for w h i c h
is a g a i n
Y
and b i r a t i o n a l .
proposition
(U)
it is o b v i o u s l y
the C h e v a l l e y
be
is a p r e v a r i e t y
by II. 6.1~ , Y m u s t
the s u b s e t
Let p
fact.
assume
~ is c l e a r l y
the c o r r e s p o n d i n g
y - y ~ X - n(y) •
noting
Also,
if X is affine,
L e t U + X be
Nakai
an a l g e b r a i c a l l y
there
surjective
we can
and p r o p e r ,
isomorphism.
Lemma,
is p r o j e c t i v e ,
theorem
of dimension
on X,
over
T h e n X is a s c h e m e . Proof:
previous
a curve
(This
applying and
the
then
varieties.)
is affine,
is a
hence
Applying a
V. 4
240
scheme.
S i n c e ~(y)
E X is a s c h e m e l i k e
The reader will
n o t e this t h e o r e m
point,
is e a s i l y
case w h e r e X is an a r b i t r a r y p r e v a r i e t y 4.10: thoerem normal
Without making
above variety
can be i n t e r p r e t e d are all d e f i n e d
Also, the u s u a l p r o o f s that
nonsingular
surfaces
the c a s e of a l g e b r a i c however. surface
specific
Recall
and a n o n s i n g u l a r
as s a y i n g
showing
are always
spaces.
w e gave
definitions,
One
examples
divisors
in the c a t e g o r y
not b e
neither
that the on a
points.
carries
of s c h e m e s over
to
too o p t i m i s t i c
in the I n t r o d u c t i o n
three-fold
to the
one.
we note
that W e i l
projective
should
(II.6.6) •
extendible
of d i m e n s i o n
by schemelike
[XVa]
w e are d o n e
of a s i n g u l a r
of which were
schemes.
V. 5
5.
241
Completions
of M o d u l e s
Definition
5.1:
of H o m o m o r p h i s m s
Let X be a n o e t h e r i a n
and F and G q u a s i c o h e r e n t onX,
Horn
(F,G)
~ r =Hom_
~y
is d e f i n e d
The h o m - s h e a f
(F,G).
This
so w e can d e f i n e
O f course,
for every Y ~ X etale,
functor
of q u a s i c o h e r e n t
sequence
we can
derived
also define
Hom ~(F,G)of
(F,G))
functors
Ext q
on X has
enough
(F,-) , q >__ 0, of
the
the d e r i v e d functor
functors
Hom
(F,G).
in the f o l l o w i n g 5.2:
Ext n ~ X (F,G) These
are two
X proposition.
With
the n o t a t i o n
above,
there
is a spectral
E(F,G) ~2 'q = HP(x,
Proof: there
injection
W e need
Extq(F,G))_-~ ExtP+q(F,G)
to show
is an i n j e c t i v e
G ~ I such
separated,
we can assume
as in 1.4.9,
we can
J w i t h ~ , J injective, to showing
that
that
for all q u a s i c o h e r e n t
quasicoherent
that HP(x,
L e t n : Y ~ X be an etale
Then
Hom
(F,-).
Proposition
and
F(Y,
sheaves
~x
on X,
of F and G
is a sheaf and if F and G are coherent, X s o
The c a t e g o r y
functor H o r n
compared
space
~
(FIy,GIy) .
injectives
of the
by,
algebraic
x
is H o r n
the
~X-mOdules.
separated
Hov~m(F,I))
covering
~X-mOdule
for all q u a s i c o h e r e n t
I and an
Since X is n o e t h e r i a n
that Y is affine
and an i m m e r s i o n
G
= 0, p > 0.
of X.
find an i n j e c t i v e
sheaves
and ~ is affine.
quasicoherent
G ~[C,J. modules
H e n c e we H on Y,
~y-module are r e d u c e d
V. 5
242
HP(x, H%omO X ( ~ , ~ . H ) )
= o, p > o.
But n, H~om ~ v ( ~ * F , H ) = Hg_m ~ ( F , ~ , H ) . on X and for X affine
(This a s s e r t i o n
is the usual adjointness
relation.)
H e n c e HP(x, H ommom(~x(F,~,H)) = H P ( x , ~ , H o m i ~ y ( ~ * F , H ) ) . is affine, If ~
this is zero
and the p r o p o s i t i o n
5.3:
We
separated
1
space, w e can w r i t e
5.2 carries
fix the following n o t a t i o n
A is a n o e t h e r i a n
ring,
Since
(I.4.16) .
is a formal algebraic
definition
is local
the same
over easily to this case.
for the rest of this section.
I an ideal of A, X an algebraic
space,
and finite type over Spec A% F and G are two c o h e r e n t
~X-mOdules
the i n t e r s e c t i o n
of w h o s e
supports
is p r o p e r over Spec A.
X~ X ~ Spec (A/I) and X is the c o m p l e t i o n of X along X 0. A Spec A A A i:X + X is the canonical map. Recall F = i'F, G = i*G. (Prop. 2.3) Lemma
i)
5.4: There exists a c a n o n i c a l A A Ext q A(F,G),
isomorphism
i*(E~xt% ~x
(F,G))
q > 0.
~OX ii)
There
E(F,G) HP(x,
exists a canonical map of spectral A A ~ E(F,G) Ext q
sequences
which
in the ~P'q terms is the map "2 p A A A (F,G)) ~ H (X, E~xt q A(F,G) c o m p a t i b l e w i t h
the above isomorphism. Proof:
This is a h o m o l o g i c a l
that of EGA Oii 1.12.3.4,
algebra c a l c u l a t i o n
5 for flat maps of ringed
identical
spaces.
to 1
V. 5
243
Proposition
5.5:
For all n ~ 0, E x t ~ y
(F,G) is an A - m o d u l e
of
finite type and its I-adic c o m p l e t i o n is c a n o n i c a l l y i s o m o r p h i c to AA Ext~A(F,G) . X Proof: C o n s i d e r the spectral sequence E(F,G) of Prop. 5.2. As m e n t i o n e d
above,
Ext q
(F,G)
is a c o h e r e n t module.
Its support
~x is c o n t a i n e d ~
the intersection
is proper over Spec A.
H e n c e by the F i n i t e n e s s
the E2P'q terms of E(F,G) implies
of the supports
are A - m o d u l e s
of F and G and hence Theorem
of finite type.
IV.4.1, This
that all of the E p'q terms are of finite type so the abutment r
n E x t ~ (F,G) is of finite type. 2k Since E x t q ( F , G ) i s o f f i n i t e Ext q ~ O x ( F , G ) A ~ i*(EXt~x(F,G)) isomorphism Applying
of L e m m a
2.3.
is
an i s o m o r p h i s m
Combining
Functions
theorem
3.1,
there
A A i s o m o r p h i s m H p(x, E~xt~(F,G))
Thus in the spectral A A E p 'q(F,G)
by Prop.
there
this with
the
5.4(i) w e have an i s o m o r p h i s m E~Xt~x (F,G) ~-- E x
the Holomorphic
p > 0, a c a n o n i c a l
type,
A A sequence E(F,G),
is,
for
%^^ (F,G)
each A
~ H p(X,E~xt~X(F,G))
we have
A = E p,q (F,G) A = E 2P'q(F3G) ~ ~ w h e r e A is the I-adic c o m p l e t i o n A
of A. N o w consider
the map E(F,G)
A A ~ E(F,G)
of L e m m a
5.4(ii).
A A is a
E(F,G)
A so we can apply the functor ~ A to all the terms of A to get a new spectral sequence. Since all the terms of
E(F,G)
are A - m o d u l e s
flat A - m o d u l e
A the functor ® A is identical A and this new spectral sequence can
of finite type,
here to the c o m p l e t i o n
functor
V. 5
be
244
denoted All
E(F,G) A.
the
I-adically
terms
complete.
the c o m p l e t i o n EP'q 2
terms
A A of E(F,G) are A - m o d u l e s
this
Hence
functor map
gives
of
the u n i v e r s a l a map
finite
mapping
type
so are
properties
A A E(F,G) A + E(F,G) .
of
In the
is A A
~ x ~ % i~l^~ ~cx ~ ^ ~~l~x which
was
abutments
shown
above
EXt~x(F,G)A
to be
an i s o m o r p h i s m .
n A A ~ Ext~(F,G)
Hence
the m a p
is an i s o m o r p h i s m ,
on the n
V. 6
6.
245
The G r o t h e n d i e c k
Existence
Let A be a noetherian
adic ring w i t h
Let
ideal
A and Y = Spf A,
let Y = Spec A, Y' = Spec A/I along Y') .
Theorem
f:X + Y be a m o r p h i s m
of d e f i n i t i o n
(the c o m p l e t i o n
of finite
type w i t h
I and of Y
X an
A and X be the c o m p l e t i o n
a l g e b r a i c space, A A A L e t f:X ~ Y be the e x t e n s i o n
of f.
A sheaf on X and F its c o m p l e t i o n Proposition F is p r o p e r
6.1:
over Y.
In this Then
of X along X' = Y' × X. Y F i n a l l y let F be a c o h e r e n t
A on X.
situation,
the c a n o n i c a l
P. :Hi(X,F) l
suppose
the support
of
morphisms
~ Hi (X,F) A A
are isomorphisms. Proof:
The A - m o d u l e s
b y the finiteness completions
theorem
in the I-adic
case of the H o l o m o r p h i c Proposition that
6.2:
the i n t e r s e c t i o n
the c a n o n i c a l
Hi(X,F)
are m o d u l e s
(IV.4.1)
so are equal
topology.
Functions
of finite to their
The p r o p o s i t i o n
Theorem
type over A separable
is then a special
(3.1).
l
L e t F and G be two c o h e r e n t
0X-mOdules
of their
over Y.
supports
is p r o p e r
map HOm~x(F,G)
A A ~ H o m xA(F'G)
u ~ u
A
such
Then
V. 6
246
is an i s o m o r p h i s m . (respectively Proof:
Further,
if f is closed,
surjective) iff u is i n j e c t i v e
The
first a s s e r t i o n
A then u is i n j e c t i v e (surjective).
is a p a r t i c u l a r
case of
5.5 since
Hom/~__(F,G) is an A - m o d u l e of finite type, so equal to its s e p a r a b l e x A completion. To see the second, w e first c l a i m that u is i n j e c t i v e (surjective)
iff there
u is i n j e c t i v e and c o k e r n e l support
a closed
this,
set,
= f-l(y.), closed
say, v a n i s h e s j t h e n
is d i s j o i n t
if f is c l o s e d
every nonempty
= Spec A/I.
from V(I)
of X on w h i c h
K = IK so the
= X'.
f(C)
N Y'
of V in X.
= @.
topology Hence
But
f(C)
to X.
T h e n C D X'
=
is c l o s e d
and
To
since Y = Spec A is
so I is c o n t a i n e d
f(C),
Then
u is i n j e c t i v e .
any such V m u s t b e i d e n t i c a l
set of Y m e e t s Y'
in its I - a d i c
o f A and Y'
T h i s is true since the k e r n e l K
A If K,
let C be the c o m p l e m e n t
and s i n c e X'
separable
set V ~ X such that X' c V and
on V.
is an open n e i g h b o r h o o d
However, prove
(surjective)
Q are coherent.
of K,
X - V(I)
is an open
so C p i s
in the r a d i c a l
empty.
A Thus u is i n j e c t i v e Thus w e h a v e ~X-mOdules with
whose
(surjective) iff u is. I A a functor F ~ > F m a p p i n g the c a t e g o r y
of c o h e r e n t
support
support proper
is p r o p e r over Y to c o h e r e n t ~ - m o d u l e s ^ X over Y. By the above lemma, this functor
establishes
an e q u i v a l e n c e
subcategory
of the second.
of the
first c a t e g o r y w i t h
a full
V. 6
247
Theorem EGA
6.3:
(The G r o t h e n d i e c k
III.5.1.4).
I an i d e a l morphism
Let A be
of A, Y'
the
functor
~X-mOdules coherent
= V(I),
and X a n y a l g e b r a i c
A A Y = S p f A = Y/y,, Then
an a d i c
support
~-modules
A F ~ > F is an
with
important
Corollary
noetherian
Theorem, ring,
f:X ~ Y a s e p a r a t e d space,
X'
= f-l(y,)
Cf.
Y = S p e c A,
finite = Y'
type x
X,
A A A A A A Y X = Y X X = X / X ,, f:X ~ Y the c o m p l e t i o n of f. Y A F~>F is an e q u i v a l e n c e of the c a t e g o r y of c o h e r e n t
with
The most
Existence
6.4:
proper support
case
Suppose
equivalence
over
S p e c A, w i t h
proper
over
the c a t e g o r y
S p f A.
is the X is p r o p e r
of the categories
o v e r Y.
Then
the
functor
o f c o h e r e n t C~X-mOdules
and C o h e r e n t 6 ~ - m o d u l e s . Proof
of
6.3:
is a l q e b r i z a b l e ~X-mOdule
I
In this p r o o f
we will
if it is i s o m o r p h i c
F with
of
support
proper
say a c o h e r e n t C ~ - m o d u l e
to a c o m p l e t i o n
o v e r Y.
We
A F of a c o h e r e n t
first prove
Lemmas
6.5-6.9. Lemma for
6.5:
L e t F'
any h o m o m o r p h i s m
and G' b e
u:F'
~ G',
two a l g e b r i z a b l e O ~ - m o d u l e s . Ker
(u) , Im(u),
Then
and C o k e r ( u )
are a l g e b r i z a b l e . Proof:
L e t F'
A = F, G'
A = G where A
with
proper
support.
Then
F and G are c o h e r e n t ~ X - m O d u l e s
A
u:F ~ G is of the
.& A
A
form v:F ~ G where
V. 6
248
v : F + G is some m a p by 6.2.
A Ker(v)
since F ~
A F is exact
of Supp(F)
and so is p r o p e r
Similarly
for Im(u)
Lemma
6.6:
is i s o m o r p h i c
and the support over Y.
(Ker(v)) A
of Ker (v) is closed Hence
Ker(u)
subspace
is algebrizable.
and C o k e r (u) •
I
L e t 0 ~ H ~ F + G ~ 0 be an exact
coherent ~-modules
to
sequence
such t h a t G and H are a l g e b r i z a b l e .
of
T h e n F is
algebrizable. Proof:
Suppose
C with proper which
by
A A H = B and G = C for c o h e r e n t ~ X - m O d u l e s
support
•
Then F d e f i n e s
5.5 is i d e n t i c a l
to E X t ~ v
an element
(B,C).
of E x t ~
B and
(B,C) ^ ^
L e t A be the c o h e r e n t
A
~X-mOdule of Ex
(determined
% (B,C).
Then Supp A c
is i s o m o r p h i c
to F.
Corollary
6..7:
O~-modules.
up to isomorphism)
Then
representing
(Supp B n Supp c)
this
element
so is proper,
^
and A
I L e t u:F ~ G be a h o m o m o r p h i s m if G, Ker(u)
and Coker(u)
of c o h e r e n t
are algebrizable,
so
is F. Proof: Lemma
Immediate 6.8 :
the c o m p l e t i o n
A-module P roo.f: by 3.1,
A
A
~,(F')
Y its completion.
I
A ~ g,(F)
of finite
type
A and Z
L e t g:Z ~ X be a p r o p e r Then
is an a l g e b r i z a b l e
L e t F be a c o h e r e n t
A A g,(F)
6.6.
of Z along Z' = Y' × Z.
and g:Z ~ ~ F',
6.5 and
L e t h :Z ~ Y be a m o r p h i s m
A
Y-morphism
from
~Z-mOdule
for every
algebrizable
~-module. such
that F' = F.
Then
i
V. 6
249
Lemma X'
6.9:
a closed
A
A
L e t X be a s e p a r a t e d
subset
A
of X, A
noetherian
f:Z + X a p r o p e r
A
A A
Z = Z X X = Z/Z , and X a c o h e r e n t i d e a l of ~ X s u c h
algebraic
morphism,
Z'
f:Z ~ X the c o m p l e t i o n
be
that
~X/M)
in X,
Then
for e v e r y
such
that
Proof:
The
Say X = Spec
B,
TO
I-adic
topology
canonical finally
statement and X'
Let M
complement
is an i n t e g e r
on X so w e
for s o m e
B is a d i q u e
let B 1 b e
the
and I 1 = IB I.
Then X 1
Z 1 = Z × Xl,
A let X 1 b e
there
f.
of
f is an i s o m o r p h i s m .
of the c a n o n i c a l
is l o c a l
= V(I)
of d e f i n i t i o n map.
F,
~ U of
of
map
n > 0
PF:F ~
A A f,f*F
An by M .
see this,
ideal
if U is the o p e n
f-l(u)
~A-module
and c o k e r n e l
we can assume
of B.
with
coherent
the k e r n e l
is a n n i h i l a t e d
Also
the restriction
= f-l(x'),
A
X = X/X,,
Supp
space,
|
separable
of X 1 a l o n g X~,
of d e f i n i t i o n
completion
of B in t h e
noetherian
ring
B 1 and h : X 1 ~ X the
= h - l ( x ') is i d e n t i c a l induced
X is affine.
I of B.
B 1 is an a d i q u e
Put X 1 = Spec
fl:Zl ~ X 1 the
X the c o m p l e t i o n
assume
and I is an i d e a l
Then
I I.
ideal
can
to V ( I I ) .
map,
which
Put
is p r o p e r ,
and
A A Z 1 = Z 1 × X 1 the completion X1
A
of Z 1 a l o n g
Z~ = f[l(x{) A
is an i s o m o r p h i s m Finally,
and
AA
fl the c o m p l e t i o n
f.
A
is a c o h e r e n t
ideal
Hence
of O x
and 1
A
Then h:X 1 ~ X A
so Z 1 ~ Z is an i s o m o r p h i s m .
M 1 = h*(M)
of
A
fl = f"
V.6
250
Supp(%i/M
I) = h - l ( s u p p ( ~ x ( M ) )
Supp(~Xl/M1)
so,
l e t t i n g U 1 be the c o m p l e m e n t
in Xl, U 1 = h-l(u)
of fl is an i s o m o r p h i s m .
w h e n c e the r e s t r i c t i o n A A A l s o M and M 1 are i d e n t i f i e d
of
f~l(Ul) ~ U 1 A b y h. All A
the h y p o t h e s e s
of the l e m m a
and o n e c a n t h e n a s s u m e of d e f i n i t i o n
are then
3.1,
g
and I an ideal
A s h e a f F on X is h e n c e
0X-mOdule.
A A f,((f*(G)) ) is c a n o n i c a l l y
Q of p
ring
f l and M1
of B.
G is a c o h e r e n t
completion
Xl, Xl,
by
B is an a d i q u e n o e t h e r i a n
A W e h a v e X = S p f B and a given A G where
satisfied
equal
A
pg where pg:G -~ f , f * ( G ) .
are c o h e r e n t
(using II.5.8)
there
and their
A f*(F)
B y 2.3, to
A =
(f*(G))
and b y
But t h e k e r n e l P and c o k e r n e l to U are zero.
n > 0 such that Mnp
Hence
= MnR = 0
AnA Aria sO M P = M R = 0. Final Proof we canmerely and this For theorem
I
of
6.3:
quote
For the case
[EGA III.5.2]
is the c a s e the g e n e r a l is true
form
(f,(f*(G))) ^ so DF is the
restrictions
is an i n t e g e r
of the
treated
since Y is a f f i n e
so X is a s c h e m e
there.
case w e u s e n o e t h e r i a n
for e v e r y p r o p e r
take the c o m p l e t i o n
f:X ~ Y q u a s i - p r o j e c t i v e ,
closed
induction
subspace
A T to be the c o m p l e t i o n
and a s s u m e
T of X.
the
(We
of T along T' = T O X').
W e can assume X / @. Since applies
f is s e p a r a t e d
and t h e r e
the c o m p o s i t i o n
and of finite
is a Y - s p a c e
type,
Chow's Lemma
Z and a Y - m o r p h i s m
Z ~ Y is q u a s i p r o j e c t i v e
(IV.3.1)
g:Z ~ X such that
and g is p r o j e c t i v e ,
sur-
V.6
251
jective,
and
there
is an o p e n
s u b s e t U of X such
that
g-l(u)
~ U
is an i s o m o r p h i s m . L e t M be
a coherent
X - U and F a c o h e r e n t
ideal
of (TX d e f i n i n g
~ A-module
with
the c l o s e d
support
E proper
A
Let
A
Z be
the c o m p l e t i o n
completion
of g.
is c o n t a i n e d Y.
Then
in g
-i
6.9~
the k e r n e l annihilated
X defined One
A g*(F)
(E) .
the i n d u c t i o n 6.8,
~,A-module
hypothesis
P and R are
A of g~F)
A
g is p r o p e r ~
by
R of the c a n o n i c a l An M .
Let
, and
T be
algebrizable.
6.8. map
Then
A A PF:F + g,(g*(F)) subspace
of
injection.
S i n c e U is n o n e m p t y
and by
over
Applying
j :T + X the c a n o n i c a l
A that j*(P)
support
A A g,g*(F)
Hence
the c l o s e d
Mn A A, and R =~ j,(j (R)).
implies
whose
is p r o p e r
is a l g e b r i z a b l e .
since
some power
A A P = j,(j*(P))
support
A g*(F)
b y M n, T = S ~ e c .~'Y/
can w r i t e
so b y
the
P and c o k e r n e l by
A
o v e r Y.
h - l ( Y ') and o:Z ~ X the
• is a c o h e r e n t O ~ - m o d u l e
Thus
As h is q u a s i p r o j e c t i v e
is an a l g e b r i z a b l e
are
of Z a l o n g
subspace
A j*(R)
6.8,
are
algebrizable
F is a l g e b r i z a b l e . I
INDEX
affine
etale
affine
formal
affine
image
algebraic
covering
103
scheme
211
of a m a p
124
space
completion
of a ring
an ideal completien space
92
210 of an a l g e b r a i c
along
216
215
- -, i n t e g r a l
1z~4
components
- -,
127
Connectedness
- -, a f f i n e -
104
- is a s c h e m e
1
- -,
locally
noetherian
-,
locally
separated
-
- -, n - d i m e n s i o n a l
04
I05 97 I05
- -, n o n s i n g u l a r
1 06
-
-, n o r m a l
106
-
-, q u a s i c o m p a c t
1 05
-
-,
I O6
separated
97
algebrizableX-mOdule ample
invertible
sheaf
atom Axiom Axioms
SI,S2,S3(a),S3(b ) up
Theorem
affine
covering,
into
92
compo-
dense,
145
geometrically
124
- -, t o p o l o g i c a l l y descent, descent
125
effective theory
- -over
142
Devissage
133
diagonal
adic
32
for m o d u l e s
54
rings
207 173
component
dimension,
6,
codimension
36
-
236
Cartier
149
-, Well
169
effective
190
equivalence
240
descent
32,
closure,
geometric
124
closure,
scheme-theoretic
48
- -,
"finitely
coherent
sheaf
41
- -,
induced
- -,
quotient
categorical
72
- -, e f f e c t i v e
-,
117
-
155
etale
34
relation,
32
of s h e a v e s
80
of
a point
subcategory
Lemma
etale
nents
closed
Completeness
103
236
Lemma
(quasicoherent)
etale
representable
Chow's
cohomology
233
30
decomposition
19
Chevalley's
Theorem
curve
divisor, blowing
145
map
247
3O
A0
covering, covering
of a space
106
- -, n o e t h e r i a n
- -, r e d u c e d
a closed
subspace
- -, f o r m a l
irreducible
along
72 presented"
83 72
of
on a l g e b r a i c
72 spaces
covering,representable
113 92
253
etale topology, formal
- -, a l g e b r a i c -
-,
spaces
102 59
Lemma
157
-
-,
- -,
148
s h e a f at a p o i n t I-good
220
Theorem
202
filtration,
flat
-
topology,
42 42
open
42 property
42
condition
59
affine 58
local
-,
58
- -,
211
local
on the d o m a i n
local
ringed
schemes scheme,
affine
- -, as f o r m a l
algebraic
space
216
formal algebraic general ideal
construction
33
effective
33
locally
definitions
215
map
= morphism
map
of a d i c
of d e f i n i t i o n
217
function
field
148
- -, a d i c
function
field
148
- -,
faithfully
- -,
flat
point
generic
rank
144 of a c o h e r e n t
sheaf
-
150
geometric
closure
geometrically
124
dense
124
gluing
data
5
graded
ring
218
graded
R-module
219
Grothendieck
topology
group actions algebraic -
-,
free a c t i o n
geometric fixed
quotient
point
of -
locus
Hilbert
Basis
Hilbert
Scheme
Theorem
map
205 205 flat
207 2O7
formal
208
etale
etale
of f i n i t e
of a l g e b r a i c
covering
208 207
type spaces
92
- -, a f f i n e
108
- -
birational
144
-
closed
133
-
closed
-
-
247
-
29
-
on a n space
-,
- -,
97
rings
-, f o r m a l l y
108
immersion
covering map
101
-
etale
101
-
etale
- -
Grothendieck Existence Theorem
-
39
separated
215
sheaf
32,34
space
space
generic
128
sheaf
invertible
Jacobian
of
spaces
schemes
formal
227
closed
algebraic
of a q u a s i c o h e r e n t
Finiteness
Theorem
immersion
inductive fiter
Functions
211
schemes
Extension
Holomorphic
affine
schemes
101
surjective
- -
finite
177
- -,
faithfully
178
- -,
flat
107
immersion
1 08
178 186
108 2
-
-,
139 107
flat
- -,
locally
of f i n i t e
- -,
locally
of f i n i t e
type
presentation - -,
locally
quasifinite
107
1 07 107
254
map
of
algebraic
spaces,
map
cont.
cont.
locally
-
of
finite
presentation
107
- -,
of
finite
presentation
50
- -
of
finite
type
107
- -,
of
finite
type
5O
-
open
-
-
- -
-
-
-
-
-
immersion
108
quasicompact
45
- -
-
quasifinite
107
-
141
-
quasiseparated
109
-
-,
reduced
closed
section
of
of
immersion109 110
44
- -
surjective
- -,
union
of
43 Zariski
-
universally
closed
43
- -
universally
bijective
43
open -
sets
44
universally
closed
133
- -
universally
injective
43
universally
open
107
- -
universally
open
43
211
- -
unramified
affine
formal
schemes
211
-,
open
imbedding imbedding
formal
211
algebraic
-
of
local
of
rings,
modification
216
-
218 ringed
spaces
flat flat
schemes
39
complete
205 tensor 206
-
-,
complete
morphism
= map
Hom
206
set
23
4O
-
bijective
43
-
closed
43
etale
Noetherian
"open a
59
- -
faithfully
- -
finite
-
-
flat
-
formally
presentation
-,
53
-
locally
ring,
product
53
47
injective
radic
Space
affine
- -
19
over
Moisezon
- -,
-
44 221
condition
continuous 215,
faithfully
61
Z-open
module
proper
map
-
Mittag-Leffler
general
map
-,
43
separated
I07
211
of
radiciel
123
closed
map
-
surjective
-,
- -,
50 46
I09
imbedding
-,
47,
quasiseparated
separated
-,
of
quasifinite
-
Stein
definitions
-
47
-
105
quasiprojective
46
43
quasicompact
-
type
open
-
-,
finite
quasiaffine
108
-
of
-
quasiaffine
-
locally
-
-
spaces,
-
-
141
map
-
- -,
projective
map -
109
schemes,
-
-
separates
of
flat
etale
(fflat)
subspace
where
128
X
is
scheme"
1 31
Group
150 129 129
56 5O
Picard
56
points
61 43
of
Induction
finite
-
-
46
-
-,
equivalence
-,
generic
-,
geometric
-,
residue
of
1 44 129 field
at
129
255
points,
Schemelike
131
presheaf
30
prime
38
spectrum
projective
n-space
algebraic
over an
space
sheaf
-,
(scheme)
-, m a p
- -,
quasicoherent - -,
in the
rank,
41
sheaf etale
generic,
42 topology
representable ring,
etale
covering
adic graded
graded
of
global,
of a f f i n e
113
on the
formal
continuity
- -,
structure
Sheaf
Criterion
space,
local
Spec
92
spectrum
category
schemes
-
219
-
218
213 for 121
ringed
39 38 38
(class
of maps) of o b j e c t s )
- -, a f f i n e - -,
locally
noetherian
- -, n o e t h e r i a n - -, n o n s i n g u l a r - -,
of d i m e n s i o n
(class
-,
(property
of maps)
34
- -,
(property
of o b j e c t s )
34
ground -
-
field
(Also
strict
projective
subcategory,
- -,
Criterion
Finiteness
sheaf
-,
coherent
-, H o m and E x t
- -,
invertible
- -, q u a s i c o h e r e n t - -,
structure
30 41 241 42 42 104
- -, t o r s i o n
151
- -,
151
torsion-free
32
40
of an a l g e b r a i c 109 closed
109
open
support
109
of a s h e a f
on a
scheme -
52
- on a n a l g e b r a i c
space
surface symmetric
127 236
powers
projective
of
spaces
- equivalence - -,
221
41
subspace,
30
-, a b e l i a n
system
closed
40
space
142
35
open
subspace
- -,
124
closed
97
161 Theorem
-,
110
163
-, W e a k
object
subscheme -
space
of a m a p
p. 233)
45
44
algebraic
see
initial
45
of a m a p
147
factorization
strict
45
31
148 geometric
49
separated
Serre
-
Stein
44
reduced
Serre
-
40
-,
section
-
stalk,
-,
separated
-
41
n over a
32
-,
stalk scheme
212 212
R
stable
205
113 113
quasicoherent
42
204
- -, a s s o c i a t e d -,
sheaf
adic
- -, d i s c r e t e
-
free
of r a n k r
Isomorphism 150
of a l o c a l l y
free
- -,
of a c o h e r e n t
sheaf rank
67
113
locally
sheaf, quasicompact
space,
coherent -
-
141
on a n a l g e b r a i c
relation
effective
- -, q u o t i e n t topologically dense
188 73 73 75 125
256
topology
29
topology associated to a closed subcategory topology, etale, local vs.
-
algebraic space -
-, on schemes
102 211 102 63 59 55 132 44
205 217
space
35
global - -, affine formal schemes -, algebraic spaces -, schemes topology, fppf (schemes) topology, flat (schemes) topology, Zariski, on an -
truncation, of adic ring - -, of a formal algebraic
universal effectively epimorphic family (UEEF) variety prevariety dimension of Zariski's Main Theorem Zariski topology on schemes on an algebraic space
34 235 235 236 235 44 132
INDEX OF NOTATION A.m.n.
(A,m,n numerals)
X'(X a space) S~ xh
3 3,91 47,120 88
f., f a b
fm
114
/A~
121
Xre d Ixl
126
n
132
(~'
104
I? x
141
QCSx,CSx,MSx,ASx,AP X
113
Spf
216
GENERAL ASSUMPTIONS All the schemes (and algebraic spaces) considered are always assumed to be quasiseparated. (See 1.2.25 and II.I.9) In Chapters IV and V, all algebraic spaces are assumed to be Noetherian and separated.
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