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Lecture Notes in Mathematics A collection of informal reports and seminars Edited by A. Dold, Heidelberg and B. Eckmann, ZUrich
41 Robin Hartshorne Harvard University, Cambridge, Mass.
Local Cohomology A seminar given by A. Grothendieck Harvard University Fall, 1961
1967
SpringerVerlag. Berlin. Heidelberg. New York
All rights, especially that of translation into foreign languages, reserved. It is also forbidden to reproduce this book, either whole or in part, by photomechanical means (photostat, microfilm and/or microcard)or by other procedure without written permission from Springer Verlag. 9 by SpringerVerlag Berlin Heidelberg 1967 Library of Congress Catalog Card Number 6729865. Printed in Germany. Title No.7361.
CONTENTS Page ii
Lntroduction w I.
Definition and Elementary Properties of the Local Cohomology Groups
1
w z.
Applications of Local Cohomology to P r e s c h e m e s
16
w 3.
Relation to Depth
36
w
Functors
48
w 5.
S o m e Applications
69
w 6.
Local Duality
81
Bibliography
on A  m o d u l e s
105
Introduction W h a t follows
is a set of lecture notes
seminar g i v e n b y A. G r o t h e n d i e c k in the fall of 1961. of local
preschemes.
and my d u a l i t y Furthermore, "Elements," worthwhile short, best
cohomology
This m a t e r i a l
and g e n e r a l i z e d
at H a r v a r d U n i v e r s i t y
The s u b j e c t m a t t e r
(or relative)
at H a r v a r d
it may appear chapter
III
introduction
[5].
treatment to it.
same as the first edition,
groups
I have
of a s u b j e c t
[16]
[17].
sections
available
of his
thought
again,
since
it a
is o f t e n the
The text is e s s e n t i a l l y except
on
in e x p a n d e d
of 1962
in 1963/64
However,
theory
of sheaves
seminar
in the later
to make these notes
elementary
is his
has since a p p e a r e d
form in his Paris seminar
for a
for minor
the
corrections
and an e x p a n d e d b i b l i o g r a p h y . The study of local c o h o m o l o g y origin paper
in the observation, FAC
varieties
[i0],
already
or c o m p l e t e
local rings.
and then p r o v e
implicit
that m a n y s t a t e m e n t s
can be r e f o r m u l a t e d
statements
groups has its in Serre's
about p r o j e c t i v e
in terms of g r a d e d rings,
This allows about
one to c o n j e c t u r e
local rings,
which
then
may be of use in o b t a i n i n g b e t t e r the finiteness on p r o j e c t i v e
theorems
of Serre
varieties b e c o m e
global
results.
for c o h e r e n t
statements
Thus
sheaves
that c e r t a i n
\
local c o h o m o l o g y modules duality theorem
are
"cofinite."
for p r o j e c t i v e
varieties becomes
t h e o r e m for local c o h o m o l o g y modules. was
developed
further b y G r o t h e n d i e c k
[16], w h e r e he studies relating
~I
X,
1 we define
of an abelian
w i t h respect
right derived sections
of
sheaf
1962 seminar
to
1
theorems,
and Pic of
2.8, w h i c h
as a direct
the local c o h o m o l o g y on a t o p o l o g i c a l closed
subset
in
Fy(F),
Y,
space as the the
Y."
2 we give the first c o n s e q u e n c e s
this theory w h e n applied
groups
in his
of the functor
"with support
In S e c t i o n
is Theorem
F
to a locally
functors F
This a p p r o a c h
sections.
In S e c t i o n groups
the
a duality
local and g l o b a l L e f s c h e t z
and Pic of a v a r i e t y
its h y p e r p l a n e
Similarly
to p r e s c h e m e s :
interprets
of
The m a i n result
the local c o h o m o l o g y
limit of Ext's.
In S e c t i o n 3 we give a s e l f  c o n t a i n e d
exposition
of the n o t i o n of depth and relate
(or h o m o l o g i c a l
codimension)
it to the local c o h o m o l o g y groups
(Theorem
3.8). S e c t i o n 4 contains functors
d e f i n e d on the c a t e g o r y
Noetherian functors
ring.
In particular,
and dualizing In S e c t i o n
cations
before
"general
nonsense"
on
of m o d u l e s
over a
we discuss
dualizing
modules.
5 we give some m i s c e l l a n e o u s
and some t e c h n i c a l
section.
some
(In the lectures,
results
useful
appli
in the next
m o s t of this m a t e r i a l
came
that of S e c t i o n 4, and it is perhaps b e t t e r
read it first,
since it is c l o s e l y r e l a t e d
to
to S e c t i o n
3). S e c t i o n 6 contains are central
results
the duality
theorems w h i c h
of the local c o h o m o l o g y
theory.
R. H a r t s h o r n e Cambridge, July,
1967
Mass.
w 1.
Definition and Elementary Properties of the Local Cohomology Groups
For preliminaries
[4]
Godement
or Grothendieck
be valid for arbitrary abelian
groups Let
subspace
X
Z
[6].
topological
The results spaces
and let
F
X
space
X
we mean
be the subgroup
Fz(X,F)
sections of
whose
F
i m m e d i a t e l y that above.
Moreover,
support
Fz(X,F)
X
V C X
This is possible
V.
Z
of
be a locally closed
X.
since of
(We recall that a
B y an abelian sheaf on a topologi
simply a sheaf of abelian
Choose an open subset
Let
sheaves
is locally closed if it is the
subset.
T h e category of all abelian sheaves on
i s closed i n
to
of t h i s s e c t i o n w i l l
be an abelian sheaf on
of a ~pological
space
we refer
and arbitrary
be a topological space, let
inter section of an open and a closed cal
algebra
over them.
o f X,
subspace
of h o m o l o g i c a l
groups
on
X.
will be denoted by
such that Z
F(V)
Z C V,
~(X). )
and Z
is locally closed in
X.
consisting of all those
is contained
is independent
in
Z.
One checks
of the subset
V
chosen
one sees also that the functor
z (x, F) from
~ (X)
to
(Ab)
of F with support i__nn Z.
is l e f t  e x a c t .
Note that if
We c a l l Z
r
Z
is closed,
(X,F) t h e s e c t i o n s it i s p r e c i s e l y
2
this; if
Z Let
of
X,
Fz(X,F) = F(Z).
is open in
X,
X,Z,F,V
be as above.
U
is any open subset
the natural restriction h o m o m o r p h i s m F(V) * F ( V ~
induces
T h e n if
U)
a homomorphism
r z(X' F)  rz~u(U, F[ u) Thus we may
consider the presheaf U~r
One verifies
FZ~u(U,F
immediately
we will denote it by
~{X)
Definition. subspace,
derived
functors
Hp (X,F)
and
Let and of
r
coefficients
F
Remark: of
a
z
H PZ ( F ) , groups
to
X
F
tl~e c o h o m o l o g y
in
A s with
r Z,
is a sheaf,
and
one finds that the functor
Fz(F)
from
closed
.
that in fact this presheaf
r z(F).
F ~
is leftexact
Iu)
e., (x).
be a topological space, an abelian sheaf on
and
r Z,
X.
Z a locally T h e n the right
respectively, are denoted by
p = 0, I, ..., respectively, a n d are called (resp. , c o h o m o l o g y
a n d supports in
sheaves)
of X
with
Z.
Using the notion of a sheaf extended by zero outside
locally closed
subset
[4, II Z. 9],
we can give a different
interpretation of i n t e g e r s
of
on
r Z(F)
Z,
zero outside of
and
r z ( F ).
and let ~"Z,X
Z.
Let
Z Z be the constant
be this same sheaf extended by
Then
F z ( X , F) = Horn ( Z Z , X ,
F)
and
Fz(F) = Horn
where
(mZ, X, F)
Horn d e n o t e s t h e s h e a f of g e r m s
of h o m o m o r p h i s m s .
T h e r e s t of t h i s s e c t i o n will be d e v o t e d to v a r i o u s of t h e s e c o h o m o l o g y g r o u p s a n d s h e a v e s with s u p p o r t s
properties
in a locally
closed subspace. proposition
a)
I . ].
Let
For any
r
Z
be l o c a l l y c l o s e d in
F s ~(X)
Z
X.
Then
,
(X,F) = H ~ I
Fz(F) = Hz(F) b) of a b e l i a n
sheaves
If on
0 " F ' " F " F " " 0 X,
o  H z c x , r'~ 
and
is an exact sequence
then there are long exact sequences
.1,x
sheaf
~ z ~x,
r~

F,
H z ~ x , r,,~ ~
4
0 *H
where
the connecting c)
HP(x,F) L
If
for
Proof. H PZ
homomorphisms
F e ~(X)
: 0
z (F') " H oZ (;) " H oZ (;,,) h H Z (;,) _ Hi(;) ~) b e h a v e f u n c t o r i a l l y .
is an injective
p > O,
and
object of the category,
HP(F)
: 0
for
p > O.
T h e s e properties follow f r o m the fact that
are derived functors of
I" Z
and
r Z.
then
In fact,
and
H PZ
a,b,c
characterize the derived functors. Propo6ition
1.Z.
is the sheaf associated
H PZ (F)
u ~
Proof. H P'
With
(F)
Z , X, F
p = O,
This follows formally
= F z(F). the for
from the definitions. to the presheaf sheaves
above.
F
is injective, so is
H P Zc~j(U,FIU) = 0 p > 0.
functor s of
for p > 0.
F Iu
Then,
Moreover,
= the sheaf associated to the presheaf If
Set
is an exact functor,
f o r m a connected sequence of functors.
H~
p > O,
HE~. 7 U (u ' FIu)
equal to the sheaf associated
HP'(F)
for each
to the presheaf
s i n c e t h e o p e r a l J o m of t a k i n g a s s o c i a t e d the
as above,
for any
(U ~
FZr_~u(U, F IU)),
U; hence, all
Therefore, also,
HP'(F) = 0
But these properties characterize the right derived
Fz(F), hence
HP'(F) = HPz(F ).
for
(Excision Formula)
Proposition I. 3. closed in
X,
Then for any
and let
Proof.
~
H p (V,F Iv)
Proposition for any
are isomorphic
1.4.
be locally
and such that
F
there is a spectral
H P ( x , F)
of t h e f u n c t o r
F
=~,=
for any ~
F ~,~
If
r
C(X)
functors from
(Spectral Sequence)
HZn (X,F)
where by
X
This follows from the fact that
F z ( V , F[ V)
X,
be open in
Z
Z _C V _C X.
F ~ C(X),
H PZ ( X , F )
in
V
Let
Z
z
(X, F) and to
(Ab).
is locally closed
sequence.
E zpq = HP(x, Hq(F))
F
we mean the right derived functors
r (F).
This proposition
will follow from the theorem
and lemmas
below. Theorem
Let
A.
~.. ,
~',
~.]' be abelian categories
with enough injectives, and suppose given two leftexact covariant functor s F:
C
Suppose furthermore i. e . ,
whenever
I
that
F
takes injectives into Gacyclic object s,
is an injective object of
~. ,
RPG(F(1)) = 0
for
p > 0,
where
for each object
RPG
X
in
are the rightderived ~
the right derived functors
Rn(G
,
there is a spectral
of
F,G,
o F)(X)
~==
and
EPq=
F o r the p r o o f of t h i s t h e o r e m
G
o
I _C F
F
we refer
flasque, then
Then
RPG(RqF(X))
A n y sheaf can be e m b e d d e d
with
G.
F.
the reader of
I
t o [6]. abelian
sheaves
in a flasque sheaf, e.g. ,
the sheaf of discontinuous sections of itself. But if and
of
sequence relating
kin ~(X), t h e ca__q~29orZ A n y ~ n j e c t i v e Sheaf~s flasq~e
Lernrna I. 5.
Proof.
functors
I
is injective,
is a direct s u r n m a n d os
F,
h e n c e flasque itself.
Lemma
Proof. Z
1.6.
If
F
is flasque, so is
Replacing
X
by an open subset
as a closed subset, w e m a y
w e m u s t s h o w that if
U
Fz(X,F) is surjective, So l e t
assume
that
Z
is any open subset of

(; ~ I ~ Z r ~ u ( U , F I U ) , Z r~ U.
V
which contains
is closed in X,
X.
Then
the m a p
FZr. j(U,FIU)
where we suppose that
whose support is in
Fz(F).
i.e.,
F (~
Consider
is a given s is a section of
the zero
sheaf. F
s e c t i o n of
over F
U,
over
on
x,/
the open set their
X  Z.
domains
agrees
with
U t~ (X  Z) = U 
a'
of
F
is
c;,
and whose
over
is flasque,
there
res~ziction
to
and
This
the open set
U ~
~" ~
(~
Lemma
1.7.
Proof.
Embed
to
F
F
(;"
is
the map
If
U).
X  Z
a section
(X  Z )
under
(Z ~
on the intersection Hence,
U <J (X  Z ) ,
restriction
exists
~
F
Clearly
above.
is flasque,
in a n i n j e c t i v e
to
U
Now since
F
over
all of
whose
~"
has
Hence
then
is a section
restriction
is zero.
of
~'
whose
there
of
X,
support
I'z(F)
I,
Z,
is flasque.
Hq(x, F) = 0
sheaf
in
for
and let
q > 0.
C
be
the cokerneh 0,F,I~
Then C
I
is flasque
is also
flasque,
by lemma
HI(x,F) = 0,
HP(x,F) by induction
on
Proof which may
p,

Therefore,
1" (I) "" .I"(C)
and for
~
since
of Proposition
be written
1.5.
and we have an exact
0  ~ 1" ( F )
Hence,
C~0. Chapter
II, 3. 1 . Z ] ,
sequence
~
0
p > I,
H pl(x, C C
by [4,
) = 0
is also
1.4.
as a composite
flasque.
We have the functor functor
rz:
c(x)  (AS),
r
Z
=r
9
F
Z
where
I~ z :
C(X) " C(X)
In o r d e r t o a p p l y t h e t h e o r e m existence
of the s p e c t r a l
takes injective
quoted above,
sequence,
sheaves into
any injective is flasque,
r : C(X) "(Ab).
and
we need only show that
Facyclic
Fz(flasque
and thus deduce the
sheaves.
F
Z
But by the lemmas,
) = flasque,
and any flasque is
r acyclic.
Lemma closed in on
X,
Z,
I . 8. and let
Let
Z
be locally closed in
Z" = Z  Z'
X,
let
Then for any abelian
Z'
be
sheaf
F
there is an exact sequence 0 ~ I"z,IF )   F z ( F )  FZ,,(F )
Moreover,
if
Proof.
F
in
Z',
X,
V
we can write a
Clearly we may assume
Fz,(F)
Then
is flasque,
that
0
on the r i g h t a l s o .
Z
is closed in
is the s e t of t h o s e g l o b a l s e c t i o n s
which clearly
contained in
is open,
Z" = Z r
and
V,
Then
I ' Z , , ( F ) = t h e s e t of e l e m e n t s
is in
Z"
rz(F). i.e.,
of
Letting
Z"
F
X.
with support
V
is closed in
I Z' V.
I~(V, F 1 V) w h o s e s u p p o r t
N o w it is clear that the natural restriction m a p
in
r : F(X,F)F(V, FIV) induces a m a p ~Z((;) = 0 in
Z'.
4Z : Fz(X'F)
is to say
Hence, Now
if
F
is fla sque,
~
3 (~' s r ( X , F )
Z', a'
m u s t have
Proposition I. 9. F
is zero on
o"
" Furtherfore,
Z", i.e.,
a
to say
ha s support
our sequence is exact.
Z,,(X,F),
let
" F Z " ( X ' F )
be any abelian
o 
support
Let
i s surj ective.
Hence,
restricting to
(;.
in
Z'.
Z, Z', Z"
sheaf on
rz,(X, ~ 
X.
Hence,
if
But since ~Z
is surjective.
be as in the l e m m a ,
Then there
are
exact
and
sequences,
rZ(X,F)  r z,,r F)  HI,r
F)  HIcx
and 0 ~ F z , ( F ) ~ F z(F)
Corollary i. 9. abelian sheaf on
X,
If there
Z
" r Z,,(F) * Hz,(F) H I ( F ) ~
is closed are
exact
in
X,
and
F
is any
sequences
0  F z ( X ,F) " F(X,F) ~ r(x  Z,F) H Z1 (X, F) " 
H I(X,F)  * H I(X  Z,F) 
and 0  F z(F) ~F~j.(F I X 
Z)   H Z1 (F) ~ 0
for
p > 0
, F)~
10
where direct
j : X  Z ~
image
is the natural
X
injection,
and
j$
is the
functor.
Proofs.
Take an injective
resolution
~
= (Ii)
of
F:
0 ~ F  I0 ~ I1   . . .
Then the
I.
are all flasque
by lemma
1.5,
and so by lemma
1.8
1
we have an exact sequence
of complexes
orz,( ~ ) rz(~) ) rz,,(~))o and similarly
for the sheaf functors
give rise to the exact sequences For the corollary, and write
Z
for
Z'.
r
F
z"
written
z'
rx(F
9 The se complexes
above.
we take the particular Note that
rZf
case where
) = F,
FXZ
by the definition of the direct
image
HP(F) = 0
HPx_z(F)= R ~ ( F Ix Z).
for
p > 0, and
Proposition
1.10.
the following conditions
(i)
Let
F
functor,
and
r X is exact,
be an abelian
sheaf on
( F ) = j . ( F [ X Z) so
X.
Then
are equivalent:
F is flasque.
(ii) F o r a l l l o c a l l y c l o s e d s u b s p a c e s all
Z = X,
Y
i > O,
i.
H~(X,x F) = 0
and
i
Hy(F) =
0
of
X,
and for
II
(iii)
For all closed subspaces
Y
of
1
X,
Hy(X,F)
= O.
[
Proof. closed.
Let
subset.
Then
(i) => V C X
(ii). Suppose
F
flasque, a n d
be an open set containing
F IV
is flasq~e,
Y
Y _C X
locally
as a closed
so by the excision formula
(proposition
I
we may assume corollary for
that
V = X,
of P r o p o s i t i o n
i > 0,
since
F
i.e.,
Y is closed in
1.9, and noting that is flasque,
X.
Now using the
Hi(X, F) = Hi(x  Y, F) = 0 i Hy(X, F) = 0
we find that
for
and
i>Z,
I'(X,F) " r ( x  Y , F )  H I ( X , F )  0
B y t h e d e f i n i t i o n of f l a s q u e n e s S , U _C X,
F IU
Passing
to associated
Hy(F) = 0
also
(X, F)
= O.
i ,FlU ) = 0 Hyr_~u(U
is also flasque, so
sheaves
H y1
(Proposition
For any open f o r i > O.
1. Z), w e f i n d t h a t
f o r i > O.
(ii) => (iii) i s t r i v i a l . (iii) => (i) f o l l o w s f r o m t h e d e f i n i t i o n of f l a s q u e n e s s exact sequence,
for any closed F(X,F)
Proposition V
= X

integer.
Y,
Y,
~ F ( X  Y,F)
1.11.
Let
l e t F be a n a b e l i a n
and the
1 Hy(X,F)
Y be a c l o s e d s u b s e t of s h e a f on
X,
and let
Then the following conditions are equivalent:
n
X,
let
be an
1.3),
[2
(i)
For all
(ii)
i H y ( F ) = 0.
i<_ n ,
For all open subsets
HI(U r~ V, F )
is injective for
Proof.
(i) => (ii).
X,
the map
~ . : H i ( u , F ) ~ I
and an isomorphism
i = 0,
for all
i < n.
S i n c e t h e c o n d i t i o n (i) i s of a l o c a l n a t u r e ,
it will be sufficient to prove must
of
U
(ii)
in the case where
U = X.
So w e
show that the map oz. : H i ( x , F ) ~ H i ( x  Y, F ) I
is injective for exact sequence to show that is a spectral
i = 0,
and an isomorphism
of the Corollary
to Proposition
H~y(X, F ) = 0 f o r
i < n.
Since by hypothesis
1.9,
Using the
it will be sufficient
But by Proposition
1. 4 t h e r e
the
H yq (F) = 0
H yi ( X , F ) = 0
U,
and for H yi
associated
sheaves,
remaining
case is where
Hq(F))
Pq = H P ( x , < = E~.
for
(ii) => (i). O u r h y p o t h e s i s for any open
i < n.
sequence
i X Hy(,F)
the abutment
for
i = 0
(F) = i=n>
0
for
q<
it follows that
n,
i < n. clearly insures and for 0.
i < n. i = 0
H iy ~ u ( U , F I U ) = 0
that
Hence, and
Then for each
passing to
i < n. U,
The only we have
13
H n 1 (U, F) ~ H n 1 (U :'~ V, F) " H yn ~ u ( U ,
This exact sequence as we pass from sequence
commutes
one
U
of associated
presheaf
U ~
with the restriction
to another,
sheaves.
Hn(u, F)
But the sheaf associated
is zero,
since
let
Y
abelian sheaf on
X.
theorem
also for
i Hy(X, F),
F
X
(F)
~
So w e h a v e
0
be a Zariski space of d i m e n s i o n
T h e n for all
S e e [6,w
analogous
and
Let
be a locally closed subset of
Proof.
n > 0'
to the
~ y (F) = O.
Proposition I. 12. n,
homomorphisms
hence gives rise to an exact
H n I(F)~ R n" 1,(F[ V) " H yn
which shows
F) " Hn(u, F)
flasque
is proved
and let
F
[4, II, Th. 4.15.Z], Hi(x, F).
once one observes
so
Passing to associated sheaves gives
i Hy(F) =0.
where the
Hy(X,F)
= 0
(Proposition 1. I0)) and that t h e f u n c t o r
again a Zariski space of d i m n,
and
The same proof works
that
with direct limits and finite direct sums.
be any
i H y ( X , F) = 0,
i > n,
and for
X,
for I"
i Hyt..aU (U, F[ U) = 0 Hi(F) Y
= 0
for
i >
C ornn%ut
Y
A n y open subset
i > 0
UCXis for n.
i > n.
e s
14
Remark. cohomology
agrees
paracompact groups
For s u i t a b l y n i c e t o p o l o g i c a l with
singular
and locally contractible)*
defined above in terms
precise,
let
(respectively
Y
spaces
the constant
cohomology
one may interpret
of known relative
be c l o s e d i n
X
sheaf
and let G
in w h i c h s h e a f
on
the cohomology
cohomology.
G X).
(e. g. ,
To be
be an abelian
group
Then
H y ( X , G ) = Hi(x, XY;G)
where
the expression
of X relative to X  Y , f u n c t o r i a l isomorphism, E x a m p l e . Let subset, so that each
y s Y,
the pair
X
X
with coefficients in G. This is a canonical, c o m p a t i b l e w i t h the u s u a l e x a c t sequences. be a topological space, and
looks locally like
We c o n s i d e r in
such as above,
is homeomorphic the cohomology
Y,
Y
a closed
Y X ~Rd..That is, such that for V
there is a neighborhood
(Y r~ V , V )
have support
singular cohomology
the right is
to
sheaf
and we can calculate
of
y in
(Y ~ V X
X,
such that
{0} , Y ~ V
X
Hy(X,~).
This sheaf will
it locally.
Over a
hRd).
V
it looks like
(One can calculate this by ordinary topological m e t h o d s via the r e m a r k above: pair
The pair (]Rd, S dl),
(JR d, IR d {0} ) is of the s a m e h o m o s
type as the
so w e have l
*see [14, e x p o s ~ X I I I p.3] and [15, e x p o s e XX, p. i] . C o m p a r e also [4,II5.10.i] w h i c h says that these also a g r e e w i t h C e c h cohomology.
15
i } (~l Z ) ~ H i (m d z)9 .." Hil(IRd; Z )  ~ H iI (Sdl" Z )  " H{0 ,
We k n o w i {d
Hi(]Rd;z)
1,
Z
= 0
for
i > 0,
H dy(X,
Z)
Y.
Y
X. T h e s p e c t r a l s e q u e n c e of
Call this sheaf
H i (X, Z ) ~ 0 Y
']ry, x,
only when
~ry,x
s e q u e n c e of
i s c o n c e n t r a t e d on
Z
t h e s h e a f of t w i s t e d i n t e g e r s o f Proposition 1.4 degenerates,
i = d,
a n d we f i n d
i Hy(X, m) m Hid(x, Ty,x) = Hid(Y,
since
9
if
i s l o c a l l y i s o m o r p h i c to
along
since
Hi(s dl; Z) = 0
9
otherwise.)
Hence the sheaf
in
and
,
Y.
~"y,x )
S u b s t i t u t i n g in t h e e x a c t
P r o p o s i t i o n 1 . 9 , we h a v e
... * Hi{x,
~..) * Hi{x
 y,
7..) * Hi+ld(y,Ty,x
) ~ Hi+I(x,
Z)
~ ....
.
w Z.
Application of L o c a l C o h o m o l o g y to P r e s c h e m e s.
In this section we will apply the notions of local cohomology, as developed X
i n S e c t i o n 1, t o t h e c a s e w h e r e
is a prescheme.
We w i l l f i r s t
showing how to compute Koszul complexes. preschernes, in terms
to theorems
on projective
U and
and let
V
V
are
into
be true if sheaf of
[10,
results
of
groups
and sheaves
are very closely
w 69] r e l a t i n g
cohomology
of sheaves
space to "Ext" groups.
~X'
and
These
by means
valid on quite general
the local cohomology
of Serre
Let
Proposition Z. I. sheaf
groups
Then we will give a theorem
which interprets
space
s t u d y t h e c a s e of a n a f f i n e s c h e m e ,
the local cohomology
of the "Ext" functors.
related
our topological
X
Y
open.
X
V
be a prescherne, U

Assume
that the natural
is locally Noetherian.
for all
By Propositionl.lwe
Proof.
.
.
~ Hu(F)n .
) If
.
n "" H y ( F ) .
.
n>
F
subset,
where
injection maps
immersions.
then the sheaves
~xm~
with structure
be a locally closed
are quasicompact
~XmOdules,
sheave s of
=
X
of
U
(This will always
is a quasicoherent
Hy(F)
are quasicoherent
0.
have an exact sequence n+l  H V (F) ~
of sheaves
17
on any p r e s c h e m e SinceAthe kernel and cokernel of a h o m o m o r p h i s m sheaves
are quasicoherent,
coherent
sheaf by another
and since an extension is quasicoheren~
n
s h o w that the sheaves
Hu(F )
n
and
Hv(F )
j : U "X
is the natural
injection,
R n j , ~ F [ U).
Since
is a quasicoherent
an immersion
is separated,
following theorem,
Theorem morphism X.
B.
sheaves
on
be a closed X,
Amodule
is open in
U
X,
n
then the sheaf
Hu(F )
sheaf on
our proposition
direct
on
f : X  Y
and is just
U,
will result
We
from
and since the
F
images
Rnf~(F)
s u b s e t of
H yi ( X , F ) .
be a quasicoherent sheaf on of
F
are all quasi
Y.
Z. 2.
and for any
be a q u a s i  c o m p a c t separated
and let
See [5, I . i . 6 . 3
Proposition Y
Let
of p r e s c h e m e s ,
Proof.
are quasicoherent.
which we suppose known.
Then the higher
coherent
of one quasi
it will be sufficient to
have already seen (Corollary I. 9) that if
F IU
of quasicoherent
Let X.
i > O,
a n d III
X = Spec A
1.4.10 ]
be an affine s c h e m e , and let
T h e n for any quasicoherent sheaf Hy(F)
F
is the sheaf associated to the
M o r e o v e r , there is an exact sequence
0 ~ H y0 (X, F) ~ H 0 (X, F) ~ H 0(XY, F)  H I(X, F) ~ 0 * S i n c e the q u e s t i o n is local, it is s u f f i c i e n t to c o n s i d e r an a f f i n e scheme. T h e n the f i r s t s t a t e m e n t f o l l o w s f r o m [5, I. 1.3.9] a n d [5, I . i . 4 . 1 ] , a n d the s e c o n d s t a t e m e n t is [5, III. 1.4.17] .
18
and there
are isomorphisms
Hi(X Y ,F) % Hyi+l(X, F) Proof.
For the first
i>0
statement,
we apply the spectral
sequence
(Proposition I. 4) n
H y (x, F) < = E pq = HP(x,
By the previous
proposition,
quasicoherent. p > 0.
Therefore,
So o u r s p e c t r a l
H yq (F))
q Hy(F)
we know that the sheaves since
sequence
X
is affine,
degenerates
E pq = 0
are for
and we have
n Hy(X ,F) ~H0(X, ~y(F))
or,
in other words,
n
Hy(F)
is the sheaf associated
to the Amodule
Hy(X, F). n
The second of Corollary for
1.9,
statement since for
follows directly X
affine,
from the exact
the groups
sequence
Hi(x, F) = 0
i > 0. We n o w p r o c e e d
an affine scheme. expressed
study of the local cohomology
Since the cohomology
in terms
we will restrict
to a detailed
of the groups
n
sheaves
Hy(X, F)
our attention to the latter.
n
Hy(F)
of
are well
by the above proposition,
19
We f i r s t r e c a l l
the definition of the Koszul
[11, C h . IV] o r [5, C h . I I I , w 1. 1]). L e t and let
f
be an element
(homological)
complex
of
A.
A
complex
be a commutative
Then we denote by
of Amodules
(see
K(f)
defined as follows:
ring,
the
The module s
are
Kl(f) ~ K0(f) ~ A
;
K.(f)~ = 0
for
i> 1
and t h e m a p d : Kl(.f)" K0(f ) is multiplication
If f and if
M
by
= (fl'
f. ....
f
n
)
is a finite family
is an Amodule,
of elements
then we denote by
K,(f_;M)
of
A,
the
homological complex K{fl)
where
M

"
is considered
We d e n o t e b y
K ~_;M)
" ~A
K{fn) 
as a complex
A M
concentrated
the cohomological
in degree
zero.
complex
HOmA(K,(f_; A), M)
The homology
and cohomology
of t h e s e
complexes
will be denoted
Xc
by
H , ( f_; M) If
then there
m
and and
H (f_; M ) , r e s p e c t i v e l y . m'
is a natural
are two positive
integers
map of the Koszul complexes
such that
m' > m,
ZO
K(fm' ) . K(fm ) defined as follows: in degree
In degree
z e r o i t i s t h e i d e n t i t y m a p of
one it is multiplication
by
s
elements
also maps follows
of
A
with respect
of t h e K o s z u l c o m p l e x e s
to a module
of t h e h o m o l o g y a n d c o h o m o l o g y (letting
_ f m = (f~l . . . . .
and
One extends this
definition in the obvious way to give maps n
A,
M,
of
and therefore
of t h e s e c o m p l e x e s ,
as
fIn))n :
K,(_rm;M) H,(fm';M)
H,(fm;M)
and K * ( f m ' ; M ) ~ K (fro;M) H (fr n ; M ) '  H
(frn;M)
Thus in the first case we have inverse homology complexes
A,
and cohomology
~. 3.
Let
X; l e t f. = ( f l '
and let the variety
be an Amodule. functor s
of c o m p l e x e s
groups; in the second case we have direct
Theorem spectrum
systems
(i > 0)
systems
and of
groups.
A 9 ..,
be a c o m m u t a t i v e fn )
ring with prime
b e a f i n i t e f a m i l y of e l e m e n t s
of t h e i d e a l t h e y g e n e r a t e
T h e n t h e r e i.s a n i s o m o r p h i s m
be
Y C X;
let
of c o h o m o l o g i c a l
of M
21
lim
i  H y ( X ,
Hi(fro;M)
~I)
m
The complete notes,
p r o o f of t h i s t h e o r e m
and besides,
i s b e y o n d t h e s c o p e of t h e s e
is fairly well known.
with outlining the proof,
We w i l l c o n t e n t o u r s e l v e s
and refer to the literature
for details.
[5, Ch. n I ] The first
step is to interpret
cohomology in terms i Hf (M)
let us define
t h e d i r e c t l i m i t of K o s z u l
of C e c h c o h o m o l o g y . lirn
to be
For simplicity
Hi(f_m;M).
If
~
of n o t a t i o n ,
is a collection
in
of o p e n s e t s o f
by
~i( ~_
X,
,F)
the
ith
t o t h e f a m i l y of o p e n s e t s Finally,
if
of e l e m e n t s sets
X = Spec A, of
A,
F
and if
i s a s h e a f on
Cech cohomology ~,~
,
and if
, n,
_f = ( f l '
where
U.
1
the variety
we will denote
g r o u p of
and with coefficients "'''
t h e n w e w i l l d e n o t e by
(U.), i = 1 . . . .
X,
X
with respect
in the sheaf
fn )
is a finite family
~f
t h e f a m i l y of o p e n
is the complement
in
F.
X of
1
of t h e i d e a l g e n e r a t e d
by
f.. 1
Proposition
C.
[5, Ch. III, 1 . 2 . 3 ]
and with the notations above,
With the hypotheses
there is an exact sequence
0 ~H (M) ~ M ~ ~ O ( ~ f ,
~I)   H f 1 (M)
.
0
of T h e o r e m 2 . 3 ,
22
where
~
is t h e n a t u r a l
HI( ~ , f , Moreover,
restriction
~[)
~
map and there are isomorphisms
H +I(M)
the exact sequence and isomorphisms
T h e p r o o f of t h i s p r o p o s i t i o n
that t h e c o m p l e x coefficients ~f
(M)
for
in
~*(
~f,
l~I)
of t h e c o h o m o l o g i c a l
with the dimensions
are functorial
is elementary.
of
M.
of
~f
with
to t h e d i r e c t l i m i t
Koszul complexes
s h i f t e d by o n e .
in
One shows explicitly
Cech cochains
is canonically isomorphic
~I
i > 0
K (f 
m
;M)
,
but
That is to say, there is a
commutative diagram
o
(M) ~ f ( M ) 
)~
~( (M)
~~
0 cO(~f,~)
~~

~( ~ . f , ~ ) 
Now since taking direct limits is an exact functor, i H f (M)
a s t h e c o h o m o l o g y of t h e c o m p l e x
the exact sequence and isomorphisms proof is functorial isomorphisms
in
M,
D.
Let
by o p e n a f f i n e s u b s c h e m e s , X.
~f
*
one can compute
(M),
and one finds
of t h e p r o p o s i t i o n .
so t h e r e s u l t i n g
The entire
exact sequence and
are also.
Theorem
on
...
X
be a s c h e m e , and let
Then there is an isomorphism
F
let
~
b e a c o v e r of
he a quasicoherent of c o h o m o l o g i c a l
X
sheaf
functors
(i> O)
Z3 ~i( where
Hi(x, F )
Ffunctor
F) ~ Hi(x, F)
,~,
denotes the
from the category
ith
right derived functor of the
of a l l a b e l i ~ n s h e a v e s
on
X.
! For the proof see This theorem in the previous open affines.
[5, Ch. III, P ~ o p .
since
~,~.f
Z. 3.
have both been characterized of t h e m a p
Hi'I(x
 Y, l~i)
each
i.
for
I"(X
Hence they are isomorphic
That the isomorphism
commutes
i n t h e p r o o f of P r o p o s i t i o n
i Hf (M)
of t h e f u n c t o r
0 Hf(M}
thing to say if
M
Definition. groups. an
for
of c o c h a i n c o m p l e x e s
A
is a Noetherian
ring,
are the right derived functors of
Amodules.
Amodule,
Let {Mm)m> 0
be a n i n v e r s e
Mmw  M m
functors for
with connecting homomorphisms
is an injective
such that the map
is the zero map.
i > 0
in the category
We s a y it i s e s s e n t i a l l y
m' > m
i Hf (M)
and
C.
O u r n e x t g o a l w i l l b e t o s h o w if then the functors
Z. Z, w e c a n c o m p l e t e
Y, 1~I) f o r i = 0, 1, a n d a s
can be seen by going back to the isomorphism described
by
i n t h e s a m e w a y , n a m e l y , as t h e k e r n e l
a : M"
i > 2.
X  Y
i Hy{X,I~I)
For the functors
H i ( x  Y, l~I)
1~) by
i s a c o v e r of
N o w w i t h t h e a i d of P r o p o s i t i o n
the proof of Theorem
and cokernel
Hi( ~[r
allows us to replace
proposition,
1 . 4 . 1].
then
z e r o if f o r e a c h
It i s t h e s a m e
i Hf (M) = 0
system
for
of a b e l i a n
m_> 0,
there exists
i > 0.
24
Remark. zero,
1)
If t h e i n v e r s e
then it follows that its inverse
converse
system limit
(M
)
rn
liin ~
M
is
essentially
is zero.
In
The
is false. z)
If we have an exact sequence
of i n v e r s e
0   (M~n)   ( M i n )   ( M ' ~ )
then the middle ones are
one is essentially
essentially
zero.
 0
systems ,
zero if and only if the two outside
We l e a v e t h e p r o o f a s a n e a s y e x e r c i s e
for the reader.
Lemma f = (fl' i > 0
9..,
2.4.
fn )
Let
A
be a commutative
be a finite family
be an integer 9
of elements
Then the following statements
(i) H fi (M)  0
for all injective
Amodules
fin (H i ( f ; A ) ) i n > l i s a n e s s e n t i a l l y
(ii)
Proof.
of
ring, A, are
let and let equivalent:
M.
zero inverse
system 9
By definition,
i
Hf (M)=
l
Hi(fin M ) In
If
M
is injective,
with passage
then
to homology,
Hi(fin;M)
If
(Hi~
in
;A))in>l
Horn(
, M)
is exact.
Hence it commutes
and we have
~ H O i n A ( H i ( f i n ; A ) , M)
is essentially
zero,
then for each
m
there
is
25
an
m'
> m
s u c h that
Hi~_m;M) ~ Hi(fin';M)
is zero,
i H f (M)
and hence the direct limit Conversely,
we can imbed
suppose
H.(fm;A)
{i) i s t r u e .
of t h e s e m o d u l e s
Given an integer
in an injective module
M.
is zero.
m > 0,
Let
cr s HOmA(Hi(fm;A),M) ~ Hi(fm;M) be the i m b e d d i n g m a p .
i H f (M)
If t h e d i r e c t l i m i t
is zero,
then
m 9
there must be an is zero;
m' > m
in other words,
H.(f is zero. arrow
Since
m
cr
!
such that the image of such that the composed
;A) ~
H.(f
In
in
h~
m
(f_
w
;M)
map
M
;A) ~
is a monomorphism,
it follows that the first
is zero.
Lemma
Z.5.
Let
A
be a Noetherian
be a f i n i t e f a m i l y of e l e m e n t s
of
finite type.
system
Then the inverse
(H.(fin; N)) 1
is essentially
zero for
m>l i > 0.
A,
and let
ring, N
of A  m o d u l e s
let
_f = ( f l '
be a n A  m o d u l e
"''' of
fn )
Z6
Proof.
We proc&ed by induction
Then the only value of that
H l(fun;N)
elements
is multiplication
Noetherian
rn
by
sequence, N
and
multiplication
by
rn
n.
First
is
i = 1.
of
N
which must
of finite type.
consisting
Nmw ~ N r n , N
m
since
In other words,
there
N
rn
the map from
.
r n ~ _> r n , of
N
A
is
system
(Nrn)
i.e. , the zero
is essentially
Now let proved
,
n
modules
N
of
of finite type
N,
let
~ 2,
< n
of finite type.
Therefore,
N m , ~ N
sequence
Given
~ = (fl'
[ll,
if is
rn
Hence the inductive
Ch.
and suppose
elements
"'''
of
_f = ( f l . . . . . f
1 )" n
is an exact
m 0
zero.
be an integer
for all sequences
map.
form
is an
m 0 f
n = 1.
of those
for
be stationary,
all of the modules
m ' = r n + m 0,
suppose
One sees immediately
Now the subrnodules
annihilates
.is given,
N
frn. The map
~1 !rn
by
and rn 0 f
such that
to consider
is the submodule
annihilated
an increasing
i
on
A,
the lemma and for all
fn ) s A,
Then for each
and a module i > 0
there
~
IV],
0 ~ H0(~nn ;Hi" ( ~ r n ; N ) ) ~ H"(fm;N)l   H1 (i~na;H'n11 ~ r n ; N ) } ~ 0
To show that the middle sufficient, systems
by the remark are
essentially
inverse
system
2 above, zero.
is essentially
zero,
it will be
to show that the two outside
inverse
27
On t h e l e f t , w e f a c t o r t h e i n v e r s e (where
system map as follows
m ! > m): ~l~n t
HO(
m I
..<
;H.(g
;N))
rrl
HO(nf~ ;Hi(g
;N))
H 0 (if~;Hi (gin;N)) By the induction hypothesis,
the inverse
system
essentially
zero for
i > 0.
Therefore,
when
than
the map
~
m,
the lefthand inverse We p e r f o r m
w i l l be z e r o ,
(Hi(~m;N)) m m'
is enough greater
and hence also
system is essentially a similar factorization
is
~ o ~.
Therefore,
zero. on t h e r i g h t  h a n d
inverse
system.
Hl(nf~
'
(m'
;H.:_I
"N))
I HI (n~n;Hi I(~m;N))
H I(nf~ ,H.:_l(~m;N))
f
For given
i,m,N,
the Amodule
H. (~m;N) 11
i s of f i n i t e t y p e .
Therefore,
applying the induction hypothe sis to this module,
and to
28
the single element
f s A,
we find that for
m'
enough large r than
n
the map
fl
system
is zero.
Hence,
fl o ~
on the right is also essentially
Proposition 2.. 6. f = (fl' .... f )

be an injective
Then
By Lemmas
2.4 and 2.5.
2.7.
Let
spectrum
X,
and let
sheaf
on
X
Proof. any closed
is surjective.
Q , E . D.
be a Noetherian ring,
Amodule.
Corollary
l~i
A
zero.
A
M
be an injective
To show that
for
be a Noetherian
is a flasque
Y C X,
Hie (M) = 0
A, i
let a n d let
>
M
0.
ring with prime
Amodule.
Then the
sheaf. ~/
is flasque,
we must
show that for
the m a p
M
r(x
 Y,
Let
fl .....
i~i)
fn
generate
t h e i d e a l of
X  Y
is the union of the open sets of the family
above,
and from
F (XY, ~I)
and the inverse
be a finite family of elements of
n
Proof.
Let
is zero,
the definition of Cech cohomology
is i s o m o r p h i c
to
~o
(~[f,~I).
~,~
Y.
Then
defined
it follows that
So b y P r o p o s i t i o n
C
quoted above, there is an exact sequence
M K
But
1
Hf (M) = 0 is surjective.
F(X
for
1
Y,YWHf(M)
" 0
IVi injective by the previous proposition,
SO
m,
Z9
Remarks.
Conversely,
a s we s e e by P r o p o s i t i o n
the corollary implies the proposition
C above.
Using results
of i n j e c t i v e m o d u l e s o v e r N o e t h e r i a n r i n g s d i r e c t p r o o f of t h e c o r o l l a r y a s f o l l o w s . ring
A
on t h e s t r u c t u r e
[3], [9]* one c a n g i v e a
An i n j e c t i v e o v e r a N o e t h e ~ i a n
i s a d i r e c t s u m of i n j e c t i v e h u l l s
I
of r e s i d u e c l a s s f i e l d s P
k(p)
of p r i m e ideals
p
of A.
Since a direct s u m of flasque
sheaves is flasque, it suffices to s h o w that
I
is flasque for each
P p. A
But
I P
is a direct limit of Artin m o d u l e s over the local ring
. Hence
[
P of
is a constant sheaf P
Spec A, and zero outside.
I P
on the closed subset
V(p)
Since a constant sheaf on an irreducible
Noetherian space is flasque, the contention follows. N o w w e c o m e to the m a i n t h e o r e m of this section, relating the local c o h o m o l o g y groups on a p r e s c h e m e
Let and let
X
F
be a p r e s c h e m e ,
be a q u a s i  c o h e r e n t
be a q u a s i  c o h e r e n t let
~n =
for each
let
~X / n
Y
be a c l o s e d s u b s p a c e o f
s h e a f of
~XmOdules.
s h e a f of i d e a l s d e f i n i n g
~n.
Then
~'n
to a direct limit of Exts.
Y,
X2
Let
and for each
is a sheaf concentrated
on
n>l, Y,
and
there is a natural injection Hom~
( ~n' F) " r y ( X , F) X
For a homomorphism
of
of t h e u n i t s e c t i o n of
~
in
~n n
into
F
is determined
, w h i c h m u s t be a s e c t i o n o f
by the i m a g e F
with support
Y.
* S e e a l s o [17, II w 7] for a c o m p l e t e d e s c r i p t i o n of the s t r u c t u r e of i n j e c t i v e ~X m o d u l e s on a l o c a l l y n o e t h e r i a n p r e s c h e m e X. U s i n g t h e s e r e s u l t s a l l o w s one to a v o i d the p a i n f u l (2.3)  (2.6). It a l s o g i v e s a m o r e d i r e c t p r o o f of P r o p o s i t i o n 2.1 for X l o c a l l y noetherian.
30
Letting
F
range over the category of 0 K  m O d u l e s ,
consider on the one hand the derived functors
we
Ext,. ( ~n, F)
of
2k
H~
(~n' F), and on the other hand the cohomological functor
Hy1 (X, F), which is a p r i o r i Having a m a p
property i n
into
a derived functor in this category'*
of these functors for
by t h e u n i v e r s a l
As
not
varies,
n
i
H y ( X , F),
i = 0,
w e deduce one for
i> 0
of d e r i v e d f u n c t o r s . i
,
~
these
Hy(X,
Ext's
form a direct
system mapping
so there are homomorphisms
n
"X
n
(*)
~ H y ( X , F)
Performing these h c m o m o r p h i s m s
locally, and passing to associated
sheaves, w e have h o m o m o r p h i s m s
of sheaves
lirn n
Theorem coherent, X
xtiffx((~'n'F)
Z. 8.
If
X
(~)
is local ly Noetherian, and F quasi
then the h o m o m o r p h i s m s
is Noetherian,
i
 H y ( F )
(~) are isomorphisms.
then the h o r n o m o r p h i s m s
If furthermore
(*) are also isomorphisms.
The proof will be deferred until after s o m e auxialiary propositions and l e m m a s. *In fact, it is a d e r i v e d functor, b e c a u s e b y lemma 2.9 below, any i n j e c t i v e ~ X  m O d u l e is flasque, and b y P r o p o s i t i o n i.i0, f l a s q u e s h e a v e s are F y  g c y c l i c , so may b e u s e d to c a l c u l a t e the c o h o m o l o g y g r o u p s H~(X,F).
31
Lemma sheaves of category
[4, II, 7.3.2] Z. 9.A Let X be a p r e s c h e m e , and suppose that
(~XmOdules,
of
~'XmOdules.
Proof.
We m u s t
Then
let
U
and
G
be
G is injective in the
HOm~x(F,G
s h o w , if
F
) is aflasque sheaf.
is open in
X,
then the natural
re striction
Hom~.
(F,G)  Horn
(FIU,GIU)
x
is surjective. to
U
is
FU
Fu
F] U,
homomorphi of
Let
a n d w h i c h i s z e r o o u t s i d e of
sm of
into
denote the unique sheaf whose restriction
F[ U
G.
Since
such homomorphism
into G [ I t
U.
Then a
is the same as a homomorphism
F U _C F, a n d s i n c e
G
extends to a homomorphism
is injective, of
F
into
[4,II 7.3.3] Proposition 2. 10.ALet X be a prescheme, and let G
be sheaves of
~'__modules. X n
Ext
Proof.
(F,G)
F
EXt~x(F,G))
Hom/~ (F,G) "X
functor
H~
G.
and
Then there is a spectral sequence
<= E q = HP(x,
One can represent
any
= I"(Hom0,x(F, G))
as
a composite
32
By Lemma injectives
Z. 9 into
apply Theorem sequence
Homl~, ( F , G ) $ c o n s i d e r e d a s a f u n c t o r i n G , takes X Facyclic objects (Lemma 1.7). Therefore, we may A quoted above to deduce the existence
of t h e s p e c t r a l
of d e r i v e d f u n c t o r s .
[4, ch. II,w 4.12] Proposition 2. II. ALet X be a Noetherian topological space, and let
(F.)
be a direct system of abelian sheaves on
X.
T h e n for
I
each
p > 0,
HP(x, lira F.) = lira HP(x,F.) ~
Proof. F. a n d of 1
F,
Let
1
"~
1
F = lira F..
T o c a l c u l a t e the c o h o m o l o g y of the
take injective resolutions
of the
F.,
1
such a way that they form a direct Then
C(F)
limits
of i n j e c t i v e s .
space,
C(F. )
is a resolution
of
system,
F
r
"~
functor commutes
HP(x, Fi) =
C(F) = lira C ( F . ) . 1
by sheaves which are direct
Now since the direct limit functor is exact,
lirn
and let
Using the quasicompacity
one f i n d s that the
in
1
of a N e 6 t h e r i a n with direct limits.
we have
HP(F (C(F)))
It will therefore be sufficient to s h o w that any direct limit of injective sheaves is flasque, since then w e m a y use the c o m p l e x calculate the c o h o m o l o g y of
F.
But by L e m m a
C(F)
to
I. 5, any injective
sheaf is flasque, and an easy argurnent using qua si compacity shows
33
that a direct limit of flasque sheaves on a Noetherian space is flasque. Example.
The previous
proposition
Noetherian
hypothesis.
For example,
of l x ~ t i v e i n t e g e r s
is false without the
with the discrete
let
X
be the space
topology.
Let
F.
be the
1
sheaf whose
stalk is
into
by killing the
Fi+ 1
alone,
Then
Z
for
lim F. = 0, ~
n>
ith
i,
and
type.
Z. lZ.
X.
Let
Then there
F.
else
but
Let M,N
are
A
a tilde denotes
/ o
z neE
be a Noetherian
be two Amodules
canonical
functorial
i i E x t A ( M , N ) N  E x t , , X
where
Map
1
neE
spectrum
n < i.
stalk, a n d leaving e v e r y t h i n g
nm F(F i)=[] Z / G Lemma
0 for
ring with prime and let
M
be of finite
isomorphisms
N)
taking t h e q u a s i  c o h e r e n t
s h e a f on
X
as.sociated
to the given Amodule. Proof. that
For
H o r n ( M , N)
presentation ~f
i = 0, commutes
the isomorphism
follows from
with localization
when
[5, I, 1 . 3 . 12 (ii)].
As
M
varies
M
the fact
is of finite
in the category
i N o f m o d u l e s of f i n i t e t y p e o v e r A , ExtA(M,N) is the ith (with respect to t h ~ f i r s t variable) right derived functorAof H o m A ( M , N) . Hence, there are canonical A
homomorphisms of sheaves
of
(of functors ~XmOdules)
in
M,
from
t f
to the category
34
EXt~(M,N)~
Extl~
(M, N)
(i)
X Since
is Noetherian,
A
every
M e ~ f has a p r o j e c t i v e ~%
r e s o l u t i o n b y finitely g e n e r a t e d show
(i) is an isomorphism,
side is zero w h e n
M = Ar
free Amodules.
we need only show the r i g h t  h a n d for then they will be derived functors
i Since Ext ~ _ X Commutes with finite direct sums, to show that Extl~ ~ymodule
Hence to
(A, N) = 0
for
i > 0.
it is sufficient
In fact,
for any
F,
Hom~/ ( ~X, ~X i the Ext ~
Extl~ (A, F) = 0, since A = ~X, and vx (Recall that F) = F is an exact functor in F. are defined as derived functors
in the category
2 %
of
~XmOdules,
with respect to the second variable.)
Proof of Theorem 2.8 question is b c a l ,
In the first assertion,
so we may assume that
X
the
is the spectrum
N
of a Noetherian ring
A.
Let
~
= I
and
F = N.
Then we must
show that the h o m o m o r p h i s m s " ~ ~ i ~ lim Extm9 (A/In, N) Hy(N) n ~X are isomorphisms. By p r o p o s i t i o n 2.2, the r i g h t  h a n d
is equal to
i X ,N)N. Hy(
Moreover
using Lemma
2.12 and
noting that direct limits commute w i t h the o p e r a t i o n I, 1.3.9
side
N
[5,
(iii)], we reduce to showing that the h o m o m o r p h i s m s
n
35
are i s o m o r p h i s m s .
This is a h o m o m o r p h i s m
f r o m the category of A  m o d u l e s
of cohomological functors
into itself. For
i= 0
the functors
are isomorphic, their cornrnon value being the set of elements of annihilated by s o m e p o w e r of vanish:
I. For
i> 0
N
and N injective, both
The one on the left since the Ext's are derived functors; the
one on the right by Corollary 2.7 and Proposition I. 10.
This charac
terizes both functors as derived functors, so they are isomorphic.
;) and X as the abutments of spectral sequences, using propositions
;or the second assertion, we express Ext
I I
H i (X, F) Y
i. 4 and Z. I0.
N o w one observes that a direct limit of spectral se
quences is a spectral sequence, and that w e have functorial h o m o morphisms
of spectral sequences, which give the f~)llowing commutative
diagram:
E~q=
lirnn HP(x' E X t ~ x ( ~ n ' F ) )  ~
lirn Ext n T o show that the h o m o m o r p h i s m
(~n,F)" H y ( X , F )
of the
E pq
But this follows f r o m the i s o m o r p h i s m
(~)established above, and f r o m Proposition 2. II, since a s s u m e d Noetherian.
($)
of the abutments is an i s o m o r p h i s m ,
it will be sufficient to s h o w that the h o m o m o r p h i s m s t e r m s are i s o m o r p h i s m s 9
HP(x'HqY(F)) = E~pq
X
is
w 3.
In t h i s s e c t i o n , codimension
w e r e c a l l t h e n o t i o n of d e p t h o r h o m o l o g i c a l
introduced
give a theorem
Relation to Depth
by Auslander
(Theorem
and Buchsbaum
[1].
Then we
B. 8) r e l a t i n g t h e n o t i o n o f d e p t h t o t h e l o c a l
cohomology groups. First
let us recall
a commutative the variety I.
V(I) If
ring,
of
and I
I,
prime
If
A,
If
A
we denote by i d e a l s of
is
V(I)
A
containing
is a c l o s e d s u b s e t of Spec A. N
is an Amodule,
Ass
N
ideals
isomorphic
i s a n i d e a l of
w h i c h is t h e s e t of a l l p r i m e
w h i c h i s the s e t of p r i m e denote by
some notations and definitions.
A
ideals
?
A
such that
N
A
such that
primes
N1
of N:
~ 0.
They
contains a submodule
We
are those N1
A/~ is a commutative
a s e q u e n c e of e l e m e n t s if for each
of
t h e s e t of a s s o c i a t e d of
to
w e d e n o t e b y S u p p N t h e s u p p o r t o f N,
i = 1....
fl'
ring, and
9..,
, n,
f.
f
n
of
M A
an Amodule,
then
is said to be Mregular
is not a zerodivisor
in the module
1
/(fl . . . . . divisor in
f"1  1 )M " (In particular, this m e a n s
is not a zero
M. )
Lemma ideal,
fl
and let
S. 1. M
Let
A
be
be a n A  m o d u l e
conditions are equivalent:
a Noetherian
ring,
of f i n i t e t y p e .
let
I
be an
Then the following
37
(i) H o m ( N , M )
= 0
support is contained in
for all A  m o d u l e s
whose support is equal to No associated
{iii) ~ f s I
Noetherian
3.Z.
r i n g A.
whose
for s o m e
Amodule
of finite type N
V(I). prime
such that
Sublemma
N
V(1).
(ibis) Horn(N, M ) = 0
(ii)
of finite type
Let
of M c o n t a i n s
f
N,M
is
I.
h/iregular.
be modules
of f i n i t e t y p e o v e r a
Then
A s s (Hom(N, M)) = Supp N r~ A s s M Proof.
Using wellknown facts about associated p r i m e s of
m o d u l e s of finite type over Noetherian rings, w e have Ass( H o m ( N , M)) _C Supp (Horn(N, M)) _C Supp N Moreover,
if we express Ar,N
N
a s a q u o t i e n t of a f r e e m o d u l e
, 0
then we have
0 ~ H o m ( N , M ) ~ H o m ( A r ,M ) = M r from which follows
Ass(Hom(N,M))
C Ass M r = Ass M
Thus we have an inclusion
A s s ( H o m ( N , M)) _.CSupp N r~ A s s
M
38
Conversely,
s S u p p N r  ~ A s s M. M 1 _~ A / ~
s Supp N ,
In particular
all of Spec
A/~
has positive
rank over the integral
and therefore injection
N'
the support (since
isomorphic
N/ N must be
of
r
N/$N
N is of finite type),
domain
to an ideal of
has a submodule
M
z Ass M,
which is torsionfree
N' " M 1,
composition
~
Then since
Since
quotient module
i d e a l of A s u c h t h a t
be a prime
let
A/~
H e n c e it h a s a
a n d of r a n k o n e o v e r
A/~
Thus there
s Hem
and we can define
(N,M)
is an to be the
of t h e f o l l o w i n g h o m o m o r p h i s m s
B y our construction, the s u b m o d u l e of Horn(N, M ) generated by is i s o m o r p h i c to
A/9.
Proof of L e m r n a
,
so $ s A s s
(Horn (N,M)).
Q.E.D.
3. I.
(i) => (ibis). Trivial. (ibis) => (ii). If Horn(N, M ) = 0, and Supp N = V(I), then by the sublemrna,
V~I) r~ A s s M = A s s
associated p r i m e of M
contains
(Horn (N, M)) = ~,
SO
no
I.
(ii) :> (i). If Supp N C V(1), then again by the s u b l e m m a , Ass(Horn(N, M)) _C V(I) r~ A s s M =
by hypothesis,
so H o m
( N , M ) = 0.
39
(iii).
.....
~
(which are finite in number
since
(ii) < = > of
M
to say none of the
Let
"~ i
~l
contains
I
be t h e a s s o c i a t e d
r M
primes
is of finite type).
Then
is the same as to say their union
r
~J ~ i= 1
does not contain
i
an element element of t h e
f ~ I f s A
~ i'
is
Mregular
let
3.3. M
i s t h e s a m e a s t o s a y t h a t it i s i n n o n e
Let
A
be a Noetherian
be an A  m o d u l e
let
of f i n i t e t y p e , a n d l e t
= 0
(ibis)
be an
I
n
ExtA(N,M ) = 0
for s o m e
elements
fl
. . . . .
of finite type
Amodule
fn s I
N
of f i n i t e t y p e
i < n.
forming an Mregular
sequence.
Proof.
(i) => ( i b i s ) .
(ibis) => (ii).
H o m { N , M ) = 0, This
Trivial.
We p r o c e e d
there is nothing to prove.
Mregular.
be
i < n.
Supp N = V(I), and for all integers 3
N
for all A  m o d u l e s
such that Supp N C V(I), a n d for all integers
(ii)
ring,
Then the following conditions are equivalent:
(i) E x t A ( N , M )
such that
But to say that an
so w e a r e d o n e .
A,
an integer.
which is the same as to say that there is
w h i c h i s i n n o n e o f t h e ~ i"
Proposition
ideal of
I,
soby f
b y i n d u c t i o n on
So s u p p o s e
the lemma
there
n > 0.
n.
If
n_< 0,
Then in particuhr
exists an
f s I
which is
gives rise to an exact sequence as follows:
40
0  " M f~ M   ~ M / f M   ~
where
the first map is multiplication
and from the exact sequence for
i < n  1.
elements
f2'
Then clearly of
Therefore, ....
' f
by
of Ext's,
f.
Now from
our hypothesis Exti^ (N, M / f M ) = 0
it follows that
by t h e i n d u c t i o n h y p o t h e s i s ,
fn ~ I
f, fZ . . . .
,
0
n
there
which form an M/fMregular is an
Mregular
are
sequence.
sequence
of
n
n<
0,
elements
I. (if) => (i).
We p r o c e e d
by i n d u c t i o n
is nothing to prove.
So s u p p o s e
Mregular,
is an exact sequence.
so there
n > 0.
on
n.
If
Then in particular
there
fl
is
0 ~ M ~ M ~ M/f IM 0 Now of
fZ . . . . . I,
fn
is an M/riMregular
so by the induction hypothesis
for all Amodules integers
of finite type
i < n  1.
N
sequence
of
it follows that with support
By the exact sequence
in
of Ext's,
n
1
elements
i E x t A ( N , M / f 1 M) = 0 V(I), and for all this implies
that the natural m a p
E x t A (N, M)
is injective
for all
i < n.
map is also the zero map.
But since Hence
f l ~ I,
Ext~(N,M)
fl = 0
kills for all
so this
N, i <
n.
41
Remark.
Note that the lemma
is used only in the implication implication N, M
(ii}  > (i}
of finite type.
"Supp N _C V{l}"
of
A,
depthiM, fl ....
"fl'
Let M
A
M.
and Buchsbaum
Corollary
sequences namely,
Corollary Mregular,
3.4.
ring,
N"
let
of elements
of
I
{The Idepth
and I is
of M is what
o f M. }
Then all maximal
have the same number
of
depthiM.
3.5.
Let
A,I,M
be as above.
then depthiM = depth I M/fM
+ 1
of M,
{Note t h a t
is a local ring,
be as above.
A,I,M
be a n i d e a l
exist elements
[1} c a l l t h e I  c o d i m e n s i o n
Let
I
Then the Idepth
sequence.
we say simply depth
nor
the hypothesis
such that there
A
ideal,
elements,
n
The
Noetherian
of finite type.
f i n i t e . ) If
the maximal
A
are nilpotent in
n
which form an Mregular
and is therefore
Mregular
f
be a Noetherian
integer
n < dim M,
Auslander
one may replace
be an Amodule
is the largest
' fn s I
....
hypotheses)
of the proof.
is valid without assuming
by
and let
( i b i s ) => {ii)
Moreover,
Definition.
{with i t s N o e t h e r i a n
If
f e I
is
42
Corollary
3.6.
Let
depthlM =
where
M~
be a s above.
A,I,M inf
(depth h~
is considered
)
as a module
over the local ring
and its depth is the (usual} depth with respect
Proof.
fl . . . . .
If
f
~ I
Then
to the maximal
A~
,
ideal
f o r m an Mregular sequence,
n
then the canonical images
f , . . . ,
in
M[
~/~
, and f o r m an
1
n
regular
of t h e
f.
in
1
A
are
sequence f o r a n y ~ s
Therefore
depthlM < depth M i for each
~
s V(1).
For the converse, we will prove the following statement: n'
is an integer such that for each
n' < depth 1M".
i
~ Viii,
W e proceed by induction on n' > 0.
M~
is not associated to
__> 1, so
~A~
is not associated to an
f ~ I
M.
which is Mregular.
induction hypothesis, n'

<
the case
Then for each M;
~
n' < 0
s V(1), ,
, then
depth
i.e.,
Then by L e r n m a B. I, t h e r e e x i s t s N o w using Corollary 3.5 and the
we find
1
n' _< depth M I
n',
being trivial. So suppose
"If
depth (MIfM)
43
for each
s V(I).
Hence
n'  1 < depthiM/fM which implies
that
n' < d e p t h i M .
Definition.
Let
X
be a locally Noetherian
Y a closed s u b s e t
of
T h e n the Y  d e p t h of F,
or depthyF_ , is the
inf (depth Fx )" xzY
is an affine scheme,
say
R em_ar k. Y = V(1)
and
If F =
X
X,
A4, t h e n
and
depthyF
F
prescheme,
a coherent sheaf on
thus that the notion of Idepth depends
of the ideal
I.
Restating
X = Spec A, by C o r o l l a r y
= depthlM,
We o b s e r v e
X.
3.6.
only on the radical
t h e a b o v e t h e o r y i n t h e language o f p r e s c h e m e s ,
we have Proposition let
Y
and let
be a closed n
3.7.
X
Let
let
subset,
be an integer.
(i) E x t ~ v ( G , F ) = 0
F
for all coherent
(ibis) E x t ~ v ( G , F) = 0
sheaves
G
sheaf on X are equivalent: with
i < n.
for s o m e coherent sheaf
and for all integers
prescheme,
be a coherent
Then the following conditions
Supp G C Y, and for all integers
Supp G = Y,
be a locally Noetherian
i < n.
G
with
44
(ii)
depthyF
> n.
(iibis) depth F
for all
> n
x s Y.
x
Now we have a theorem the local cohomology
Theorem
let
Y
which relates
sheaves:
3.8.
Let
X
be a locally Noetherian p r e s c h e m e ,
be a closed subset,
and let
n
be an integer9
i (i) H y ( F )
(ii)
= 0
depthyF>
Proof.
(i)
for all
G
and for all
by i n d u c t i o n o n n > 0.
So s u p p o s e
Therefore,
G ~
X,
n ;
We proceed
in the category Y,
be a coherent sheaf on
n
is satisfied for
> n  1.
F
T h e n the following conditions are equivalent:
(i) => (ii).
depthyF
support in
let
for all i<
there is nothing to prove. condition
t h e n o t i o n of d e p t h t o
n  1,
i <
3.7,
of c o h e r e n t n 
E x t n" 1 ~X(G, F)
i.
If
n<
0,
in particular,
so by the induction hypothesis,
by p r o p o s i t i o n ~.y
Then,
n.
Ext,,
sheaves
(G, F) = 0
X on
X
with
In other words, the functor
45
from
~
Y
of i d e a l s of
to sheaves Y,
on
~rn
Let
~y
be the sheaf m
_
m,
are in the category
~ r n ' "
m' > m
is leftexact.
and for each positive integer
Then the sheaves
for
X
let
~m
Y
so the surjections
_~ y ,
0"m    0
give rise to injections
0 'Ext..
In o t h e r w o r d s ,
(
, F ) " E x t
all the homomorphisms
1 ( X
,,F)
in the direct
system
(Ext~ 'I (~rn ,F))m are injections. By Theorem ~.8 the direct limit X of this system is _~yl(F), which is zero by our hypothesis. Hence e a c h of t h e t e r m s
of o u r d i r e c t
system
is zero,
y ( ~ / ~ Y' F) : 0. We have a l r e a d y ]Bxt n,,,~..I for
f<
n  1,
so f r o m
(ii) => (i). notation,
Proposition
Suppose that
by T h e o r e m
By Proposition "m
3.7,
these
h a v e s u p p o r t i n Y,
EXt~x ( ~ /
seen that
3.7 we conclude that
depthyF
> n.
Z. 8 w e h a v e f o r a n y
m
in particular,
d e p t h y F _ > n.
Using the above
i,
vX Ext's
hence
are
0 f o r i < n,
i Hy(F)
= 0
for
since the sheaves i<
n.
y, F) = 0
QE.D.
46
Corollary prescheme, Then
3.9.
and
X  Y
Y
Let
X
a closed
be a connected subprescheme
locally Noetherian
such that
d e p t h y 0 ~ x _ > Z.
is connected.
Proof.
By the theorem, for
i = 0, 1.
our hypothesis
By Proposition
implies
1.11,
that
this implies
that
the m a p H O(x, {~X)  ~ H 0 ( x 
is bijective. observe
To complete
t h e p r o o f of t h e c o r o l l a r y ,
that a prescheme
considered
as a module
Y, ~ X )
X
is connected
over itself,
connected => H O(x, ~ )
we need only
if and only if
is indecomposable.
= H O(x  Y, ~ )
H O ( x , ~ X ), For then
X
is indecomposable => X  Y
2 ~
is connected. If
X
H 0 ( U i ' (~X
is disconnected, where
is decomposable.
Ui
are nontrivial
elements
e I , eZ
e l e Z = 0;
e I = e l,
z
is indecomposable,
sets where
different z
e Z = e Z. hence
disconnected.
eI
suppose
idempotents from
so
H 0 ( X , (~X~
For each
i s decomposable.
and such that x ~ X,
e. 1
X, h e n c e
is zero and where
),
that is,
1 = e I + ez;
the local ring
is zero at X
sum of
H 0 (X, ~'X )
e I , e z e H0(X,~x
zero,
one of the
c a n b e z e r o a t e v e r y p o i n t of closed
is the direct
are the components,
Conversely,
Then there
H0(X, ~X)
x.
Neither
i s t h e di sj o i n t s u m o f t h e e Z is zero.
Thus
X is
47
Remark. codimension
The Corollary applies in particular if
at least two and
as it does for example
if
X
intersection
in a nonsingular
Hartshorne
[8], w h o p r o v e s
of depth,
and applies
is normal, ambient
or if
X
prescheme.
the corollary
is of
Sz
of Serre,
is a complete For details,
directly
from the definition
intersections.
case of a local Noetherian
3.10.
maximal
ideal
Let
be a module
m
Let
A
and residue
ring,
be a local Noetherian field
of finite type,
Then the following conditions
the above
are
k.
Let
and let
ring with
X = Spec
n > 0
A,
U = X{m}
be an integer.
equivalent
(i) depth M > n. (ii) Exti(k, M) = 0
for
(iii) HilX, ~i) * HilU, ~i) injective
see
give the following
Corollary
M
has the property
it to the study of complete
In t h e p a r t i c u l a r results
~X
Y
for
i < n. is bijective
i = 0.
(iv) H~ (M) = 0 l m}
for
i < n.
for
i < n
and
.
w 4.
In this section, ring. of
We w i l l d i s c u s s
Amodules
will discuss
F u n c t o r s on A  m o d u l e s
A
will denote a commutative
Noetherian
various
leftexact
functors
from the category
to the category
of abelian
groups.
In particular,
dualizing functors
we
which will be used later in the section
on duality. As a matter (Ab)
of notation,
let
= the category of abelian g r o u p s = the category of A  m o d u l e s
d If
4~
= the category
is anidealof C~
If
A,
of Amodules
,
of finite type
let
= the category of A  m o d u l e s
with support in
= t h e c a t e g o r y of A  m o d u l e s support in V( e~t )
of finite type with
is any category,
we will denote by
~
the opposed
category.
Lemrna fr om
to
4.1. (Ab).
Let
T
be a contrav~riant
Then there
: T ~Hom
is a natural
A( ., T(A))
additive functor
functorial
morphism
V(
49
Proof.
For any
m
s M,
let
: A ~ M
be the morphism
m
which sends
1 into
T ( M ) ~ T ( A ) .
m.
Then
T(~m)
For a fixed element
we have an element
of
is a morphism T{M),
of HomA(M , T(A)).
as
of
m
s M
varies,
Thus we have defined a
morphi sm. ~b(M) : T ( M ) * H o m A ( M , T ( A ) )
One checks and
t h a t it i s f u n c t o r i a l
HomA(M,T(A))
morphism
~f
the natural
to
4.2.
(Ab).
Let
In f a c t ,
Amodule
is an isomorphism
Proof.
If
since the functor
T
structures,
T(M)
it i s a
if and only if
~
be a contravariant
additive functor
9 , T(A))
T
is left exact.
is an isomorphism,
Horn is.
then
On t h e o t h e r h a n d ,
T
is leftexact
suppose
T
is left
since
T
is
T h e n in the f i r s t p l a c e
T(A r) ~ H O m A ( A r ,
T(A)) = T(A) r
is an isomorphism
for any positive integer
additive.
M
exact
if one gives
Then the morphism
~b : T  * H O m A (
exact.
M.
of Amodules.
Proposition fr om
in
Now if
sequence
is any Amodule
r,
of finite type,
there
is an
50
A r  ~ A s ~ M  ~ 0
Letting
s t a n d for t h e f u n c t o r
T'
HOmA( 9
T (A)),
we h a v e a n
exact commutative diagram
0

T (M)
~
T (A s')
~
T (A r )
N
0
~
T'(M)
w h e n c e by t h e 5  l e m m a ,
Definition. denote b y
Sex(
functors from
to
T ' ( A s)
T(M) ~ T I(M)
~., 8
~,
~
explained thus:
If
~
N
~
T'(A r)
is an isomorphism.
a r e two a b e l i a n c a t e g o r i e s , )
B
we
t h e c a t e g o r y of l e f t  e x a c t ( c o v a r i a n t ) ( T h i s n o t a t i o n i s due t o G a b r i e l , a n d i s
s = s i n i s t e r , ex = exact. )
C o r o l l a r y 4.B
The categories
Sex( ~ f ~
Ab) a n d
~
m a d e e q u i v a l e n t by t h e f u n c t o r s
T ~
T(A)
I
HOmA(',
for
T e Sex( ~ f
and ~
I)
for
I s
9 , Ab)
are
51
Remark. exact functors injective, T(A)
correspond
HOmA{ 9 ,I)
is injective,
injective for all
Note that in the equivalence to the injective
is exact.
Amodules
Lemma an additive is a natural
modules.
M
4.4.
functorial
M ~
if
T
I
is
is exact,
a module
HomA(M,I)
the
I
is
i s exac_t
of finite type.
Let
contravariant
ring,
4.3,
Indeed,
On the other hand, if
since over a Noetherian
if and only if the functor
of C o r o l l a r y
~
be an ideal in
functor from
A,
~f
and let
to
(Ab).
T
be
Then there
morphism
: T ~HornA( ", I) where
I = l~n T(A/ n ) . n
Proof. some power
of
Let ~
of finite type over
Then
M s n
, say A/
n
is annihilated
In other words,
So b y L e m m a
T(M} >Horn
M
M
4. 1 t h e r e
by
is a module is a morphism
n(M,T(A/4~n} : H o m A ( M , T ( A / n ) }
A/ Furthermore,
there
is a natural
morphism
Horn A (M, T ( A / ~ n)) ... lim Horn A (M, T ( A / ~ n)) n
then
SZ
a n d t h i s l i m i t i s e q u a l to and therefore
H o m A ( M , I ),
HomA(M , ")
these morphisms,
since
IV[ i s of f i n i t e t y p e ,
c o m m u t e s with d i r e c t l i m i t s .
Composing
one obtains a morphisrn
T (M) ~ Horn A (M, I)
w h i c h i s e a s i l y s e e n t o be i n d e p e n d e n t of t h e a n d f u n c t o r i a l in
The morphism
a n i s o m o r p h i s m if a n d only if
If
for each
M s
n' > n,
T
of L e m r n a 4 . 4 i s
~
is left exact.
is an isomorphism,
~
since the functor Hom is. exact, and let
chosen above,
M.
Pr0positiDn 4.5.
Proof.
n
then
T
On t h e o t h e r h a n d , s u p p o s e
~f
is left exact, T
be a m o d u l e a n n i h i l a t e d by
is left 4M n.
Then
the morphism n I
T(M) "H o r n A ( M,T(A / ~
is an i s o m o r p h i s m ,
by P r o p o s i t i o n 4. Z.
g o e s to i n f i n i t y , we f i n d t h a t
Corollary 4.6.
~
))
Taking the limit as
n I
is also an isomorphism.
The categories
Sex( ~
fo
, Ab)
and
a r e m a d e e q u i v a l e n t by t h e f u n c t o r s T ~
lirn T(A/,~M n) n
for
T zSex(
~fo
Ab)
53 and I ~
Proof.
HOmA(" ,I)
of those elements
annihilated by
Proposition
4.7.
If
I
suppose
n
~
I
i s t h e u n i o n of t h e s e s u b 
to h a v e s u p p o r t in
In the e q u i v a l e n c e
correspond
Proof. Conversely,
~
l i r n H o m A ( A / ~ n , i ) ~ I. n t o t h e s u b m o d u l e of I c o n s i s t i n g
is isomorphic
s i n c e it w a s a s s u m e d
exact functors
I ~
We n e e d o n l y c h e c k t h a t
N o w H o r n ( A / r i M n , I)
modules
for
V( ~
).
of C o r o l l a r y
4.6,
the
to the i n j e c t i v e m o d u l e s .
is injective,
then
T
and let
isexact,
H O m A ( 9 , I)
is exact.
I= lira T(A/
,~n).
We
n
wish to show
I
is an ideal in
A,
injective. and
f:
extends to a morphism,
I s
~
theorem, of
A,
Thus
Since ,
f
~
A,
is of f i n i t e t y p e n
annihilates
the ~topology
also annihilate s
as follows:
then
~
f
f : A ~ I.
of
and
(A
01, ~ I
f:
n.
M r r~
such that ,
and
By Krull's
i s i n d u c e d by t h e r
a
being Noetherian),
for some
i.e. , there is an integer f
a morphism,
~I
i s a n i d e a l of
So suppose
m o r p h i sin.
It wiI1 b e s u f f i c i e n t t o p r o v e t h a t i f
 topology
M r e~ ~ C
and therefore
f
not, factors
54
P
fl r
But now we have an injection r
of m o d u l e s in
,
fl : and
T
we have a rnorphism
(it/
is exact.
r
r
~
~
Therefore,
~I,"
fl
I
;
extends to a morphism
m
fl
If
p : A " A /
f=fl
0
p
: A/Mr
~
r
" I
is the canonical projection morphism,
is the desired
e x t e n s i o n of
f.
Hence
then
I
is
be an injective Amodule,
let
inj e c t i v e .
Corollary 4.8. be of
i d e a l of
an
I
A,
and let
with support in
Proof. H o m A ( 9 , I). i s in~ective.
Let
Let
I In
= H~
V( ~w ).
Then
T s Sex ( ~f
9
Then by Proposition
4.7,
(I) In
Ab) I~
be the largest
submodule
is injective.
be the exact functor = lirn
T(A/~
n)
55
We n o w c o m e t o t h e s t u d y o f d u a l i z i n g f u n c t o r s .
Proposition A
(i. e . , s u c h t h a t ~i
T a Sex(
(i}
e
Ab)
,
Let
4.9.
For all
444
A/~ 9
be an ideal of finite colength in
is an Artin ring), and let
Then the following conditions are equivalent:
M a
~f
,
T(M)
is an Amodule
of
finite
type, and the natural morphism M
TT(M)
~
(defined via the isomorphism (ii) where
T
~
.
is exact,
of L e m m a
4.4) is an isomorphism.
and for each field
is a maximal
k
ideal containing
~
of the f o r m
A/
~.,
1
, there is some
1
isomorphism
T ( k ) ~ k.
(i) => (ii).
Proof. above.
Then
T(k)
Since
k
be a field of the form
is of finite type by hypothesis;
~*l., h e n c e m u s t be i s o m o r p h i c 1 Z
annihilated by n > 0.
Let
T
is additive,
TT(k) = kn ,
A/~t i
moreover,
to
kn
whence
as
it is
for some n = 1.
Thus
T ( k )  k. It r e r r ~ a i n s t o s h o w faithful, i.e., nonzero,
if
O #0,
T then
is exact. T ( Q ) ~ 0.
then there is a surjection Q ~ k . 0
,
We f i r s t Let
show that
O s
~L
T be
is
56
where
k
i s a f i e l d of t h e f o r m
A/~
Since
..
T
is exact,
1
0 ~ T ( k ) '~ T ( Q ) . But
T ( k )  k / 0, Now
T
T ( Q ) / 0.
hence also
i s l e f t e x a c t by h y p o t h e s i s ,
t h a t it i s r i g h t e x a c t .
so we need only. show
Let
O   ~ M , ,.M be a n i n j e c t i o n of m o d u l e s
f
in
Apply
T,
and let
Q
be the cokernel" T(M)
Since

T(M')
is left exact,
T


0
we have an exact sequence
0 * T ( Q ) ' T T ( M ' ) " T T ( M ) But
is isomorphic
TT
T ( Q ) = 0,
Therefore,
to the identity functor by hypothesis. and since
T
is faithful,
Q = 0, i . e . ,
T
i s exact.
(ii) > (i). If series above.
s
~..~f
, then
whose quotients are all fields Therefore,
composition Hence
M
series
T(M)
length of
M).
since
T
k
is exact,
M
has a finite composition
of the f o r m T (M)
, as
will have a finite
w h o s e q u o t i e n t s a r e of t h e f o r m
is of f i n i t e l e n g t h
A/4~ti
T ( k ) ~ k.
(in f a c t , i t s l e n g t h i s e q u a l t o t h e
57
For the second assertion where
M
is a field
k
of
(i), w e first c o n s i d e r the c a s e
of t h e f o r m
in some way, and the natural map be z e r o ,
M
k
is easily s e e n not to
~ T T ( M )
so is the middle one.
(since any
M s
~ O,
of t h e f i r s t e x a c t s e q u e n c e i n t o
If the two outside arrows
51emma,
' and
 T T ( M " )
with a canonical morphism
the second.
~L
then w e d e d u c e a n exact s e q u e n c e
0  T T ( M ' )
M
TT(k) ~
" M ~" M " "" 0
is a n exact s e q u e n c e ,
Of
Then
k " T T ( k )
is any module in
0 "M'
the
i
h e n c e it is a n i s o m o r p h i s m . Now if
together
A/~
are isomorphism,
then by
T h u s by i n d u c t i o n on the l e n g t h
has finite length), we see that the
~
natural morphism
M
~ T T ( M )
is an isomorphism
Definition. a maximal
for all
Let
ideal.
M s
A
f ~ ~,~
be a c o m m u t a t i v e
Then a functor
the equivalent conditions of Proposition functor for
A
at
~
Q.E.D.
Noetherian
T z Sex( 4.9,
ring,
~L ~ , Ab),
and
satisfying
is called a dualizing
An injective module
I
with support in
58
V( , ~
)
is called a dualizing module
T = Horn( 9 , I)
Proposition
is a dualizing functor for
4.7,
these dualizing
there
is a natural
functors
ring with maximal or
ideal
,%~
Remarks.
1)
ideal.
A
at
onetoone
Let
A
,%~
,%~
)
If
be a Noetherian
ring, A
is a local
ideal
~f
the categories
and
at
AM
Indeed,
between
functor"
understood.
Then a dualizing functor for
A~
A
"dualizing
thing as a dualizing functor for the local ring ~
(By
correspondence
, we say simply
with
, if the functor
,~
and dualizing modules.
"dualizing module, "
maximal
for A at
,~
~e~
a
is the same
at its maximal involved are
isomorphic. Z) Amodule.
Similarly, Then
is dualizing for categories 3) and
I
A/~
n m o d u l e
,W~ I
n
I A
~i Let
let
A
be a local ring,
has a natural
structure
and of
I s ~ ~module,
i f a n d o n l y if i t i s d u a l i z i n g f o r
~.
an and
I
Again the
involved are isomorphic. A
be a Noetherian
and Amodule.
For each
n
HOmA (A/~
I).
if and only if for all large is dualizing for
A/~
n
,~
ring,
a maximal
n
1, 2,
Then
I
is dualizing for
enough
n
(and hence for all
at the maximal
....
ideal
let
I
ideal,
~
n
/
be the A
n
at n), Indeed,
59
observe
that if
HomA(M,I) all
~ H o m f ~ ~
M s
Hom
M s
n
A/
is annihilated
(M, In).
n
Thus
is exact for
A/
HOmA(k,I) ~ Hom
k =A/~
an injective
essential
extension
M r~ N = 0,
then
Definition. is an essential language,
is exact for
I)
n,
n modules
of f i n i t e t y p e .
for any
f o r t h e s a k e of c o n v e n i e n c e ,
An injection
M~
if whenever
N
n,
where
If
we also call
M
is an
hull,
P
is a submodule
I
where
an injective
of
is an P
(EckmannSchopf
such that
is injective.
of
By abuse
[3, C h . II, p . 2 0 f f . ] )
Amodule
has an
isomorphism.
A
be a Noetherian
I is dualizing for
A
if and only if it is an injective
field
k.
M
M.
[2]; s e e a l s o Then every
hull
Let
hull of the residue
4.10.
I
an injective
hull of
unique to within (nonunique)
Proposition
of A  m o d u l e s
Amodule,
M ~ I
be a ring with identity.
an Amodule
the definition of
N = 0.
extension
Theorem.
injective
then
hull.
Definition.
A
,
Thus the result follows from criterion (ii) of Proposition 4.9. We n o w r e c a l l ,
Let
n
~
enough
(k,I) n
A/An
by
HomA(
if and only if for all large
(M, In)
Moreover,
~f
local ring.
Then
of
6O
Proof. Hom(k,I)
~ k,
show that of
I
First
k ~ I
such that Hom(k,Q)
V( ~
), I
is an essential k ~Q = 0.
s o Q = 0.
= 0.
But
I,
Then any homomorphism because
k ~ I
Hom(k,I)
~
k,
Corollary ideal.
a prime. Z oo'
Then
A
let
(3,
implies
in
~ k,
has that
I.
To
be a submodule
(3
Hom(k,I)
also
I
we must
support I
in
is injective,
into
is an injective I
must
extension,
is a dualizing
A
hull of
k.
have
its image
in
k
is a field.
and
Thus
module.
be a Noetherian
has a dual/zing
ring,
module
at
= pZ
,
and ~
, unique
A =Z
automorphisms
,
andlet
module
for
are
~ A
at
~
where
p
is the group
the units in the ring of
padic
P i n t e g e r s.
Let functor.
A
k,
isomorphism
Then a dualizing
whose
k
Let
Let
since
suppose
4. 11.
(nonunique)
Example.
I
k
k.
is an essential and
extension,
dualizing
of
Then
is dualizing.
we can embed
hence
hull of
On the other hand,
to within
I
Then
Now I
is an injective
a maximal
that
and so in particular,
have
so
suppose
be a Noetherian
local ring,
We list without proof various
and
D
properties
of
a dual/zing D,
most
of
is
61
which follow f r o m the above d i s c u s s i o n .
see
For more details,
Matlis [9].* 1.
D
gives an isomorphism _f C~
its dual category gives
a
11
DZ Z.
systems,
D(M),
b e t w e e n s u b m o d u l e s of and vice versa,
with
length.
M,
for any
It
and
M e
i s i s o m o r p h i c to the i d e n t i t y .
D transforms
d i r e c t s y s t e m s of
correspondence
~f
into inverse D
T h u s , p a s s i n g to l i m i t s ,
a n d vice versa.
give s a
b e t w e e n d i r e c t l i m i t s a n d i n v e r s e l i m i t s of m o d u l e s
of f i n i t e l e n g t h o v e r 3.
~f
D is exact and preserves
correspondence
q u o t i e n t m o d u l e s of Finally,
O
of t h e c a t e g o r y
A.
In p a r t i c u l a r ,
the functor
D = Horn( ", I),
is a dualizing m o d u l e , gives an a n t i  i s o m o r p h i s m (ACC) = t h e c a t e g o r y of
Rmodules
(DCC) = t h e c a t e g o r y of A  m o d u l e s
where
I
of th e c a t e g o r i e s
of f i n i t e t y p e , a n d M satisfying the following
three equivalent conditions: (i) M h a s t h e d e s c e n d i n g c h a i n c o n d i t i o n , (ii)
M i s of c o  f i n i t e t y p e , i . e . , H o m ( k , M ) i s a
finitedimensional vector space over (iii) M in this correspondence, of
~
is a s u b m o d u l e D(~) = I;
k, a n d of
M
has
In , for s o m e
D(1) = Hom(I,l) = ~.
c o r r e s p o n d to s u b m o d u l e s
of
support
I
in
n. T h e ideals
by %.he m a p
*See also Macdonald [21], w h o e x t e n d s t h e r a n g e i z i n g f u n c t o r to " l i n e a r l y c o m p a c t " m o d u l e s .
of
the
dua i
62
9,. " ~
D(.~/ Oi, )
, I
a n d the ideals of definition c o r r e s p o n d to the s u b m o d u l e s of
of
finite length.
Proposition
4. l Z .
of l o c a l N o e t h e r i a n
rings,
B.
B " A
and let
be a s u r j e c t i v e
morphism
be the maximal
i d e a l of
Then 1) If
to
Let
D
Amodules Z) If
is a dualizing functor for
is a dualizing functor for I
is a dualizing module for
is a dualizing module for Proof.
The first statement
HomB(
then
I9
restricted
A. B,
then
I' = H o m B ( A , I )
A.
definition of a dualizing functor the functor
B,
9 , I)
follows immediately
(see Proposition 4.9 (ii)).
is a dualizing functor for
associated dualizing m o d u l e is
from the
lirn H o m B ( A / ( ~ A )
A,
and its
n, I) = I',
n
since any morphism
of
~
A.
of
A
into
I
annihilates
Hence
some power
63
We n o w g i v e t h r e e
specific examples
Example
A
a field
k 0,
extension
of
1.
be a Noetherian
and suppose that the residue k 0.
Then any module
dimensional
vector
functor
by
D
Let
of dualizing functors.
space over
k 0,
local ring containing
field
M s ~
f
k
is a finite
is a finite
and we can define a dualizing
D(M) = HOmk0(M, k0) The correspondingd u a l i z i n g module is I = lirn Horn
{A/,~,~n, ko ) k0
w h i c h c a n be t h o u g h t of a s the m o d u l e of c o n t i n u o u s h o m o m o r p h i s m s of
A
with the
4~adic
Example
2.
Definition. residue
field
k,
topology into
k 0.
T h e c a s e of a G o r e n s t e i n
A Noetherian
local ring
is called a Gorenstein
ring. *
A
of d i m e n s i o n
n,
ring if
EXtA(k'A) = I kO for ii/=nn *See also the paper of Bass [13], and [17, ch. V w 9] for more information about Gorenstein rings.
with
64
Examples.
Any regular
local ring is Gorenstein.
Gorenstein
ring is CohenMacaulay.
local ring
A
is called a CohenMacaulay
its dimension:
See Cor.
Proposition n.
where
Let
A
X
i ~ n,
~
be a Gorenstein
r i n g of d i m e n s i o n
E x t n (M, A)
and its associated Spec A,
Proof 9
First
and Y = {~t
}
= 0.
is a n exact s e q u e n c e
in
~'k
9
Indeed,
i.
M e
M = k.
'
and for
~
~ E x t I ( M ' , A )
If t h e t w o o u t s i d e g r o u p s a r e z e r o ,
~2
,
, then there is an exact sequence 9
ExtI(M,A)
~.,,f~
~ 0
by induction on the length of
for all
M s
if
ExtI(M,,,A) ~ E x t I ( M , A )
for a n y
n
H e , (A)  H y ( X ,
is the closed point.
we show that for any
Exti(M,A)
n
dualizing module is
0 ~ M' ~ M ~ M "
for
if its depth equals
Then the functor
is dualizing,
Thus
ring
that a Noetherian
3.10.)
4. 13.
M
any
(We r e c a l l
Any
M,
,
so is the middle one.
one shows that
= 0
and all
i ~n,
s i n c e by h y p o t h e s i s
it i s t r u e
O'x),
65
It f o l l o w s n o w t h a t t h e f u n c t o r
M
is exact for
~
M s
Extn(M,A)
~~
On the other hand,
hypothesis,
so we have a dualizing functor.
by Theorem
2.. 8, i s
k
I = lirn Extn(A/ ~ k In particular,
Proposition dimension
n.
4.14.
Let
n
A
 k
by
Its dualizing module,
,A) = H ~
H ~n
we have shown that
Extn(k,A)
(A)
(A)
is an injective
be a Noetherian
Then the following conditions
are
hull of
local ring of
equivalent:
(i) A i s G o r e n s t e i n . (ii) A i s C o h e n  M a c a u l a y ,
Proof.
(i) => (ii)
(ii) => (i). for
i < n,
Since
by Corollary
proof of the previous and
all
i < n.
is left exact,
A
is dualizing.
proposition.
is CohenMacaulay,
3.10.
proposition
~
H~(A)
by the previous
Therefore,
T : M
n
and
Therefore, that
Extl(k,A)
we can show as in the
Extl(M,A}
= 0
for all
the functor
Extn(M,A)
and so by Proposition
= 0
4.5 and Theorem
2.8,
k.
66
E x t n ( M , A ) m& Horn (M, H ,n~
n
H ~
But
(A)
was assumed
an exact functor, a n d
to be
Therefore,
dualizing.
T
is
T(k) ~ k.
It r e m a i n s to s h o w that T
(A))
for
Ext1(k,A) = 0
i > n.
Since
is exact, the functor
M
~': ~ Extn+l (M, A)
is left exact for
M z
~
f
, and so, as above,
E x t n + 1 ( M , A ) ~ H o r n (M, H ,n+ ~ 1 (A))
But
n+l H , ~ (/%) = 0
Extn+l (M, A) = 0
Thus funct
by Proposition I. IZ, since for all
M
z
f vt~
n = d i m (Spec A). , and therefore the
or
M is left exact.
~
P r o c e e d i n g thus by induction on
that
E x t i ( M , A) = 0
A is
Gorenstein.
Exercises. dimension
n,
E x t n + Z (M, A)
for all
i > n
and all
i > n, M s
~.f
one show
s
Thus Q.E.D.
1) and let
Let
A
be a CohenMacaulay
local ring of
be an ideal genera ted by a maximal
67
A  sequence.
n
Show that
H~(A)
is an essential
and therefore that the dimension is independent of ~ Z) n
H~
(A)
Let
A
e
3) S h o w that if q u o t i e n t of a r e g u l a r
module,
A
power
series
in
n
namely
connecting
B
~I
we know there
4. 1 4 ( i i ) ,
by an ideal A
~
(i.e., a
which can be
is a Gorenstein
ring.
x ]] b e t h e ring of f o r m a l n of b o t h
n H ~ (A).
So b y t h e u n i q u e n e s s
is some noncanonical
of the
isomorphism
them. in terms
of t h e t h e o r y
be the m o d u l e of continuous differentials of
this case will be the free Let
Show that
This falls under the situation
T h i s f a c t c a n be i n t e r p r e t e d Let
ring.
so we have at hand two dualizing modules,
Hom contk(A,k ) and
~njective hull,
Hom(k,A/~
a n d d e d u c e t h a t in P r o p o s i t i o n
A = k[[x 1 .....
variables.
of the above examples,
exercise.
is a complete intersection
local r i n g
Let
space
,
n H~a (A) injective.
generated by a Bsequence), then
3.
A/~
of
e = 1 for a Gorenstein
be as in the previous
it is sufficient to a s s u m e
Example
of t h e v e c t o r
S h o w also that
is anindecomposable
extension
f n = An(f~l)
called the module
Amodule
by the
generated
nth exterior
by
power
of (continuous) ndifferentials.
dx of
of differentials.
A/k , l
,
f~l
Then
9
9
9
which in ,
d~"~
9
n
which i s n
is the
68
free A  m o d u l e generated by a dualizing module.
Res:
dx I ^
... A dx n . So
H ~n
(~n)
is
N ow t h e r e s i d u e m a p * i s a c o n t i n u o u s h o m o m o r p h i s m
n
H,~(~n)
 k
,
canonically defined, which gives rise to a canonical i s o m o r p h i s m
~(~n)
_~, Horn contk(A, k)
between the two dualizing modules.
*The d e f i n i t i o n of the r e s i d u e map is rather subtle. In the t e r m i n o l o g y of [17, ch. VI, w 4], it is the trace map, a p p l i e d to the last term of a residual complex, w h i c h can be i d e n t i f i e d w i t h ~ (~n).
w 5.
S o m e Applications
In t h i s s e c t i o n w e g i v e s o m e a p p l i c a t i o n s developed
so far.
Ext group";
In particular,
various
of d i r e c t i m a g e
p a r t on b a s e c h a n g e s ,
Proposition let
Y
that on
X
> n,
Let
and let
with support in
for all
Y.
X let ~.fy
be a locally Noetherian F
be a c o h e r e n t be the category
n
Proof.
Proposition
First
case
X
localization
X
of c o h e r e n t
such
sheaves
isomorphism,
n
Horn Cx(G, Hy(F))
note that since depth y F
= 0
for all
5.7.
Thus
may not vanish. proposition,
prescheme,
sheaf on
Then there is a functorial
E x t ~ X (G, F) ~
, F)
the
f ,,~y,
G s
_Ext _ ~ ,Iv ( G
especially
w i l l be u s e d i n S e c t i o n 6.
5.1.

of b a s e c h a n g e ; a n d t h e c o h e r e n c e
S o m e of t h i s m a t e r i a l ,
be a c l o s e d s u b s e t ,
depthyF
we study "the first nonvanishing
properties
sheaves.
of t h e t h e o r y
i < n, n
To establish
and for all
is the first integer
> n, 
G s
the sheaves
le ~..f y,
for which
the sheaf isomorphism
modules
and show that this definition commutes
to a smaller
affine.
Extn~x(G,F)
of t h e
we make a definition for the corresponding
is affine,
by
We t h e n i n v o k e p r o p o s i t i o n
with Z. Z
in
70
and L e m m a
2.12 to pass f r o m m o d u l e s to sheaves, and glue together
the i s o m o r p h i s m s
w e have defined over the affine pieces of
So suppose that ring
A,
A,
and let
and
N
seen ttRt
an
X
is the p r i m e s p e c t r u m of a Noetherian
Y = V(J), Amodule
i ExtA(M,N)
X.
= 0
F=
N,
whose for
where
Jdepth
i < n
J
is an ideal in
is
~ n.
and for
M s
Then we have f ~j.
Thus
the functor n
M ~
is left exact,
ExtA(M,N )
and so by Proposition
4.5 there is a canonical
functorial i somorphism
E x t A (M, N)
~ H e m A (M, I)
where lim__~ E x t A k
I =
Now if B = AS ,
S
is a multiplicative
then the isomorphism
is the localization Ext's
of thisr one,
X
system
in
A,
obtained as above for since localization
and
if
Bmodules
commutes
with
and direct limits.
Proposition let
(A/Jk, NI = H (NI
n > 0
be an integer,
such that
points
5.Z.
Let
be a locally Noetherian
and let
Exti~v(G, F) = 0
z s Supp G
X
be coherent
G, F
for
such that depth
i < n. F
z
= n.
Let
7.
Assume
prescheme,
sheaves
on
be t h e s e t of that
Z
71
has no embedded
points.
Then
n
Ass EXt~x(G,F
Lemrna let
Z
5.3.
Let
) = Z
X, Y, F
be the set of points
y z Y
be as in Proposition 5. i, and such that depth
F
= n. Y
Then
A s s Hn(F) C ~ y
and moreover, of
z
points,
is in
if
Z
~
z
t
i s any point of
A s s H~(F).
Z,
then s o m e
In particular, if
Z
specialization
has no e m b e d d e d
then
Hn(F) = Z y
Ass
Proof. be the closure
Let of
z.
z
be any point of
Y,
Then by Proposition
and let
Y'CY
5.1,
n
Ext ~X (
Therefore,
,, F) ~ H o r n
using the expression
of a Horn g i v e n in s u b l e m m a
3.2,
(
,,
(F))
f o r t h e s e t of a s s o c i a t e d we have
H n A s s Ext n~ X ( ~y,, F) = Y' ~ A s s  Y (F)
primes
72
Now if depth
X, xC
z s Ass F
z
= n,
and observe
and
z s Z.
that depth
F
z
0
(Use Corollary can only increase
3.6 and Proposition
3.7,
when one specializes
z.) S u p p o s e on t h e o t h e r h a n d t h a t
.x n xC
s o i t s s e t of a s s o c i a t e d
nonempty. and also
z ~ Z.
If
z'
z' s Y',
is one of its a s s o c i a t e d i. e. ,
z'
1~ > n,
primes
primes,
is a specialization
Proof of Proposition 5.2. Proposition 3.7, depth
Then
Let
m u s t be then
of
Y = Supp G.
z'
z. T h e n by
and, as in the proof of the lemrna,
we find that Ass Ext
n
'X
But since we assumed applies to show
Ass
Corollary Extn~ (G,F) v X
5.4.
that
(G, F ) = Y r~ A s s H n ( F ) = A s s H n ( F ) . Y Y
Z
has no embedded points,
Hn(F) = Z
y
the 1emma
the proof.
U n d e r the h y p o t h e s e s of P r o p o s i t i o n
has no embedded associated
Next we study the behavior Ext's,
which completes
5. Z,
primes.
of t h e l o c a l c o h o m o l o g y g r o u p s ,
and dualizing functors under base change.
will be useful in Section 6 on duality.
These results
n
A s s y H (F),
73
Proposition topological
5.5.
spaces; let
l(y); Y' = f and let
Let Y
f : X' * X
b e a c o n t i n u o u s m a p of
be a closed subspace
F' be an abelian sheaf on
of
X,
X'.
and let
Then there
is a spectral sequence Epq
Proof. functor Now
p n = Hy(X, Rqf,(F ')) => H~(X', F')
We i n t e r p r e t
l"y o f., f,
where
takes injectives
the functor f.
is the direct image functor.
into injectives,
exact functor
f
theorem
s t a t e d i n S e c t i o n 1,
A,
the spectral
Corollary q > 0,
[5, Ch. 0, 3 . 7 . Z].
sequence
then the spectral
of a c o m p o s i t e
functor.
R q f , ( F ') = 0
sequence degenerates,
of
for
and
H i (X', F')
i. Example.
Then
which gives the existence
If i n t h e p r o p o s i t i o n
Hiy(X, f,F') ~ for all
since it is adjoint to the
Thus we can apply the
of d e r i v e d f u n c t o r s
5.6.
l"y, a s the c o m p o s i t e
y ' = y r~ X ' ,
supportin X',
Let
X'
be a closed subprescheme
and for any abelian
H cx', FIx'
sheaf
F
on
of X
X. with
74
J all
Corollary
5.7.
be an ideal in
B,
i
Let
B ~ A
and let
M
be a morphism be an
of r i n g s ,
Amodule.
let
Then for
we have isomorphisms
Hij(M B) ~_ H i j A ( M ) B
where a superscript considered
B
X = S p e c B.
exact in this case since [5, Ch. I I I . ,
Corollary
Remark. remarkable
"by restriction
We a p p l y t h e p r e v i o u s
X' = S p e c A a n d
of s c a l a r s .
corollary
"
to the case where
an
is
f~
is
affine morphism
1.3.2].
The result
2.8 to express
of C o r o l l a r y
5.7 becomes
more
of N o e t h e r i a n
assumptions
when we can
the groups involved as direct limits
F o r t h e n it s t a t e s t h a t t h e r e is a n i s o m o r p h i s m
lirn E x t Bi ( B / J
n , M B )  ~ ~
n
which is surprising isomorphic
denotes that module
The direct image functor
f : X' ~ X
in the presence
use Theorem of E x t ' s .
applied to an Amodule
as a Bmodule
Proof.
,
lira EXtA
(A/(JA) n , M)
because the Ext groups are in general
b e f o r e t a k i n g t h e lknit.
For example,
the natural map Ex i
,
n
CB/J , B/J
 Ext iB.j B/J,
B/J
if
not
A = M = B/J,
75
is in general term
is
not an isomorphism
0,
since the righthand
However,
shows that the first nonvanishing
Proposition let
i > 0,
and the lefthand one not.
proposition
rings,
for
N
type, and let
5.8.
Let
B ~ A
be a Bmodule, n
be an integer.
iV[
the following
Ext's will be isomorphic.
be a morphism an
Amodule,
of N o e t h e r i a n b o t h of f i n i t e
Then the following conditions are
equivalent:
where
(i)
E x t B~ ( N ,
M B) = 0
(ii)
i ExtA(N A,M) = 0
N A = N: ~ ) B
Furthermore,
for all
i< n
,
for all
i < n
,
A .
if the conditions are
satisfied,
then there
is
an i somorphi sm
n
n
ExtB(N, M B) ~ EXtA (NA, M)
Proof.
Let
J
be a n i d e a l of
T h e n it f o l l o w s t h a t Supp NA = V ( J A ) . Theorem for all i
HjA(M)
3.8, i~ = 0
the
n,
B
such that
Supp N = V(J).
Now by P r o p o s i t i o n
condition (i) is e q u i v a l e n t
to saying
B. 7 a n d
Hij{M B) = 0
and the condition ~ii) is equivalent to saying
for all
i < n.
equivalent by Corollary 5.7.
But these latter two conditions are
76
From
Corollary 5.7 and T h e o r e m
depthjM B = depthjAlVi. then this c o m m o n
3.8 one also deduces that
If the conditions (i) and (ii) are satisfied,
depth is
_> n.
Therefore by Proposition 5.1
there are i s o m o r p h i s m s n
E x t B ( N , M B) ~ H o r n B (N, H
(MB))
and n
,~
ExtA(N A, M)
n
" H o m A ( N A, HjA(M))
But by Corollary 5.7, again, the modules are isomorphic, hence our if
B "A
Extn's are isomorphic.
is a morphi.sm of rings; if
presentation; and if
M
H j ( M B)
N
H jnA (M)
and
(In general,
is a B  m o d u l e of finite
is an A  m o d u l e , then there is an i s o m o r p h i s m
H o m B ( N 'M B ) _~ HomA(NA~vi )B
Proposition J D J'
be two ideals
finite type. respect
5.9.
Denote
to the
of
A,
by a hat
J'adic
A
Let
and let ^
i
topology.
~
~
H i
^ ( J
M
)A
ring,
let
be an Amodule
the operation
i s o m o r phi s m
Hj(M)
be a Noetherian
Then for all
of completion i
there
of with
is an
77
Proof.
Using Theorem
groaps as direct limits
Z. 8 t o r e p r e s e n t
o f E x t ' s,
the local cohomology
we need only show that the natural
maps
EXtA(A/jn
are isomorphisms. is of f i n i t e t y p e ,
But since
9
,
so to t e n s o r
^
i
Ext
Proposition 5. i0.
U.
to
U
Let is
F F 0,
M
,M) (~)A ~
of p r e s c h e m e s , be a p r e s c h e m e ,
and let
F0
be any coherent sheaf on
conditions are equivalent:
so tensoring with
of w h e n t h e d i r e c t i m a g e o f a
X
Let
Y = X  U,
and let
J'
it with
This proves the proposition.
sheaf, under a morphism
an open subset, let
n
since J D
Now we study the problem
on
and since
is the same as to tensor
it with
is an isomorphism.
coherent
A,
g r o u p on the r i g h t i s a n n i h i l a t e d by
= A/J n A
~I )
is flat over
M ) = Ext A ( A / J
On the other hand, the n
2k
( ~ / ~n
we have
Extt (~/~n A
J
M) . E x t i
n
be an integer.
X
is coherent. let
U
be a c o h e r e n t
be sheaf
whose re striction
Then the following
78
(i)
RJ,(F0)
is coherent
for all
i
<
where
n,
j : U ~ X
is the inclusion map. H~(F)
(ii)
Proof.
is coherent for all
i <
n.
By Corollary I. 9 there is an exact sequence
0'H~F)__ Fj,(F0)II!~F)0
,
and t h e r e a r e i s o m o r p h i s i n s R q j , ( F O) ~ y H q+l(F)
for
q > O.
kernel
Now the statement
and cokernel
an extension [5,
Ch.
of a map
of a coherent
of the proposition of coherent
sheaves
sheaf by a coherent
follows are
since the
coherent,
and
sheaf is coherent.
O, 5 . 3 ] Moreover,
there
is also the following
result,
which we will
not prove. Theorem. prescheme, equivalent (iii)
[16, Corollary VIIIII3] A If X is locally embeddable in a nonsingular
then conditions
Z
and
to the following
condition:
for all
,
depth whe r e
(i)
z e U
(Fo) z + codim
is the closure
of
z
in
(ii)
(Y r~ Z , X.
of proposition
Z)>n
are also
79
Examples. coherent _> 2
A necessary
is that for all
in
Z,
2) f:X
1)
and let
since for Suppose
"'Y
be a proper
Rqf0,(F0)
Indeed, f, o j,
z e Ass
j,
Y
where
f to
are
Y
for all
on
U.
let
Noetherian,
Then the i < n.
as the composite
f0~ ~
injectives
satisfied,
is locally
of
carries
into injectives,
functor so there
is
sequence
Now the sheaves
o) are
Rqj,(F 0) E pq Z
Thus the
pr opo sition.
p,
since
f
It f o l l o w s t h a t t h e a b u t m e n t i<
5.12
be the restriction coherent
be
( F 0 ) z = 0.
of proposition
morphism,
j,(F0)
be of codimension
depth
EzPq
and for all
that
Y :% Z
F 0,
we can interpret
and
a spectral
F 0,
the conditions
f0 : U  " Y
sheave s
z e Ass
condition
coherent
terms
are
for
q
coherent
is a proper m a p
by the
n,
sheaves
for
q < n
[5, Ch. III, T h e o r e m 3.Z. I].
i a f0,(F0)
terms
<
are coherent for
n'. Problem.*
closed
subset.
_~y(F) = 0
Let If
for all
n
equations,
be a prescheme,
is an integer,
i >
It i s s u f f i c i e n t
X
n,
find conditions
Y under
be a which
and for all quasicoherent
sheaves
Y
by
that
but not necessary.
*This problem
and let
is studied
be locally For
in
describable
example,
[19].
if
X
n
is the cone
F.
80
over a n o n  s i n g u l a r the cone,
plane cubic curve,
a n d if
then the condition is satisfied for
necessarily
Y
n = I,
is a g e n e r a t o r but
Y
of
cannot
be d e s c r i b e d by one equation at the vertex of the cone.
w 6.
Local Duality
In t h i s s e c t i o n w e p r o v e a c e n t r a l cohomology theory, over a regular forms
projective if for
F
the duality theorem.
local ring (Theorem
of t h e d u a l i t y t h e o r e m
These theorems
is a coherent
It i s s i m p l e s t
6.3); however,
valid over more
[10, n ~
Theorem
s h e a f on p r o j e c t i v e
there is a perfect
Hi(X,F) X
to state
we also give
general
local rings. of S e r r e ' s
1], w h i c h s a y s t h a t
rspace
X = IP r ,
then
pairing
ri Ext ~ x ( F ,
n ) " k
of finitedimensional vector spaces over sheaf of
of t h e l o c a l
s h o u l d b e t h o u g h t of a s l o c a l a n a l o g u e s
duality theorem
i _> 0,
theorem
rdifferential f o r m s on
k,
r
where
]P , i s o m o r p h i c to
~
is the ~ x (  r  I).
E v e n the proof, which uses the axiomatic characterization of derived functors, is similar. W e will apply the duality t h e o r e m to determine the structure of the local cohomolog~r groups
i H~
(M)
over a local ring, and w e
end by giving an application of duality to a t h e o r e m on algebraic varieties.
We b e g i n b y r e c a l l i n g the Ext groups
briefly Yoneda's
(for d e t a i l s a n d p r o o f s ,
interpretation
s e e [7, e x p o s ~ 3]) .
of
82
Let
~
be an abelian category with enough injectives.
A complex in
~,
is a collection
together
with coboundary maps
all
d n+l 9
K
n, by
fn:
K n . L n + s
such that for
We d e n o t e t h e c o h e m o l o g y
morphisms are
(pn)ns
s
of a c o m p l e x
K
into
L
gn
We d e n o t e b y
elements If complex
f e Horn A K,
S
~
the
in t h e
group of
such that for all
s n+l + (i) p
n 9
dE
the g r o u p of h o m o t o p y c l a s s e s
of
(K, L).
such that
0
of m o r p h i s m s
ns
Two such morphisms
s.
pn 9
L)
o_ff K
if there exists a collection
n+sI
i s a n o b j e c t of
c : A ~ K 0,
is exact.
of degree
= dL
HS(K,
(fn)
a morphism
p n : K n  L n + s  1
of m o r p h i s r n s

,
H o r n S ( K , L)
said to be homotopic
fn
~
with the coboundary maps
We d e n o t e b y
of
in
is a collection
which commute
t w o c o m p l e x e s.
map
d n : K n ~ K n + l
are two complexes
K, L
of d e g r e e
L
f, g
of objects of
Hq(K). If
into
d n = 0.
(Kn)n s
~
Kn = 0
,
a resolution for
n < 0,
of
A
together
is a with a
such that the sequence
A
~
A resolution
K 0 ~
K 1
(K, ~ )
of
A
i s s a i d t o be e x a c t i f
n~,
83
Hq(K) = 0 Ki
for
q > 0.
is said to be injective
if each
is injective. Now let
of
The resdution
A,
A
and let
the complex
b e a n o b j e c t of
L
be a n y c o m p l e x .
( H o r n (A, Ln))
we define an element
of
determines
Theorem
~
Corollary. and
L
of
f.
H o r n ( A , L)
f ~ HornS(K, L),
by composing Hem
c
(A, L ) ,
H S ( H o m (A, L ) ) ,
with and
which depends
Thus we have defined a natural
HS(K, L) " H S ( H o m
E.
a n d if t h e c o m p l e x maps
Given an
be a resolution
s,
cs :
natural
class
of
(K, ~)
We d e n o t e b y
is a cocycle in
an element
only on the homotopy map for each
nzZ"
, let
Horn(A, L) s
One shows that this element hence
~
s
If L
(K, e ) consists
(A, L))
is an exact resolution entirely
of injectives,
of
then the
defined above are isomorphisms. If i n p a r t i c u l a r
i s an e x a c t i n j e c t i v e
B
resolution
is another of
B,
object of
then there are
canonical i somorphi s m s
HS(K, L)" ExtS(A, B)
Now we apply this theory below.
A,
to the situation we will meet
f0
84
Proposition
6.1.
Let
~ ,
~'
having enough injectives, and let
T:
an additive covariant leftexact functor. of
~
,
be a b e l i a n c a t e g o r i e s ,
If
~"
~'
A
and
be
B
are objects
then there are pairings
RiT(A) X E x t S ( A , B)  Ri+ST(B)
for all
i, s
Proof.
(L, U)
of
RIT (A)
and
and
Pick exact injective resolutions
A
and
RiT(B)
T (L). If f
then
B,
respectively 9
(K,~)
and
T h e n we c a n c a l c u l a t e
a s t h e c o h o m o l o g y of t h e c o m p l e x e s
is a m o r p h i s m of
T (f) is a m o r p h i s m of
T(K)
K into
into
L
T(L)
of d e g r e e of d e g r e e
T(K) s, s,
w h i c h on p a s s i n g t o c o h o m o l o g y g i v e s a m o r p h i s m
f , : RiT(A) . Ri+ST(B)
for any c l a s s of
s. f.
One s e e s e a s i l y t h a t
f~
d e p e n d s o n l y on t h e h o m o t o p y
Thus using the isomorphism
of th e c o r o l l a r y a b o v e ,
we h a v e d e f i n e d a pairing
RiT(A) for all
i, s.
X ExtS(A, B)  R i+s T(B)
85
6. Z.
Corollary ideal, and let
Let
A
be a r i n g , l e t
be A  m o d u l e s .
M,N
~
be an
Then there are pairings
i ( M ) X ExtJ(M'A N} ~ H i+j H~ ~ (N)
for all
i,j. Let
Proof.
of
T
be t h e f u n c t o r
M
~
H 0 (M)
The
H~(M)
for
i > 0
are right derived functors
by C o r o l l a r y Z. 7.
T h e o r e m 6.3. of dimension
n
(Duality) Let
A
(See proposition 4.13),
be a G o r e n s t e i n r i n g let
~
ideal, let I = H n (/%) be a d u a l i z i n g m o d u l e , a n d l e t functor
Horn ( ", I).
Let
M
Hi (M) X E x t A i ( M , A )   ~ I
gives r i s e to i s o m o r p h i s m s
~i
:
i
H ~ ( M ) ~ D(ExtA'I(M,A))
and r
: ExtA i(M'A)^
be t h e m a x i m a l D
be a m o d u l e of f i n i t e t y p e .
the pairing
(*)
from
A  m o d u l e s into i t s e l f , a n d a p p l y the p r e v i o u s
t h e c a t e g o r y of proposition.
T
~ D(H~(M)) i
be t h e Then
86
In particular, {A p a i r i n g
if
A
is complete,
L X M =~ N
L ==~ H o r n ( M , N )
and
isomorphisms.
then
{*)
is a perfect
is said to be perfect
if the maps
M =" H o r n { L , N )
it induces
are
pairing.
both
)
Proof.
We first
consider
the case
i = n,
and show that
the map n
~n : H ~ (M) ~ D(Hom(M,A))
is an isomorphism. since
H ~ (A) = I = D ( A ) .
H n (M) Hence all
Indeed,
and
is an isomorphism
Considered
D 91
are
by a now familiar
M
Jn
as functors
rightexact
argument,
in
for
M = A,
M,
both
and covariant:
is an isomorphism
n
for
of finite type. To show that the remaining
4.
are
isomorphisms,
we
1
use the axiomatic is exact, derived form
characterization
the functors functors
D(EXtA1 ( " , A))
of
a connected
of derived
D{Hom{.,
sequence
A)).
functors.
for
i < n
and
are
D
left ~
i H4~ { 9 )
The functors
of functors,
Since
~
is an isomorphism 9 n
Hence,
to show that the other
~
are
isomorphisms,
we need only
i show that the functors
of
H
In fact,
CM), since
i.e, A
that
Hi~ { 9 ~
for
i < n
H i . IP)  0 for
is Noetherian
are
P
leftderived
p r o j e c t i v e and
and since we are
interested
functors
i < n. only
87
in
is of f i n i t e t y p e ; s i n c e
of f i n i t e t y p e , w e m a y a s s u m e
M
Hi
is additive,
i H ~ (A) = 0
we need only show that
This follows from the fact that
A
~. 1
i.
is an isomorphism
for all
D
i<
n.
Thus
is CohenMacaulay.
Applying the dualizing functor
for
to the isomorphisms
~i'
we obtain the isomorphisms
D(~bi): D D ( E X t A "i ( M , A ) ) ~ D(Hi~ (M)) A
But
DD(M) = M
for any module
M
of finite type.
(Indeed,
A
the functors
M ~
exact covariant; isomorphic
and
they are isomorphic
for all
i somor phi sm s
DD(M)
M ~
M
are both right
M = A,
for
hence they are
of f i n i t e t y p e . ) H e n c e we h a v e a l s o t h e
M
~ i" Q.E.D.
We n o w c o n s i d e r
an arbitrary
Noetherian
(subject however to the weak restriction regular
of t h e m o d u l e s
finite type, and are easier define
~
to express
i H~(M),
is the maximal in terms
Ti(M) = D(HL(M)),
where ideal.
of the duals,
where
D
A
t h a t it be a q u o t i e n t o f a
local ring) and prove two propositions
structure
local ring
which describe M
the
is a module of
M o s t of t h e s e p r o p e r t i e s so f o r e a c h
i
we
is some fixed dualizing
88
functor. TO
Then the
Ti
is right exact,
functors
of
T
and the
Ti
for
additive functors
i > 0
in
M;
are the left derived
0 ,
Proposition
6.4.
(quotient of a regular M
are contravariant
be an Amodule
1)
Let
A
be a Noetherian
local ring) of arbitrary
dimension,
of f i n i t e t y p e a n d o f d i m e n s i o n
Hi~(M) = 0
and
local ring
Ti(M) = 0
for
and let
n > 0.
Then
i> n ^
Z)
The modules
3) For each
4)
dim
Ti(M)
i,
are
of finite type over
A.
dim^ T i ( M ) < i. A
T n ( M ) = n.
(In p a r t i c u l a r ,
HI
(M) / 0. )
i Proof. A/Ann
M,
1) M
which is of dimension
we may assume HI(M)
= 0
expres sing n
is also a module
that
for
A
elements,
by Theorem
of some dimension is a surjective
r
n.
5.7
Then we see that
either by using Proposition
in terms
For the remaining
Thus by Corollary
is of dimension
i > n
Hi~ (M)
n.
over the local ring
1. lZ , o r b y
of a l i m i t o f K o s z u l c o m p l e x e s
of
Z. 3. statements,
of which
map of Noetherian
A
fix a regular is a quotient.
local rings,
local ring Then
B
B " A
and using Corollary
5.7
*In fact, the functors T i, w h i c h really have nothing to do w i t h local cohomology, can be interpreted as E x t l ( ' , R ") for a suitable dualizing complex R on A. See [17, ch V w 2].
89
and Proposition calculated theorem
over 6.3,
Ti(M)
4. lZ w e s e e t h a t A
or over
Thus we may apply the duality
B.
and find
T i (M) ~
Ext ri(M, B)
To prove
Z)
it is also over
B u t an
Ext
Ti(M)
B,
that since
and hence
of modules
A,
~f
is of finite type over
is of fini~ type over
t~
for all
i.
3),
we note that
and that the dimension over
or
Ti(M)
of
B .
t~
B . so
But it is also a
h e n c e i t i s of f i n i t e t y p e o v e r
T o prove
calculated
M
'
of finite type is again of finite type,
is of finite type over
m o d u l e over
ring,
observe
fi)
Ext r'1 ^ B
"
S
A~
is the same whether
A .
is also a regular is the same,
Thus we have reduced
local
whether the problem
to the following: Lernma r;
let
for all
M
6.5.
and
Let
N
be a regular
be modules
l o c a l r i n g of d i m e n s i o n
of finite type over
A.
Then
i,
dim
Proof.
Ext A i (M, N) < i
To say that an Amodule
is to say that for any prime L~
A
i szero.
If
~
~
is sucha
C A
L
i s of d i m e n s i o n
of coheight
prime,
then
A~
> i,
< i
the module is a regular
9O
local ring of d i m e n s i o n homological
dimension
< r  i. is
B y Serre's t h e o r e m
< r  i,
so
N)7  ExtAr  t T hu s
its g~obal
,N 7 )  0
d i m ExtA~{M, N) < i.
To prove assertion
4) o f t h e p r o p o s i t i o n ,
we have
IB By assertion
1)
non vanishing that is
Z
which we have already
Ext
group.
Applying
i s j u s t t h e s e t of p o i n t s
n:
since
for any
]B
is regular,
y ~ Spec B .
proved,
this is a first
Proposition
y s S u p p ~4 depth
whose dimension
]By = d i m
In p a r t i c u l a r ,
Z
5. Z, w e f i n d
]By = d i m y
has no embedded
points,
SO
Tn(M)
Ass
which shows that dim
~f = n ,
dim
= Z
T n ( M } = n.
every point of
Z
But furthermore,
is associated
since
to
~f,
so
Z
n.
This proves
A
is the set of points of assertion
5)
ring
of dimension
of the next proposition.
Notation. Noetherian
A s s t /% M
If A,
0t
is an ideal of dimension
we will denote by
0I,
n
n
in a
the intersection
of
91
t h o s e p r i m a r y i d e a l s in a N o e t h e r i a n d e c o m p o s i t i o n of a r e of d i m e n s i o n p r i m e s of
~
( T h e s e p r i m a r y i d e a l s a l l b e l o n g to m i n i m a l
, h e n c e a r e u n i q u e l y d e t e r m i n e d . ) If
of p r i m e i d e a l s of 7.
n.
A,
we w i l l d e n o t e by
w h i c h a r e of d i m e n s i o n Proposition 6.6.
ring
9, which
Z
n
Z
is a set
t h o s e p r i m e s in
n. Let
A
be a c o m p l e t e l o c a l N o e t h e r i a n
( q u o t i e n t of a r e g u l a r l o c a l r i n g ) of a r b i t r a r y d i m e n s i o n ,
and let
M
be a n
A  m o d u l e of f i n i t e t y p e of d i m e n s i o n
n > 0.
Then 5) A s s Tn(M) = (Ass M)n 6) If ~
s A s s Tn(M), t~
w h e r e for any N~
(Tn(M)) = t ~
Amodule
N,
over the local ring
then (M) ,
iN)
d e n o t e s t h e l e n g t h of
AI
7) A n n Tn(M) = (Ann M)n 8) T h e r e i s a n a t u r a l m a p :
M
"~
TnTn(M)
w h o s e k e r n e l i s t h e s e t of e l e m e n t s of dimension
<
n.
M
whose support has
9a
5) w a s a l r e a d y
Proofs. let
~
s Ass Tn(M).
proved above.
To prove
Then,
letting
B
be a complete
r
of which
A
is a quotient,
l o c a l r i n g of d i m e n s i o n
rn
Tn{M) ~ E x t B
6),
regular
(M, B)
and so Tn(M)~
But b y
5),
~
___
EXtrB?{MI ,
i s of d i m e n s i o n
l o c a l r i n g of d i m e n s i o n
r  n.
n,
B~ )
so
B7
is a regular
H e n c e by P r o p o s i t i o n
4.13,
the
functor rn
is dualizing,
and so preserves
proves
and also
6)
7),
lengths and annihilators. since the primes
the only ones which could be associated 8)
Tn(M)
by P r o p o s i t i o n
~ = Tn(A).
the natural map
since by x s ker ~.
< n, 5)
are
functor in
M,
so
4.2 there is a canonical isomorphism n
dimension
Ass Tn(M)
Ann Tn(M).
is a left exact contravariant
T (M)
where
to
in
This
*
By means
~r.
If
x
of t h i s i s o m o r p h i s m
is an element
then any morphism
above,
,
HomA(M,~)
Ass ~
of
M
we can define whose support has
f : M * ~
i s of p u r e d i m e n s i o n
annihilates n.
Hence
x,
93
G o n v e r s o l F, l e t is of dimension
n.
x
be an element
is a morphism
Or,
equivalently,
letting
by
x,
N
M
(x) / 0,
To show that
to show that there
of
whose
it w i l l b e s u f f i c i e n t
f : M~
such that
be the submodule
of
where
j : N ~ M
is the canonical
then there
will be an
restricted
to
From
N
generated
map
,
injection,
f 8 Tn(M) = HomA(M,
is not zero,
f(x) ~ 0.
M
it w i l l b e s u f f i c i e n t t o s h o w t h a t t h e n a t u r a l
T n ( j ) : T n ( M )  T n ( N )
support
i.e. ,
is not zero. ~)
For
such that
f
f(x) / 0.
the exact sequence,
M~ M / N   ~ 0
j
0~N
we deduce an exact sequence
T n (M) T AJ) T n
But by dim
3)
above,
Tn(N) = n,
Tn0)
must
dim since
contain an isomorphic of their whose
dimensions.
~ = Tn(A)
only associated
A
4),
andby
Therefore,
n.
T n 1(M/N)
which is impossible
Tn(N)
is just those elements
ker
would
because of
M
< n.
is a complete
is torsionfree prime
nl,
is of dimension
c o p y of Hence
If
M/N)
since if it were zero,
support has dimension Example.
then
T nI(M/N)< N
be nonzero,
* T n" 1
local domain
of rank
is the zero ideal,
1.
Indeed,
of dimension b y 5),
hence it is torsionfree.
n,
its
94
Moreover,
to be the zero ideal in
taking
j u s t t h e r a n k of = rank
for any
N,
Let
(N)
is
Hence rank
complete be aAlocal ring of dimension
A
be a dualizing functor, and let
D
N .
~
A = 1.
Definition. let
Amodule
A,
Then
n,
n
~ = Tn(A) = D(H,m
is called a m o d u l e of dualizing differentials for
(A)). A.
It is determined up to i s o m o r p h i s m . If
Exercise. the map
~
M
8)
of
is a CohenMacaulay
above is an isomorphism.
A ~ T n T n ( A ) = H O m A ( ~ , Problem. * Let in
A,
and
M
module,
e)
show that
In particular,
is an isomorphism.
A
be a Noetherian ring,
a m o d u l e of finite type.
J
an ideal
Consider the local
cohomology module s i i n H j I M ) = lirn E x t A ( A / J , M ) n
Find finiteness
statements
assertion
of Proposition
2)
without using duality. projective
limits
Lefschetz
relations
of a n A r t i n  R e e s 6.4.
type,
to generalize
But they should be stated
One should also have commutativity
(cf. t h e i n v a r i a n c e of c o m p l e t i o n s
theorem
of Z a r i s k i ,
with and the
along a subvariety).
*This p r o b l e m is s t u d i e d in [18]. See also [16, e x p o s e XIII] for a m o r e p r e c i s e s t a t e m e n t of the p r o b l e m .
95
Now we c o m e to d u a l i t y t h e o r e m s
for nonregular
local rings.
F o r a c o m p l e t e C o h e n  M a c a u l a y l o c a l r i n g , it w i l l be s u f f i c i e n t to replace
Extnl(M,A)
Extni(M, ~),
where
i n t h e s t a t e m e n t of T h e o r e m ~
6.3
by
i s a m o d u l e of d u a l i z i n g d i f f e r e n t i a l s .
If o u r l o c a l r i n g i s not C o h e n  M a c a u l a y ,
h o w e v e r , t h e r e w i l l be a
f i n i t e n u m b e r of m o d u l e s
...,
to i s o m o r p h i s m
~ = ~0,
~1,
~n
d e t e r m i n e d up
by t h e l o c a l r i n g , a n d w e r e p l a c e
Extn'l(M,A)
in T h e o r e m 6 . 3 by t h e a b u t m e n t of a s p e c t r a l s e q u e n c e w h o s e terms are
ExtP(M, ~q).
Let n
E pq
A
be a c o m p l e t e N o e t h e r i a n l o c a l r i n g of d i m e n s i o n
( q u o t i e n t of a r e g u l a r l o c a l r i n g ) .
t he a s s o c i a t e d d u a l i z i n g f u n c t o r
D,
m o d u l e of d u a l i z i n g d i f f e r e n t i a l s f o r type over
A
by
of r a n k 1
if
A
Z)
Fix a dualizing module I and let A.
~ = D ( H ~ (A))
Then
~
and be a
i s of f i n i t e
of P r o p o s i t i o n 6 . 4 , a n d i s t o r s i o n  f r e e
is a d o m a i n .
Since the functor
r i g h t e x a c t on t h e c a t e g o r y of A  m o d u l e s ,
H n~a~( 9 )
is
t h e r e is a c a n o n i c a l
i somorphi sm
M
(A)
for all
M
of f i n i t e t y p e
n
n (M)
(the u s u a l a r g u m e n t
n
H ~ (~2) H ~ ( A ) ( ~ A H ~
n
.
. . ).
(A), I)
Therefore,
96 and we see that there
is a natural
map
(n)  i C o m p o s i n g this m a p with the pairings of Corollary 6. Z, w e can define pairings for all
i ni
(*)
Hi~(M) X Ext A
Theorem (where
k > 0
6.7.
i<
The following conditions are equivalent
is an integer): (i)
nk<
the pairings
= 0
In particular, the pairings A
are perfect
for all
n  k <
for alI
($)
i < n.
are perfect for all
i
if and only
the proof of Theorem
6.3.
is Cohen Macaulay.
First first consider
Proof. the case
4n
Moreover, ~
n
We m i m i c i = n,
for
(M, ~))
M  A,
since
both sides are right exact covariant is an isomorphism
We
and the morphism
: H n (M) ~ D ( H o m
This is an isomorphism
so
(~c)
n. (ii) H ~ ( A )
if
(M,e)~I
for all
M
D(~)=
functors
of finite type.
To show that i " H i ~ (M)   D ( E x t A ' i ( M , ~ ))
DD(H~a (A))= H i (A). in
M,
97
is an isomorphism
for
i H4m ( 9 )
functors
n  k ~ i < n,
n
are left derived functors
that
HiMt(A} = 0
(ii).
That the pairing is perfect for
for
n  k<
the proof of Theorem Conversely, n
k~
i < n,
H~(A) = 0
assume
n
( 9 ),
n  k < i < n
condition
k~i<
(i).
i.
e.
our condition
f o l l o w s a s in
follows from Theorem
Therefore,
S. 8, s i n c e
l o c a l r i n g of d i m e n s i o n
of w h i c h
r for
Let
B A
is a quotient.
T h e n by T h e o r e m
_C J).
directly,
be a c o m p l e t e
(we m a y a s s u m e ,
B
A
n.
Instead of proving the theorem 6. S.
I = HomB(A,J)
i < n.
for
for
n.
we deduce it from Theorem
b e a dualizing m o d u l e
H i (A),
Then
if and only if its depth is
Second Proof.
that
H~
This is precisely
ni E x t A (A, ~ ) = 0
The last statement is CohenMacaulay
i < n.
of
6.3.
is dual to
for
we need only show that the
regular Let
J
by Proposition
4. 1Z,
6.3 and Corollary
5.7
t h e r e is a p e r f e c t p a i r i n g
i
{1)
H~(M)
for all If
M
i,
I
and for all
is an Amodule,
Amodules,
ri
X Ext B
M
(M, B) * J
of f i n i t e t y p e o v e r
then both partners
subBmodule
of
J
(or
B).
to t h i s p a i r i n g a r e
s o t h e i m a g e o f t h e p a i r i n g w i l l be in
is the largest
A
I.
(In f a c t ,
which is also an Amodule.
)
98
Thus to obtain a duality the modules
express
intrinsically
over
There
rings
statement ExtB'I(M,
intrinsic B)
over
in terms
A,
we need only
of things
defined
A.
is a spectral
sequence
associated
with a change
of
[IZ, Ch. XVl, w
E zpq : Ext,(M, Ext,(A, This reduces
the problem
to calculating
which (except for their
numbering)
of
dual to
B,
since they are
B))
i M xtB(,B)
>
the groups
do not depend
H~q(A):
Extq(A,
B),
on the choice
In our situation
we
have 0
for
q < r  n
for
q = r  n
for
q > r
Ext q (A, B)
0 Assuming r  n < q< partially,
condition
(ii) o f t h e t h e o r e m ,
r  n + k.
Therefore,
Extq(A, B) = 0
the spectral
sequence
for degenerates
and we find
Ext rB i (M,B) Ext A"I(M, ~)
for
i > n  k.
theorem.
Substituting
in
(1)
gives
condition
(i)
of the
99
The converse
(i)  > (ii)
is proved as in the first proof.
N o w we c o m e to t h e m o s t g e n e r a l proof will generalize Theorem dimension ~i
n ni
= D(H~
whose
duality theorem,
t h e s e c o n d p r o o f of t h e l a s t t h e o r e m .
6.8.
Let
A
be a c o m p l e t e
( q u o t i e n t of a r e g u l a r
(2%.)) for
l o c a l r i n g of
local ring).
i = 0, I, ..., n.
Let
Then there is a spectral
sequence
(M)) Proof. let
B
which
A s i n t h e s e c o n d p r o o f of T h e o r e m
be a c o m p l e t e A
regular
is a quotient.
6.7 above,
l o c a l r i n g of d i m e n s i o n
T h e n by T h e o r e m
r,
of
6.3,
ExtB'i(M,B ) = D(Hi~(M))
for all
i.
Furthermore,
there is a spectral
sequence
i EZPq= ExtAP(M, Extq(A, B)) => E x t B ( M , B )
Since
Extq(A, B)
is dual to
Hr'q(A),
I
0
for q for
Ext B (A, B) = 0
for
we h a v e
i<
r
n
i = r  n +q i > r
(see above),
I00
Now by s u b s t i t u t i n g a n d j u g g l i n g t h e i n d i c e s one o b t a i n s the s t a t e m e n t of t h e t h e o r e m . Remark. EXtB(h/l, B),
One c a n g i v e a n o t h e r i n t e r p r e t a t i o n
intrinsic over
A,
e x a c t i n j e c t i v e r e s o l u t i o n of Let
KA
be the c o m p l e x
to t h e l e f t .
Then
B
as follows.
Let
K(B)
in t h e c a t e g o r y of
Horn B ( A ; K ( B ) ) ,
of the g r o u p s be a n
Bmodules.
shifted
KA
is an injective c om pl e x over , d e t e r m i n e d to w i t h i n h o m o t o p y , a n d
r  n A,
places unique ly
i<0
f
0
for
f~ i f o r (In p a r t i c u l a r ,
KA
if a n d only i f
A
i>0
i s a n i n j e c t i v e r e s o l u t i o n of
is Cohenh4~caulay. )
~=~0
Now
ri ni Ext B (M,B) = H (HOmA(M,
w h i c h i s , by d e f i n i t i o n , t h e h y p e r e x t g r o u p
KA) )
ni Ext A (M,KA).
A s a n a p p l i c a t i o n of the t h e o r y of d u a l i t y , we p r o v e the following theorem, Theorem
c o n j e c t u r e d by L i c h t e n b a u m .
6.9.
over a field
k,
F
H n ( x , F)
k.
on
X,
Furthermore,
Let
X
of d i m e n s i o n
be a q u a s i  p r o j e c t i v e n.
scheme
Then for any c o h e r e n t sheaf
is a finitedimensional vector space over
the following conditions are equivalent:
*K A is a d u a l i z i n q [ 1 7 , c h . V w 2].
complex
for
A,
in t h e
sense
of
101
(i)
All irreducible
components
of
X
of dimension
n
are nonproper.
(ii)
Hn(x,F)
(iii) H n ( x , is a very ample
ffx(m))=
Since
as a locally closed
X
X'
for all quasicoherent 0
for all
sheaves
m >> 0,
sheaf induced by some projective
Proof.*
Let
= 0
X
is quasiprojective,
subscheme
be its closure
is the restriction
IP r .
where
sheaf
X.
fix(l)
we can embed it space
Then any coherent
of a coherent
on
embedding.
of s o m e p r o j e c t i v e
in
F
F'
on
]pr.
sheaf X',
F
on
and there
an exact sequence
Hn(X ',F') .Hn(X,F) ~ Hx'x(n+l "X',F')
The last termis dimensional the middle
zero,
vector term
by Proposition
space over Hn(X, F)
k
1. l Z , a n d t h e f i r s t i s a f i n i t e [10, n~
is also.
Theorem
TI~ proves
1], s o
the first
assertion. (i) of
X.
Let
irreducible Then
=> (ii). Y
X'
be a projective
b e a f i n i t e s e t of c l o s e d p o i n t s ,
component
X'  Y
immediately
Again let
of
X',
also satisfies
t h a t it i s s u f f i c i e n t t o t r e a t
(i),
one from each
X C X'  Y C X'.
and such that condition
completion
a n d one sees
the case
X=X'Y.
*Another proof, not using local cohomology, has since b e e n given by Kleiman [20], and a third proof, using purely local methods, is g i v e n in [19].
is
10Z
Since
H~
commutes
i t is s u f f i c i e n t t o p r o v e
(ii)
with direct limits for coherent
(Proposition
sheaves
F.
Z. 11)
There is
an exact sequence
n
!
Hy (X , F) ~~ H n ( X ', F) " Hn(X, F) ~ 0
where
F
is any coherent
is the same as showing
s h e a f on
~r
X'.
surjective,
To show
H n ( X , F ) = 0,
or equivalently,
that the
map
H O m k ( H n ~ X ',F), k) ~
of the dual vector spaces is injective.
H O m k ( H n (X' F), k)
We i n t e r p r e t
these vector
spaces by m e a n s of local and global duality theorems. be e m b e d d e d in projective space s h e a f of
rdifferential
forms
V = pr V.
on
and let
Let
X' be the
Then by Serre's global
duality theorem,
HOmk(Hn(X', F), k)
=e ~
Ext rn
(F,~)
V For the local cohomology group, o b s e r v e t h a t b y t h e e x c i s i o n formula (Proposition i. 3),
II '
i
. Yi
F):/Ii Hti( .. Yi )
103
where V;
Y = {Yi } ' 4~M i
andwhere
i s the l o c a l r i n g of
ffYi
its maximal ideal, and
Fyi
the s t a l k of
F
Now by the l o c a l d u a l i t y t h e o r e m 6 . 3 , a n d the f a c t t h a t i s a d u a l i z i n g f u n c t o r f o r m o d u l e s with s u p p o r t in
Yi at
in Yi"
HOmk( 9 , k)
V(4Ad.) 1
(See E x a m p l e
3 of
w4) n
!
HOmk{Hy(X , F ) , k )
~
(9 , @'Yi Yi
I I Extrn
i Moreover
~
) Yi
(and the j u s t i f i c a t i o n of t h i s s t a t e m e n t r e q u i r e s the
e x a m i n a t i o n of the r e s i d u e m a p , a n d of the r e l a t i o n b e t w e e n l o c a l
and globalduality),the transportedmap ~c
I I Ext[n
(~
i
, ~
Yi
)
~
Yi
Ext rn(F,~)
0' v
i s the one i n d u c e d on the d e r i v e d f u n c t o r s by the n a t u r a l m a p (for a n y two s h e a v e s
F, G
on
V)
HOm~v(F,G)~ HOmyy (Fyi, Gyi)'H~ i To show t h a t
~
is injective, observe that
i s t h e e n d i n g of a s p e c t r a l s e q u e n c e
(Fyi,6yi) Yi Ext
(F, ~ )
(see P r o p o s i t i o n 2. 10) w h o s e
E pq t e r m i s
EPq= HP(v, EXt~v(F,~)) *The compatibility of local and global duality is proved in [17, ch VIII, Prop. 3.5].
104
But since the support Ext%(F,~)
= 0
of
for
Ext
F
is contained
q<
rn
r  n,
in
of d i m e n s i o n
n,
so
H0(V, Extr~ n
e )
X'
(F, ~))
V E x t ~:v' i n ( F , ~ )
Now
to prove, global
is either
or a first
must
be zero.
support
of pure
Ext.
~
In the latter
dimension
at all the points Thus
(ii) < = >
in which case there
nonvanishing
section has
thus if it vanishes
zero,
Yi
under
n
is nothing
case,
any
by Proposition
the map
~ ,
5. Z:
it
is injective.
(iii). O n e implication is obvious.
follows f r o m the fact that for
F
T h e other
coherent a n d for all
m
large
enough, one can find surjections ( m ) P
~ F " 0
X
( i i i ) =>
that if then in
X
(i).
One sees
is projective
Hn(X,
~X(m
V = ]pr
this
that it is sufficient
and irreducible
)) / 0 Hn
easily
for all
m
of dimension >> 0.
to show n
over
Embedding
X
is dual to
But this sheaf m
Ext
is n o n  z e r o by Proposition 5. Z, hence for all
large enough its global sections are nonzero. Q.E.D.
k,
105
Bibliography i.
M . A u s l a n d e r and D. Buchsbaurn: "Codirnension and Multiplicity, " Annals of M a t h e m a t i c s , vol. 68 no. 3 (1958) pp. 625657.
Z.
Ecl~xnann and Schopf: " U e b e r injektive Moduln, " A r c h i v der Mathernatik 4 (1953). P. Gabriel: " D e s Cat4'gories A b e h e n n e s , " B u l l . Math. F r a n c e 90 (1962) 3 2 3  4 4 8 .
Soc.
~'
.
R. G o d e m e n t : "Topologie Algebrlque et Th6orie des Faisceaux, " Herrnann, Paris (1958).
.
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