Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann
444
E van Oystaeyen
Prime Spectra in Non-Commutative Algebra
Springer-Verlag Berlin-Heidelberg • New York 19 75
Prof. Freddy M. J. van Oystaeyen Departement Wiskunde Universiteit Antwerpen Universiteitsplein 1 2610 Wilrijk/Belgium
Library of Congress Cataloging in Publication Data
Oystaeyen~ F van, 1947Prime spectra in noneonmm~tative algebra. (Lecture notes in mathematics ; 444) Bibliography: p. Includes index. 1. Associative algebras. 2. Associative rings. 3o Modules (algebra) 4. Ideals (algebra) 5. Sheaves~ theor~j of. I. Title. If. Series: Lecture notes in mathamatics (Berlin) ; 444.
QA3.L28
no. 444
[0J~251.5]
510'.8s [5~'.24] 75 -4877
AMS Subject Classifications (197'0): 14A20, 16-02, 16A08, 16A12, 16A16, 16A40, 16A46, 16A64, 16A66, 1 8 F 2 0 ISBN 3-540-07146-6 Springer-Verlag Berlin. Heidelberg. New York ISBN 0-387-07146-6 Springer-Verlag New York • Heidelberg • Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin • Heidelberg 1975. Printed in Germany. Offsetdruck: Julius Beltz, Hemsbach/Bergstr.
CONTENTS
Introduction
1
1.
Generalities on Localization
4
1.1.
Kernel Functors
4
1.2.
L o c a l i z a t i o n at a Prime Ideal
8
II.
Symmetric Kernel Functors
16
11.1.
Localization at Symmetric Kernel Functors
16
11.2.
Q u a s i - p r i m e Kernel Functors
24
11,3.
Reductions
31
III.
Sheaves
42
111.1. Spec and the Zariski Topology
42
111.2. Affine
48
Schemes
Primes in Algebras
IV,1.
Pseudo-plaees
IV°2.
S p e c i a l i z a t i o n of Pseudo-places
62
IV.3,
P s e u d o - p l a c e s of Simple Algebras
67
IV.4.
Primes in A l g e b r a s over Fields
71
IV.5.
Localization at Primes,
81
V_~.
Application
V.1.
Generic Central Simple Algebras
86
V.2.
Two Theorems on Generic
9O
V.3.
The Modular Case
VI.
over Fields
58
IV.
of Algebras
over Fields
and Sheaves
: The Symmetric Part of the Brauer Group
Crossed Products
58
86
96
__Appendix : L o c a l i z a t i o n of A z u m a y a Algebras
101
VI.1.
The Center of Qo(R)
101
VI.2.
L o c a l i z a t i o n of A z u m a y a Algebras
109
VI.3. A z u m a y a Algebras VI.4.
over Valuation Rings
Modules over A z u m a y a Algebras
References
for the A p p e n d i x
References Subject Index
115 119 122 123
LEITFADE,N
IVy- ................
II
/
"-- I I I
V
VI
ACKNOWLEDGEMENT
The author is indebted to Professor Dave Murdoch at the U n i v e r s i t y of British C o l u m b i a for reading the m a n u s c r i p t and for many helpful suggestions. Part of this r e s e a r c h was done at Cambridge University and I am thankful to the people of the mathematical department for the h o s p i t a l i t y enjoyed there. I was able to continue with this work while serving in the Belgian army and I thank my superiors
for
not trying too hard to make a soldier out of a mathematician. I am obliged to the U n i v e r s i t y of Antwerp for facilities and the necessary financial support. I thank Melanie for reading the illegible and for careful typing.
SYMMETRIC L O C A L I Z A T I O N AND SHEAVES
Introduction
Present notes are mainly concerned with two topics zation and p s e u d o - p l a c e s of algebras over fields.
: symmetric locali-
The first three sec-
tions deal with localization theory while in the remaing sections the accent is on pseudo-places.
This split up is r e f l e c t e d
in a shift of in-
terest from prime ideals to completely prime ideals.
A f t e r a brief summary of the basic facts about kernel functors,
loca-
lization techniques e x p o u n d e d by P. Gabriel and 0. Goldman are adapted, in Section i!, so as to yield a s a t i s f a c t o r y ideal theory.
In Section
III we construct a presheaf of n o n c o m m u t a t i v e rings on the prime spectrum, Spec R, of a left N o e t h e r i a n ring.
If
R
is a prime ring then this pre-
sheaf is a sheaf and Spec R is then said to be an affine scheme. though Spec is not n e c e s s a r i l y functorial, cause of the many local properties is an ideal of
R
studying it is worthwile be-
that still hold.
For example,
where QA(R)
if A
such that the Zariski open subset X A of Spec R is such
that the a s s o c i a t e d localization functor QA has property by 0. Goldman,
Al-
(T) discussed
then X A is an affine scheme too, in fact X A ~ Spec QA(R),
is the ring of quotients with respect to QA"
Local proper-
ties like this are related to the question whether the extension of an ideal
A
of
R
to a left ideal Qo(A) of Qo(R), for some symmetric kernel
functor o, is also an ideal of Qo(R).
Therefore the local properties
of the sheaf Spec R lean heavily on the ideal theory e x p o u n d e d in II. 1.
The r e l a t i o n between symmetric localization at a prime ideal of a left N o e t h e r i a n ring and the J. Lambek - G. Michler torsion theory, [21], is explained in II. 2. functor o R _ p at a prime ideal
cf.
It turns out that the symmetric kernel P
of
R
is the biggest symmetric kernel
functor spired
smaller
than the J. Lambek
by this we define
be the biggest
Goldman's
supporting
quasi-prime functor prime
left
prime
ideals
prime
ideal
some
special
Using
functor
module.
The
modules.
sense
ideal.
is strongly
to be natural
should
functor
ideal while
cases,
indeed
correspond
the c o r r e s p o n d e n c e
prime
ideals
A,
subrings
of the k - a l g e b r a
of a place
of a field.
specialization
applies
fields,
the c o m m u t a t i v e
over
The general
fields,
which
thus
theory
us to construct
in the n o n c o m m u t a t i v e
stalks
of Primk(A)
case.
there
is added
are several
symmetric
about
Interrelations
between
the
a sheaf Primk(A)
functors these
sheaf we con-
concept
and their
primes
in alge-
on
A.
but this
functors associated
to l o c a l i z a t i o n
kernel
in
are ring h o m o m o r p h i s m s
The kernel
interest
to every
Therefore
of p s e u d o - p l a c e s
turn out to be the q u a s i - p r i m e s
between
of the
generalizing
information
kernel
completely.
case Prim k is a functor Al~k ~ Sheaves,
true
ted to a prime.
Connell.
so as to yield
allowing
always
However
by I.G.
a
and moreover,
may be d e s c r i b e d
to
of a
one because
a quasi-prime
A,
related
then be the analogue
we get a g e n e r a l i z a t i o n
introduced
induced
a prime
is the better
does
In-
to c o n s i d e r
functors
of certain
IV.
2., to
and q u a s i - p r i m e there
there
in II.
that the c o r r e s p o n d e n c e
of algebras
cause
functor,
be noted
s~der p s e u d o - p l a c e s
ideals.
~p.
It should
completely
over
theory
concept
of a prime [12]
torsion
than the kernel
latter
It seems
of Goldman
GamkA on a k - a l g e b r a
bras
kernel
smaller
as a " g e n e r a l i z a t i o n "
in the
Michler
a quasi-prime
symmetric
by a q u a s i - s u p p o r t i n g
-G.
In
is not
at the to prime
at primes
be-
which may be associa-
are the main
subject
of
5..
A further Section
V.
generic
of the theory
To every finite
from fields galoisian
application
and surjective
pseudo-places.
property
:
abelian places
The
of p s e u d o - p l a c e s
group
G
to crossed
skew field
is given
we construct product
Dk(G)
(Dk(G)
in
a functor
skew fields
and
= ~(G)(k))
has
~(G)
a
every crossed product
(G,I/k,{Ca,T}) , where
i/k is an abelian ex-
tension with Gal(i/k)
m G, defined by a symmetric
{C ,TIo,T e G}, is residue algebra of ~k(G) do-place.
In this way a p a r a m e t r i z a t i o n
factor set
under a galoisian
of certain
subgroups
pseuof
the Brauer Group is obtained.
In the Appendix,
Section VI, the theory of symmetric
tion is applied to an Azumaya algebra be proved there that symmetric cended"
to kernel functors
Moreover, properties
localization
R with center
kernel functors
localiza-
C.
It will
on M(R) may be "des-
on M(C) when they have property
at a prime ideal
P
of
one could hope for and therefore
R has all the good
Spec R is as close as
one can get to an affine sheaf in the commutative
case.
The J. Lambek,
coincides
the symmetric bra,
G. Michler torsion theory Op at °R-P in case
and Op has property
symmetric
kernel functor
tral extension
of
R,
R
(T).
The ring of quotients
of Azumaya algebras
ple algebras.
algeat a
then every ideal
is an ideal of Qo(R).
over valuation rings
related to the theory of unramified
Qa(R)
with
algebra and it is a cen-
so if o is a T-functor
A ~ T(o) has the property that Qo(A) Localization
P
is a left Noetherian Azumaya
o is an Azumaya
(T).
pseudo-places
is closely
of central
sim-
I. G E N E R A L I T I E S
I.
1. K e r n e l
Functors.
All rings
considered
mean
left m o d u l e
sided
ideal.
les.
A functor
following
Let
R
c(M)
= M,
defined free
Denote
submodule
g(M)
N
is a s u b m o d u l e
f(~(M))
of
M
c ~(N)
we h a v e
it is G - t o r s i o n
free
if ~(M)
the
class
A kernel
of t o r s i o n
functor
a filter
is a G - t o r s i o n
has
the
following
If A e T(~)
T(~)
module. properties
and
Ideal
the
Module
stands
category
functor
will
for t w o -
of R - m o d u -
if it h a s
the
if
B
M.
. = N n a(M).
is said to be g - t o r s i o n = 0.
A torsion
objects
~ on M(R)
= 0 for all M • M(R).
of
a(N)
M • M(R)
R/A
by M(R)
is a k e r n e l
then
is a s s o c i a t e d
element.
be u n i t a r y .
functor
objects.
B e
to M(R)
a unit
:
M • M(R),
by g i v i n g
~(M/~(M))
2.
be a ring.
a f r o m M(R)
Let a be a k e r n e l
1.
will
If f • H o m R ( M , N ) , t h e n
3. For a n y
to h a v e
all m o d u l e s
properties
1. For e v e r y 2.
and
are a s s u m e d
ON L O C A L I Z A T I O N
and the
is c a l l e d
To an a r b i t r a r y
consisting
of left
The
T(~),
filter
theory class
is in fact of t o r s i o n
idempotent
kernel
ideals
sometimes
if
if
functor
A
of
R
called
~ there such that
a topology,
:
is a l e f t
ideal
of
R
such that A c B then
T(a).
If A , B • T(~)
t h e n A n B • T(~).
3. For e v e r y A • T(a)
and a n y x • R t h e r e
exists
a B • T(a)
such
that
B x c A.
4.
Let M • M ( R ) ,
then
x • g(M)
if and o n l y
if t h e r e
is an A ~ T(a)
such
t h a t A x = 0.
Any
filter
ve,
defines
Conversely,
T
of l e f t
a topology to s u c h
ideals in
R
a filter
of
R,
such T
having
that
there
R
properties becomes
corresponds
1,2,3,
listed
a topological a functor
abo-
ring.
~ on M(R),
defined
by o(M)
= {x • M, A x
funetor with T(o) functors talk
the
a topology
set F(R)
of kernel
N
of
This t o p o l o g y
M
for which
is called
F(R) may be p a r t i a l l y for all M • M(R).
the quotient
induced
ordered
Gathering
PROPOSITION
1.
Equivalently
1. o • F(R)
is idempotent.
then
M
N
by the o - t o p o l o g y 4. If B c A are left torsion,
Remark.
then
being in
M
R
A o • F(R)
M/N
to
induces
of 0 in
M
the
is o-torsion.
is exactly
T(o).
o < T if and only
from
and
[12] we obtain
The
set
if o(M) c T(M)
the following
if both
in
coincides of
K
and
I
are o - t o r s i o n
mo-
R
M, with
then
the topology
the o - t o p o l o g y
such that A • T(o)
in
induced
in N
N.
and if A/B
is o-
B • T(o).
If o • F(R)
is idempotent,
then T(o)
is a m u l t i p l i c a t i v e l y
clo-
sed set.
An I e M(R)
is said to be o - i n j e c t i v e
0 ~ K ~ M ~ M/K ~ 0 with M/K being there
is an f • HomR~M,I)
A o-injective f to
M
module
verified
to be f a i t h f u l l y If
I
I
if, for every
o-torsion
extending
exact
sequence
and any f e HomR(K,I) ,
f to M.
is f a i t h f u l l y
o-injective
if the e x t e n s i o n
T of
is unique.
It is easily
:
module.
o-open
ideals
it p o s s i b l e
:
is a o - t o r s i o n
3. Let M,N • M(R),
in
results
module
kernel
in M.
by putting
2. If 0 ~ K ~ M ~ I ~ 0 is exact dules,
between
makes
on M(R).
o is a kernel
for the n e i g h b o r h o o d s
the o - t o p o l o g y
M = R, the o - t o p o l o g y
type,
functors
taking
This
correspondenee
of the p r e s c r i b e d
in every M • M(R)
submodules
for some A • T}.
The o n e - t o - o n e
and topologies
about
Taking
= T.
= 0
that a n e c e s s a r y
o-injective
is o - i n j e c t i v e
is that
then every
and sufficient I
is o-torsion
f • HomR(A,I)
condition
for
I
free.
with A • T(o),
extends
to
an ~ • HomR(R,I) ; this condition is clearly also a sufficient one.
Unless otherwise
specified, o will always be an idempotent kernel
functor from now on. If M • M(R)
is o-torsion free then there exists a faithfully o-injec-
tive I • M(R), containing
M,
then unique up to isomorphism, of
M,
X I
of
E
E
with the property o(X/M)
containing
M
This
I
is
it can be c o n s t r u c t e d as the extension
in some absolute injective hull
dules tive
such that I/M is G-torsion.
of
M,
= X/M.
maximal among submoThe f a i t h f u l l y o~injec -
will be denoted by Qo(M).
The d e f i n i t i o n of Qo(M)
may be considered as the direct limit of the system
{H°mR(A'M)' where TAB(f) If
M
~A,B
: H°mR(A'M) ~ H°mR(B'M)' A n B • T(o)},
=f]B.
is not o-torsion free then we put Qo(M)
= Qo(M/o(M))
and the di-
rect limit i n t e r p r e t a t i o n yields at once that Qo is a covariant and left exact functor on M(R),
(if the R-module
defined in the usual way, cf. ning R/o(R)
as a subring.
[12]).
structure on the direct
Moreover,
limit is
Qo(R) is a ring contai-
The ring structure of Qo(R)
dule structure of Qo(R) and it is unique as such.
induces the R-mo-
The ring Qo(R) toge-
ther with the canonical ring h o m o m o r p h i s m j : R ~ Qo(R), provides us with a satisfying localization technique. of Qo is not garanteed.
P R O P O S I T I O N 2.
Recall from
[12]
In general, right exactness :
The following statements are equivalent
1. Every M • M(Qo(R)) 2. For all A • T(o), 3. Every M • M(Qo(R)) 4. For all M • M(R),
:
is o-torsion free. Qo(R)j(A)
= Qo(R).
is faithfully o-injective. Qo(R) ® M m
Qo(M).
R 5. The functor Qo is right exact and commutes with direct sums.
Let
a •
R-module dules map
F(R)
P
M,M'
sion
rows P
sequence
cf.
,p
M'
~M
projective [12],
that
in P r o p o s i t i o n
Note
contains h
ideal
for
An
of Qa(R)
Noetherian
that,
then
a B • T(a)
0
The
second
A • T(a),
then
which the
L.
Silver
one
property becomes
of the
is g e n e r a t e d Q~(R)
with
that
that
free
R-mo-
an R - l i n e a r
P/P'
the
An
is ~ - t o r -
diagram
:
if and o n l y
is left
F(R)
then
properties
having
one
a T-functor.
by a left
It is
if e v e r y
Noetherian
equivalent
a •
Qa is
listed
of the
In this
ideal
it is s u f f i c i e n t
is a - p r o j e c t i v e
~M
since
is a - p r o j e c t i v e .
proper-
case
of R / a ( R ) ,
so if
is too.
commutative
because,
diagram
that from
every
A • T(a)
a morphism
:
~A
• 0
B' • T(a).
in P r o p o s i t i o n
trivial
R
2 is c a l l e d
~ B' --.---~ B c
is a - t o r s i o n
for a T - f u n c t o r
such
exact
if
idempotent
M'
A/B'
a-torsion
such
it c e r t a i n l y
a to be a T - f u n c t o r
: A ~ M we d e d u c e
where
P
functor.
~0
if any
ties
R is left
P' ~ M'
However,
2 holds.
left
: given
P' of
Qa is r i g h t
in P r o p o s i t i o n
every
if
kernel
is c o m m u t a t i v e .
if and o n l y
mentioned
map
,p'
is a - p r o j e c t i v e .
exact
idempotent,
M' ~ M ~ 0 t o g e t h e r
is a s u b m o d u l e
is an R - l i n e a r
is a b s o l u t e
A • T(a) right
there
exact,
well-known,
necessarily
to be a - p r o j e c t i v e
an e x a c t
then
and t h e r e
with If
is said and
P ~ M,
be a, not
under
2 states
extension
that
every
to the r i n g
left
ideal
of q u o t i e n t s
Qa(R)
a.
started
investigating
the
correspondence
between
prime
ideals
of
R
not
in T(o)
in g e n e r a l ,
one
these
sets
from
example
of rings this
tion
8 of
where
2.
in s e c t i o n
a cross-cut A.G.
i.e.,
In the
The
R
i.e.
functors
may
of t o r s i o n D.C.
at a Prime
an ideal
sequel,
does
hold;
R - P.
linked
In the
be a s s o c i a t e d
Murdoch
used
may
be d e d u o e d a class
we r e t u r n ring,
to the
to
locali-
localiza-
noneommutative to a p r i m e
by A.
and the
that,
between
is to d e t e r m i n e
way
theories
shown
this
is a c o m m u t a t i v e
Goldie,
case
ideal. J.
Lambek,
author.
Ideal.
P
of
R
is p r i m e there
If R - P is a m u l t i p l i c a t i v e ideal
of fact
correspondence In case
been
correspondence
question
is in a n a t u r a l
Heinicke,
prime
if and
only
if R - P is an
is an x • R such set t h e n
P
that
is said
to be a
.
R will
always
be a left
Noetherian
ring,
unless
other-
specified.
To a p r i m e
ideal P of
R
the m u l t i p l i c a t i v e
: {g e R, r ~ P i m p l i e s
important Define
part
Op e
that
Michler the
47).
if Sl,S 2 • R - P t h e n
s l x s 2 e R - P. completely
a one-to-one
system,
kernel
Localization
m-system~
such
P
It has
as a m a t t e r
II.
ideal
By d e f i n i t i o n ,
G(P)
p.
of Qo(R).
[31],
the o n e - t o - o n e
several
G. M i c h l e r ,
wise
(cf.
at a m u l t i p l i c a t i v e
We p r e s e n t
ideals
establish
[12],
at a p r i m e
however,
I.
cannot
of ideals,
problem
zation
and p r i m e
in G o l d i e ' s
F(R)
by its
injective
that hull
HomR(M,
E(R/P))
jective
module
: 0). E
T(Op)
# ~ for
Op c o i n c i d e s E(R/P)
as
is a s s o c i a t e d .
localization
filter
[A : r] n G(P)
proved
rg ~ P},
theory,
r • R.
O. G o l d m a n follows.
defines
In
[21], functor
(M • M(R) a kernel
For any M e M(R),
T E ( M ) = r~ {Kernels
of R - l i n e a r
[10],
of left
the k e r n e l
of R/P • M(R),
This
of.
consisting
every with
set an
[11].
ideals J.
A
of
Lambek,
determined
is t o r s i o n functor
put
maps
set p l a y s
M ~ E}.
R
G. by
if
T E by an
in-
Hence, for
TE-torsion
left N o e t h e r i a n
TR/P
are
rings
tative
way ring
F(R)
contains is said
is the
following
= 0 and this
Op c o i n c i d e s
is c a l l e d
an ideal
R
to be s y m m e t r i c
3.
which
shows
that
with
closed.
PROOF.
implies
sely,
if A • T(o)
cient
to p r o v e
with
and
that
is p o s s i b l e Since
because
C,A • T(o)
Transfer
of the
for k e r n e l
for all ~ • V and
{or,
and
o(A/B)
of e l e m e n t s
= ~ o
the
ideals = A/B. that
(M),
that
• V, t h e n
v
o(A/B)
R
is left
C.l • T(o)
if e v e r y
A • T(o)
functor
This
= A/B
o
then
that
it is suffiA = Ra I + . . . + R a n
C = n C. we o b t a i n i l • T(o) h e n c e B • T(o).
CA
for f i l t e r s y i e l d s For a set
{Or,
kernel
filter
a partial
Obviously
~ ~ o.
filter
functors
is a s s o c i a t e d
if p • F(R)
{or,
we d e f i n e
o ~ ov
kernel
functors
is s y m m e t r i c . and w r i t e
by all
finite
~ • V}.
and
Now
T(~)
for
products
to a s y m m e t r i c
is i d e m p o t e n t
sup of
ordering
It is i m m e d i a t e
functors
generated
C A c B.
v • V} c F(R)
M • M(R).
kernel
Conver-
C.l a.l C B, w h i c h
inf of a set of i d e m p o t e n t
o is the
if T(o)
set.
Noetherian,
such
for all v • V t h e n
the
if and only
Putting
for e v e r y
be the
that,
p ~ o; this
only
ker-
of ideals.
A bilateral
that
inf of s y m m e t r i c
Let T(o)
property
an i d e m p o t e n t
is a m u l t i p l i c a t i v e
Since
ordering
if T ~ o
in u T(~).
if and
in a n o n c o m m u -
a basis
is s y m m e t r i c
T(o)
(see before).
the
has
in T(o).
v • V} be a set of s y m m e t r i c
the filter T(Ov).
o with
that
Choose
definition
is i d e m p o t e n t
F(R)
B c A is such
inclusion
i~f o v by o(M)
this
o •
it f o l l o w s
functors
R - P,
filter
functor
is also
B • T(o).
a 1 , . . . , a n • A.
its
ideal
if it is i d e m p o t e n t .
A bilateral
Idempotency
that
a bilateral
of
at a prime
: to the m - s y s t e m
such
is m u l t i p l i c a t i v e l y
let
functor
localization
is a s s o c i a t e d
PROPOSITION
from
by H o m R ( M , E )
the k e r n e l
of i n t r o d u c i n g
functor
A o •
o+=
given
= TE(R/p).
Another
nel
modules
functor
p ~ o v for
all
10
Now, ted.
to an m - s y s t e m
Let M • M(R)
The
topology
of
R
T(R-
containing
in the
absence
bilateral closure
but
JR-
R- P a symmetric
and
define
P) c o r r e s p o n d i n g an
ideal
of the
P'
sense
p(R)
ponent ideal
and
o R _ p(R)
of t h e A
of
zero
R
of G o l d m a n
but
we m a y
as d e f i n e d
associate
in
for
In t h a t
[12], m a y
s R m c N,
= 0 for of t h e
R,
some left
the
symmetric.
by
kernel
Note
lower
Finally,
that,
idempotent
s • R- P yields
[24].
ideals
o R _ p is s t i l l
case
and
s • R-P}.
Observe
be d e f i n e d
the u p p e r
symmetric
o R _ p is a s s o c i a -
s • R - P.
condition
not n e c e s s a r i l y
are r e s p e c t i v e l y
ideal
by some
idempotent.
p(M) = n {N, N c M,
T h e n J R - P is i d e m p o t e n t
sRm
to o R _ p c o n s i s t s
left N o e t h e r i a n
in the
functor
= {m • M,
(s) g e n e r a t e d
not n e c e s s a r i l y
~R-
~R-
o R _ p(M)
kernel
:
m • N}.
also
that
(R- P)-com-
to an a r b i t r a r y
functors
o A a n d A o,
i.e.,
oA and both sible
depend
d i n g to t h e metric and
since
o 0 is Op a n d
T(o)
the
r • R,
only F(R)
The
: r] n G(P)
is e q u i v a l e n t entailing
define
A o = sup{o R_P,
of
will
A
in
be u s e d ,
o 0 as b e i n g
upon
the
smaller
ideals
than
relation
o.
the
Though
Indeed,
kernel
o R _ p c a n be e x p r e s s e d
are p l a u -
III).
functor
For
correspon-
o D is the b i g g e s t
o 0 is c l e a r l y
it f o l l o w s
the
both
(in s e c t i o n
in T ( o ) ;
closed
between
R.
P n A}
bilateral
immediately
Lambek-Michler as f o l l o w s
sym-
that
torsion
theory
:
Op0 = OR _ p.
If A • T ( o ~ ) [B
first
is m u l t i p l i c a t i v e l y
symmetric
4.
the
based
functor
symmetric.
PROPOSITION
PROOF.
o •
filter
kernel
p n A],
o n l y o n the r a d i c a l
definitions,
an i d e m p o t e n t
= inf{o R_P,
with
then # ¢.
B n G(P)
0 Op ~ o R _ p.
A
contains Since
B c
# ¢, h e n c e
Moreover,
an i d e a l [B
: r]
B • T(Op), it f o l l o w s
B • T ( R - P) a n d
if x • o R _ p ( R / P )
then
i.e. that
for e v e r y B • T(Op)
a l s o A • T ( R - P), C x
c p for
some
11
ideal this
C e T(Ryields
P) a n d
some
x 6 p and ~
o R _ p ~ TR/P.
Now,
x • ~.
Thus
= O, i m p l y i n g
because
TR/P
sRx c p for
that
= Up,
some
o R _ p(R/P)
and
s • R- P and
= 0 which
o R _ p is s y m m e t r i c
yields
we g e t
0
o R _ p < Op.
Symmetric play
kernel
the m a i n
me-spectrum paid
left
ideals
noted
left
left
ideals
5.
PROOF.
"if" p a r t
(section
symmetric (T),
due
of
for
R
for a f i x e d
sheaf
to t h e
The
that
idempotent
is the
principal
in T ( R - P).
closed,
if it is
if it is m a x i m a l
idempotent
o •
kernel
pri-
price
theories
fact
or s i m p l y
funtors
on t h e
III).
torsion
is c r i t i c a l
some
these
not n e c e s s a r i l y
is o - c l o s e d ,
The
A is a c r i t i c a l
the R - m o r p h i s m o(R/A)
hence o.
A
ideals
ring
s 6 R - P are R
f.i.
F(R).
The
funetor
among
set of
o will
be de-
by C'(o).
PROPOSITION
onto
of
ideal
prime
property
advantages,
of a s t r u c t u r e
of u s i n g
by some
ideal
A left
o-closed
critical
advantages
a left
several
construction
Noetherian
generated
in T(o).
proper
have
of i n v e s t i g a t i n g
say t h a t
not
in the
of a left
f o r the m a n y
difficulty
We
role
funetors
B
is t h e n
it is a l s o
~.
R/B
is u - t o r s i o n ,
A • C'(o),
TR/A(R/A)
B contains
A
properly
while
A
B
since
that
R-submodule contains
for
some
A
then
Consider R
mapped
properly
idempotent
= R/A.
yields
and
functor
: (R/A)/o(R/A) The
that
is T R / A - C l o s e d .
B ~ T(TR/A)
of
R/B ~ (R/A)/(B/A)
: 0 and t h i s A
if A e C ' ( ~ R / A ) .
is G - c r i t i c a l .
R = B and o(R/A)
o(R/A)
= 0, we h a v e
if and o n l y
be the
A e C'(o) but
hence
thus
B
# 0 then
because
free,
ideal
Suppose
let
If o ( R / A )
in T(o)
u-torsion
contradicts
is t r i v i a l .
: R ~ R / A and
under
Therefore
Because
~
left
latter
o < TR/A.
Moreover,
B 6 T(o)
if
c T ( T R / A ) , con-
tradiction.
Let R-linear
A
be o - c r i t i c a l map
~s
: R/[A
and
let
s
: s] ~ R/A,
be an e l e m e n t defined
not
by x m o d [ A
in
A,
then
: s] ~ x s
the mod A,
12
is a monomorphism.
It follows
the pr o p e r t y
= R/B and hence
in T(o), cible sI
left
~ A,
lated
or
o(R/B)
[A : s] • C'(o). ideals
A
B
s 2 ~ B such that
if and only
Critic a l
and
left
if the
left
class
ideal
of
A
ideals
PROPOSITION are
of related
6.
be a prime
P
among
A n G(P)
[8], that
I(R/B)
irredu-
if there A
and
are
B
are re-
isomorphic.
as being the m a x i m a l
prime
left
ideals.
then A •
elements
This
in its i d e a l i z e r
if x • R A - A
exist
implies~
A R in R,
[A : x],
and by
class we get A : [A : x] and the set follows
ideal
P
of
R
ideal
of
R.
easily.
Critical
are of p a r t i c u l a r
The
following
prime
interest.
statements
left
ideals
prime
not
left
intersecting
ideal
of
R
G(P).
containing
P
and
prime
left
ideal of
R
containing
P
and
= ~.
and r e l a t e d
properties
may be c o n n e c t e d
to this.
the prime
ideal
P
ning
Proposition
blem because
of
R;
may be found
in [21].
Let o = o R _ p be the characterize
8 fails
between
The f o l l o w i n g
symmetric
the o - c r i t i c a l
to give a s a t i s f y i n g
the c o r r e s p o n d e n c e
problem
localization
left
solution
ideals
for this
C'(o R _ p) and C'(TR/P)
prime
of a "critical kernel
left
funetors,
ideal"
cf.
[12].
is strongly
connected
at
contaipro-
is not
enough.
The concept Goldman's
and
ideal not
= ~.
3. A is a critical
known well
V. Dlab
left
: s] has
:
2. A is an i r r e d u c i b l e
P.
Indeed
a prime
1. A is m a x i m a l
This
cf.
irreducible
is c o m p l e t e l y
containing
equivalent
A n G(P)
also,
I(R/A)
is a m u l t i p l i c a t i v e
Let
of R/[A
[A : s 1] = [B : s 2] and that injectives
A
B
are said to be r e l a t e d
in the e q u i v a l e n c e
fact that A R - A left
R
submodule
[A : s] is a m a x i m a l
Recall of
with A R = {x • R, Ax C A}. maximality
every
ideals may also be c o n s i d e r e d
in an e q u i v a l e n c e a critical
that
with
Let o • F(R) be idempotent.
13
A support nonzero
for
o is a o - t o r s i o n
submodule
is c a l l e d
a prime
such that
T S = o.
Clearly,
if
S
S is a n y (up to
S
kernel
functor
free
support
ideal
we h a v e
7.
for o t h e n
in R, t h e n
that
has
to be
be
such
a support
nonzero
injective.
Moreover,
A ~ e S
exists time
let
A
F(R)
for o
If o is p r i m e
same
and
for every
homomorphism
there
at the
idempotent
that,
is o - t o r s i o n .
exists
any
for o which
F(R)
S
S/S'
if t h e r e
T S = o.
support
Let o e
R-module
f o r o, t h e n
R-module
isomorphism)
PROPOSITION left
S' of
is a s u p p o r t
to a o - t o r s i o n
free
from and
S if
a unique
is o - i n j e c t i v e .
be a o - c r i t i c a l
:
1. A is T R / A - c r i t i c a l . 2. T h e
quotient
3. T h e
induced
module kernel
PROOF.
The
nonzero
submodule
A properly The
last
first
and
o = inf{TR/A,
PROOF.
The
of R/A
A ~ e
is prime.
follows
from proposition
of some
it is a - o p e n
F(R)
and
immediately
is
f o r o.
left thus
from
idempotent
ideal R/A
1 and
5. of
Secondly, R
which
is a s u p p o r t
every
contains
f o r o.
2.
if and o n l y
if
A • C'(o)}.
fact
Then
TR/A
is i m a g e
follows
that
for e v e r y A • C'(o) T ~ o.
functor
as such,
8.
is a s u p p o r t
statement
statement
PROPOSITION
R/A
there
o(R/A)
a n d thus
= 0 f o r a n y A e C'(o) o < inf{TR/A,
is a C • T(T)
A 1 • C'(o)
such
that
C c A 1.
T ( TR/A1
cannot
hold,
hence
converse,
define
C'(O)
to be the
R maximal
in an e q u i v a l e n c e
T(o).
implies
A e C'(o)}. Since
C ~ T(o)
that
Let T ~ a w i t h we m a y
For t h i s p a r t i c u l a r A 1 it f o l l o w s o = inf{TR/A,
class
A • C'(o)}.
set of m a x i m a l of r e l a t e d
o ~ TR/A
find
an
that
To p r o v e
the
o-closed
left
ideals
irreducible
left
ideals.
of
14
Since ~R/A is idempotent for any A e C'(o), o is idempotent too.
COROLLARIES.
If o • F(R) is idempotent then o = T M where
M
is the di-
rect sum of the n o n - i s o m o r p h i c quotient modules R/A for all A • C'(o). Furthermore, A • C'(a).
o = T N where N = ~ Qo(R/A), the direct sum ranging over all It is clear that M (or N) cannot be a support for o if there
exist at least two factors in the sum, whence the following results.
An
idempotent o • F(R) is a prime kernel functor if and only if Q (R/A) ~ E for all A • C'(o). A l t e r n a t i v e ways of looking at critical left ideals are e n c o u n t e r e d in [19],
[32]; they may be described as left annihilators of the elements
of i n d e e o m p o s a b l e
injective modules,
so they are related to what
is cal-
led an atom in [32]. For completeness sake, for
R
let us recall that the left A r t i n i a n c o n d i t i o n
is equivalent to every critical prime left ideal being a maximal
left ideal of
R.
A r t i n i a n conditions will be avoided in the present
context.
The c o r r e s p o n d e n c e between prime ideals of
R
Qo(R) has been studied in case o : ap in [21], get useful Let
P
Then
pect to G(P)
R
In order to
R,
Then G(~)
The image of
P
set G(P)
under R ~ R/Op(R) will be denoted
= (G(P) + Op(R))/Op(R), R
From
and by s t r a i g h t f o r w a r d argumen-
satisfies the left Ore condition with respect
if and only if R/Op(R)
pect to G(~).
with a s s o c i a t e d m u l t i p l i c a t i v e
R.
is said to satisfy the left Ore condition with res-
tation one derives that to G(P)
[31].
if for any x • R, g E G(P), there exist x' E R and g ' e G ( P )
such that g'x = x'g. by ~.
[13],
results one has to impose the left Ore condition on
be a prime ideal of
as before.
and prime ideals of
satisfies the left Ore condition with res-
[21] Proposition 5.5.,
it follows that
R
satisfies
the left Ore condition with respect to G(P) if and only if the elements of G(P) are units in Qop(R).
This is also equivalent to Qop(P) being
the Jacobson radical of Qop(R); and Qop(R/P)
is then isomorphic to the
15
classical
ring of quotients
Moreover,
Op has property
aim of the following in case
R
Special references
D.C. MURDOCH
is a simple Artinian ring.
The
section is to derive more or less similar results (prime) ring, with respect to localiza-
T-funetors.
for Section I.
[8]; P. GABRIEL
A.G. HEINICKE
[30],
(T) and Qop(R)
is a left Noetherian
tion at symmetric
V. DLAB
QcI(R/P).
[9]; A.W. GOLDIE
[13]; J. LAMBEK
[19],
[24]; D.C. MURDOCH,
[31]; H. STORRER
[32].
[10],
[11]; O. GOLDMAN
[20]; J. LAMBEK,
F. VAN OYSTAEYEN
G. MICHLER
[26],
[12]; [21];
[2?]; S.K. SIM
II.
II.
1. L o c a l i z a t i o n
Unless
at S y m m e t r i c
otherwise
tric T - f u n c t o r .
specified,
The canonical
ring homomorphism. of j(A) hand,
Kernel
R
B
is a left of
THEOREM
For e v e r y
9. A
PROOF.
of
B
R,
u-torsion, h a v e that it f o l l o w s B = B ce.
that
left i d e a l
= Qo(R)
Qo(R)Cb
Qo(R)j(C)j(x)
j(x) e A e and x e A ee. j(x) e Q o ( R ) j ( A ) n j(R).
B
and h e n c e
C ' C x c A.
COROLLARY
1.
Qo(1)
Let
Since
On the o t h e r
For e v e r y
left
2), h e n c e
note
Qo(R)/j(R)
By p r o p e r t y from
is (T) we
Cb : j(C)b
= Bce,
entailing
first that
Conversely
and t h e n p r o p e r t y
Thus we may w r i t e
j(x)
an ideal
C
= Z' qiai w i t h
in T(o)
and Cx c A + o(R).
!dempotency
(T) y i e l d s
let x e A ec, i.e.,
of o i m p l i e s
be a left ideal of
R.
such that
Cqi c j(R)
By the left N o e t h e r -
we can find an ideal C' in T(o),
I
Qo(R)j(A)
is said to be the
B ee ; B.
let b e B.
c Qo(R)j(A)
• j(A)
T h e n Cj(x) R
is a
If x e A ° then Cx c A for some ideal
for all
for
by A e.
B e = j-I(B)
of Qo(R),
statement
Now choose
ian p r o p e r t y
the e x t e n s i o n
such that Cb c j(R).
qi • Qo (R)' ai • j(A). i.
then
j : R ~ Qo(R)
= Q o ( R ) b or b e Q o ( R ) ( B n j(R))
A ec = j - I ( Q o ( R ) j ( A ) N j(R)). Hence
R,
(see P r o p o s i t i o n
To p r o v e the s e c o n d
C e T(o).
of
and o is a symme-
w h e r e A o = {x • R, Cx c A for some C e T(a)}.
is a C e T(o)
Qo(R)j(C)
morphism
w i l l be d e n o t e d
the f i r s t a s s e r t i o n ,
there
Functors.
R.
A ec = A
To p r o v e
A
ideal of Qo(R)
to
FUNCTORS
is left N o e t h e r i a n
R-module
to a left ideal of Qo(R)
if
KERNEL
For a left ideal
contraction
ideal
SYMMETRIC
such that
that C'C • T(o)
C'o(R) = 0 and x • A o.
It is e a s i l y v e r i f i e d
that
= Qo(R)j(1).
COROLLARY
2.
There
is o n e - t o - o n e
correspondence
between maximal
left
17
ideals
Proof that
of Qo(R)
of the
and
last
elements
statement
A e is a m a x i m a l
containing dicht
A
M c is p r o p e r
in
A
an
and t h e r e f o r e
ideal
mormorphisms morphism
restricts if
of
R
is a m a x i m a l
M ce
left
by p r o p e r t y
It is o b v i o u s
a proper
and thus
Consequently R/K ~ 0
~ Qa(R/K),
follows,
A
since
left
But
then
most
of the
R
then
M c C A for
Thus
M = A e.
but A e is not sequence
not a l w a y s
yield
if Q~ is exact.
This
following
M,
contra-
of Qa(R)
an e x a c t
does
even
of
ideal
= A e would
ideal
(T).
t h e n A a is an ideal
of Qo(R).
: Q~(R)
in what
M
to
A e C'(a).
M ce c A e or M c A e follows.
0 ~ K ~ R ~-L
Qa~
that
of Qa(R)
and M c ~ T(a)
is an ideal
cessarily
central
R
: Suppose
ideal
Conversely,
some A e C'(o)
If
left
properly,
M # A e.
of C'(o).
results
ne-
of r i n g
ho-
a ring
homo-
problem
is
apply
to s e c t i o n
III.
THEOREM
1.
10.
Let
Qa(T(R))
= ~(Qa(R))
2. The u n i q u e JT
R-linear
: R ~ QT(R)
induced
PROOF
T ~ a be a r b i t r a r y
1.
The
calization
and
QT(Qa(R))
map
Qo(R)
to Q~(R),
in Qo(R)
and
follows,
sequence
0 ~ ~(R)
an exact
sequence
:
able and
to p r o o f
equality
some
A,B
ohosen
This
be
entails
that
that
to be Axc
respective
Axc
ideals
of
T(R/T(R))
the
then
:
canonical
for the r i n g R-module
~ R ~ R/T(R)
T(Qo(R/T(R)))
while
functors,
structure
structure.
~ 0 yields
under
lo-
~ Qo(R/~(R))
is i m m e d i a t e .
B e T(o)
extending
homomorphism
by t h e i r
exact
Bx = 0 for may
~ QT(R)
~ Qo(R)
kernel
m Q (R).
is a r i n g
QT(R)
0 ~ Qo(~(R)) If we are
symmetric
Pick
= 0 then
an x e T ( Q o ( R / T ( R ) ) ) .
R/m(R) R
T(Qo(R)) c Qo(T(R))
we get
for
some A e T(o).
BA c B, h e n c e
= 0 and thus
x = 0.
Then Since
BA x = 0.
Moreover
18
R/o(R)
n T(Qo(R))
: T(R)/o(R)
yields
inclusions
R/~(R) ¢-~ Qo(R)/T(Qo(R))C--~ and t h e r e f o r e
Q~(Qo(R))
give
should
= 0 it f o l l o w s
and so the R - m o d u l e
Q~(R)
a Qo(R)-module
structure
structure,
c o i n c i d e w i t h the s t r u c t u r e
QT(R).
Let JT be the u n i q u e
and let ~,~ be e l e m e n t s
le s t r u c t u r e .
= C~.JT(n)
J
Finally,
BAh x = 0 w i t h
: C.~JT(n),
linear.
PROPOSITION
11.
Suppose
lows.
that
R.
Hence,
- ~J
in jy
then
= C JT(~n), Qo(R)-modu-
(~) • o ( Q T ( R ) )
a right
that
ideal
= 0,
Qa(~(R))
and a left R-
Bx = 0 for some
R
that
T-functor,
ideal
property
since
pe for some left
~x • T ( Q o ( R ) ) .
R
C
a left N o e t h e r i a n
o-closed
is a o - p e r f e c t
Now pe is an ideal of Qo(R)
Suppose ABc
of it,
of the
left to p r o v e
T h e n pe is a p r i m e
By the left N o e t h e r i a n
for some C e T(o).
We are
if e v e r y p r o p e r
ideal A e of Qo(R).
PROOF.
to
extending
Then J T ( C ~ )
by d e f i n i t i o n
entails
Let o be a s y m m e t r i c
ideal of
uniquely
o-
for some A • T(o).
to a p r o p e r
prime
C~ c R/o(R).
BA • T(T)
R is said to be o - p e r f e c t
closed
map Qo(R) ~ QT(R)
and ~ • Qo(R)
w h i l e A~ c R/o(R)
DEFINITION.
extends
i n d u c e d by r i n g m u l t i p l i c a t i o n
By 1. it is o b v i o u s l y
If x • T ( Q o ( R ) )
B • T(T),
of QT(R)
f o r m this that J ( ~ )
is Qa(R)
is an ideal of Qo(R). module.
Q (R) is f a i t h f u l l y
w h i c h by the u n i q u e n e s s
R-linear
such that
We d e r i v e
in o t h e r words,
that
of Qo(R).
We may f i n d a C e T(o) but also J ~ ( C ~ n )
Q~(R)
m Q (R).
2. Since T > o and o ( Q T ( R ) ) injective
:
for
ideal
R
P
extends
be a o-
ideal of Qo(R).
R,
we h a v e that CP in
we a s s u m e d
ideals A , B of Qo(R).
A C B c c (AB) c c pee = p and t h e r e f o r e
of
r i n g and let
is not c o n t a i n e d because
A
ring.
R
P,
o
c p
P = Po fol-
to be o-perfect.
T h e n we h a v e
A c or B c is c o n t a i n e d
in P,
that
19
y i e l d i n g that A ce = A or B ee = B is contained in pe.
COROLLARY.
With the above assumptions
:
there is a one-to-one
corres-
pondence between proper prime ideals of Qo(R) and prime ideals of which are o-closed. ideals
P
This is easily seen by v e r i f y i n g that proper prime
of Qo(R) restrict to o-closed prime
A,B are ideals of
R
ly A • T(o) yields
such that A B c
Let
1. For every ideal
pC then
ideals of
R.
(AB) e c pCe = p.
Indeed, if Consequent-
B e c p and B c pC while A ~ T(o) yields AeB e c p,
thus A e or B e is c o n t a i n e d in
P R O P O S I T I O N 12.
R
A
R
P
e n t a i l i n g that
A
be a o-perfect ring, then
of
R,
or
B
is in p C
:
rad A e = (tad A) e
2. There is a one-to-one c o r r e s p o n d e n c e between o-closed left P-primary ideals of
PROOF.
R
and left pe-primary
ideals of Qo(R).
The previous p r o p o s i t i o n yields that rad A e is intersection of
the extended ideals pe with p e n
A e.
Hence
(rad Ae) e = n {P, p n A and P ~ T(o)}.
If (rad Ae) c c (rad A) ° then (rad A) e = (rad Ae) ce : tad A e will follow. Therefore,
take x • P for all P n A such that
be an arbitrary prime ideal in T(o),
P
such that P0 D A.
there is an ideal C O • T(o) for which CoX c P0" minimal prime ideals c o n t a i n i n g
A
is finite,
However,
Then,
if x ~ P0'
Because the n u m b e r of
P
containing
since x (and therefore c e r t a i n l y Cx)
ned in all o-closed minimal prime
Let P0
there exists an ideal
C e T(o) for which Cx c p for every minimal prime that P e T(o).
is o-closed.
ideals containing
A,
A
such
is contai-
it follows that
Cx C rad A and x e (tad A) o.
2. Recall that an ideal
I
implies B C I or A c rad I.
of
R
is said to be left primary if A B c
Since
R
I
is left N o e t h e r i a n it follows that
20
rad I is a prime
P,
and
I
is called a left P-primary
CI ° C I for some ideal C e T(o).
Then P ~ C forces
ideal.
Again,
I o = I and using 1.
the proof becomes easy, following the lines of the proof of P r o p o s i t i o n 11.
Remark.
If
R
is left Noetherian,
closed ideal of
R
and o being idempotent,
is contained in a maximal u-closed
a maximal element in the set of o-closed ideal.
For, let
A
and
B be ideals of
ideal.
ideals, then R
P
such that A B c
and B ~ P, then we have that A + P and B + P are in T(o). (A + P)(B + P) c p contradicts
P ~ T(o).
in the set of o-closed ideals} determines symmetric,
then T(o)
The set C(o) o
then every oLet
P
be
is a prime p with A ~ P Hence
= {P, P maximal
completely in case o is
is the set of left ideals of
R
c o n t a i n i n g an ideal
which is not contained in any element of C(o).
LEMMA 13. If
P
Let
R
be an arbitrary ring and let o be a T - f u n c t o r on M(R).
is a left ideal of
dules
X
R
then Qo(P)
in Qo(R) containing P/o(P)
is maximal
in the set of R-submo-
such that X/(P/o(P))
is a u - t o r s i o n
module.
PR00F.
Denote P/a(P) by P.
Qo(Qo(R)/~)
Property
(T) implies that
= Qo(R)/Qo(P)
= Qo(R/P),
and we may derive the following exact sequence
:
0 ~ Qo(P)/~ ~ Qa(R)/p ~ Qo(Qo(R)/~)
The R-module Qo(Qo(R)/p)
~ 0.
is u-torsion free, thus o(Qo(R)/P) c Qo(p)/~,
but since Q a ( p ) / ~ is u-torsion equality follows. ximal with the desired property.
O b v i o u s l y Qo(P)
is ma-
21
We r e t u r n
to the
o is a s y m m e t r i c
DEFINITION. that
[AC
such
that
THEOREM
An
where
R
is a left
Noetherian
prime
ring,
if for all
C • T(o)
and
T-functor.
ideal
A
: A] • T(o),
of
R
i.e.,
is a o - i d e a l
for a C • T(o)
there
exists
we h a v e
a C' • T(o)
C ' A c AC.
14.
Let
T-funetor. lent
case
R
For an
be left ideal
A
Noetherian
and prime,
of
following
R,
the
let
o be a s y m m e t r i c
statements
are
equiva-
:
1. A is a o-ideal. 2. A e is an i d e a l
PROOF.
Because
A e n R = A o.
there
o is a T - f u n c t o r
Consider
x = ~ qiaiqi Therefore,
of Qo(R).
with
C'qi
the
AeQo(R).
extended
If x e A e Q o ( R )
a i e A and q i , q i e Qo(R)
c R for
is a C" e T(o)
all
such
i,
that
ideal
for C"a.
and
A e is Qo(A)
then the
we m a y
sum being
a well-chosen
C' • T(o).
c AC'
o-ideal
by the
and
write finite. Moreover,
condition
for
A. Finally, The
there
foregoing
proof
of the
Conversely,
is a C e T(o) lemma
fact
then
that
suppose
states
1.
that
such
thus there Note
Ae/AC
A e is an
is o - t o r s i o n ,
is a C' • T(o) that
the
that
implies
(AC) e = Q o ( R ) A C
such
implication,
that
x • Qo(A)
for all and t h a t
i,
thus
CxCA.
finishes
the
2.. ideal
= Aec
~ fortiori that
Cqi c C"
of Qo(R).
= AeQo(R)C
A/AC
C ' A c AC
Then
= A e,
is o - t o r s i o n and
thus
1 ~ 2, is o b v i o u s l y true
A
entailing
that
is a o-ideal. for a r b i t r a r y
T-func-
tors.
COROLLARY.
In case
every
ideal C e T(o)
contains
a central
element
22
generating matrix tor
an ideal
rings
over
in T(a),
then
commutative
obviously,
prime
rings
R is a - p e r f e c t .
are
a-perfect
Therefore,
for any
Tdfunc -
o.
The
a-ideal
condition. case
condition,
We
investigate
~ = ~p for
respect
to G(P)
if Rs • T(ap)
(written
some
the
prime
then
for
correlation
ideal
is e q u i v a l e n t
all r • R,
such
that
s'r • Rs,
exists
such
that
s'r
viously s' •
implies
[Rs
dition
: r].
yields
that The
that
(P,A)-eondition, that
s'A C As.
PROPOSITION left
Ore
Then,
A
16.
idempotent
kernel
is such proper
PROOF.
that ideal
R
respect
Conversely s • G(P) that
we have
to G(P)
ideal
the
exists
exists
there
condition exists left
14 is e q u i v a l e n t
ob-
an
Ore
con-
to the
an s' • G(P)
such
proved
for
prime some
ring
prime
satisfying ideal
A e is an ideal
prime
is a p r o p e r
Ore
Indeed,
there
s • G(P)
satisfies
be a left N o e t h e r i a n that
with
and r • R t h e r e
true.
such
condition
i.e.,
left
is also
functor
converse
~ ~,
Ore in
s • G(P).
r • R,
there
left
conditions
Ore
all
the
R
(P,A)-eondition,
the
for
every
s • G(P)
this,
left
: r] n G(P) for
these
the
then
ring.
pe
of
R.
of Q~p(R).
Let
a = t 0 is a T - f u n c t o r . of QT(R)
P
the
T be an
If P e C(T 0)
= Q~(R)P
is a
of Q~(R).
Since
T ~ o, we have pe c Q T ( R ) p
is an i d e a l
The
be a left N o e t h e r i a n
the
then
Let
QT(P)
R
R.
1 of P r o p o s i t i o n
for e v e r y
with
satisfies
PROPOSITION
or,
assumption
condition
Let
If ~p is s y m m e t r i c
[Rs
: r's.
Summarizing
condition
if
extra
of
resembles
between
Rs • T(ap)
for a r b i t r a r y
i.e.,
15.
P
with
an s' • G(P) s',r'
element-wise),
containing
an
inclusion
c QT(p),
Qa(R)~-~
I = QT(P)
pe and t h e r e f o r e
I
QT(R).
Then
n Q~(R)
restricts
to
P
or
R,
hence
23
I = Qo(R) or I c = P.
COROLLARY.
But as 1 @ QT(P)
If o R _ p is a T-functor,
it follows that I = pC.
while
R
satisfies the left Ore con-
dition w i t h respect ot G(P), then pe is an ideal of Q R - P and pe is the i n t e r s e c t i o n of Qo(R) with the J a c o b s o n radical of Qop(R).
One easily
checks that if o R _ p is a T - f u n c t o r and if the elements of G(P) are units in Q R - p(R) then Rs e T(o R _ p) for all s • G(P), also we have that Op = o R _ p and
PROPOSITION
R
17.
satisfies the left Ore condition.
Let
R
be a left N o e t h e r i a n prime ring satisfying the
left Ore c o n d i t i o n with respect to G(P) 0 Suppose that Op = Op(= o R _ p). Then : 1. The
for some prime ideal
P
of
R.
(P,P)-condition holds.
2. The J a c o b s o n radical of Qop(R)
is equal to pe and Qop(R/P)
is a simple
A r t i n i a n ring. 3. P = n {A, A e C'(o)}.
PROOF.
P is known to extend to an ideal under localization at Op, cf.
[21].
It has been noted that this yields the Op-ideal condition and that
c o n d i t i o n 1 of P r o p o s i t i o n 14 transforms to the the additional hypothesis that
R
satisfies
(P,P)-condition under
the left Ore condition.
2. This is a consequence of [21]{ see the remarks on p. 14 and p. 15. 3. From C o r o l l a r y 2 to T h e o r e m 9 we derive that ideal of Qop(R)
if and only if M c e C'(Op).
M
is a maximal left
Now, because of the se-
cond statement, we may write pe : n {A e, A ~ C'(Op)} and thus, by contraction,
P : (n Ae) c = n A ec = n A.
Property 3 above, will reappear in the next section.
24
II.
2. q u a s i - p r i m e
In t h i s that
section
Functors.
quasi-prime
kernel
e a c h a R _ p is q u a s i - p r i m e .
dence
between
implicite
set
S
ring
of
R
: R] w i l l
then
[A
tains
: B]
every
whenever
A
A • C'(o)
and
let
[A
18.
funetor that
ideal
in
certain
left
of
is c a l l e d
R
which
will For
If
B
[A
a restricted
[A
is
be a l e f t any
sub-
So
is a left
and because
that
then
R
functor.
A.
~ is s y m m e t r i c
subsets
ideals
a way
correspon-
{x • R, xS c A}.
in
it f o l l o w s
in s u c h
partial
section
kernel ideal
the
prime
this
contained
A
[A
ideal
: S] c o n -
: S] is in T ( o )
: TS]
= [[A
kernel
: S]
functor
: T].
if
[A : R] • C(o).
1. A s y m m e t r i c
e a c h A • C'(a)
the
Since
If S,T are
implies
and
Throughout
: S] d e n o t e
are d e f i n e d
ameliorates
be a s y m m e t r i c
contained
kernel
[A + P : R]
set up.
~ will
functors
This
functors
is a n ideal.
is.
PROPOSITION
there
kernel
functor
is a P • C(~)
is r e s t r i c t e d
such that
[A
if a n d o n l y
: R] c P a n d
= P. 2.
found
kernel
be the b i g g e s t
ideal
A symmetric
if f o r
prime
in G o l d m a n ' s
Noetherian
[A
Kernel
an i d e a l
If f o r
every
I • T(~)
P • C(a)
such that
[A
and
every A • C'(~)
there
: I]
= P then
C(~)
P = [A
: R] has
the d e s i r e d
c a n be
= {P} a n d
= aR_ P •
PROOF.
1.
If ~ is r e s t r i c t e d
Conversely, Hence
A + P ~ T(~)
tradicts 2.
lows
I
since
P • C(~).
Let P,P'
If
let A e C ' ( a )
ideal
because
P
and
[A
: R] c p a n d
otherwise
choose
such that
and
a = a R_ p follows.
with
N o w A • C'(a)
• C(a)
is a n
then
P
would
yields
A • C'(a)
P1
P1 are b o t h
= [A
in T(~)
P c A and such that
: I] n
elements
be
[A + P
[A
: R]
of C(a).
: R]
property.
= p e C(~).
too a n d t h i s
P ~ [A P C A,
: R] i.e.,
= P then Hence
con-
follows. P = [A
: R].
P = P1 f o l -
C(~)
= {P} a n d
25
COROLLARY. Next
o R _ p is r e s t r i c t e d
proposition
PROPOSITION then
the
19.
generalizes
part
Let
for o R _ p and
following
o stand
statements
1. O is a r e s t r i c t e d 2. The
extension
PROOF. left
The
kernel
pe of
Jacobson
ideals
P
of Qo(R),
if pe
= e A e then
is r e s t r i c t e d .
p e ( N Ae) ee The
1.
ideal
is a u - p e r f e c t
PROOF.
1.
and
closed
:
Jacobson
is the
by the
eorollary
n'{Ae~ A
e C'(~)}.
radical
to T h e o r e m
if P = n {A, A 6 C'(o)} (N Ae) e = n A e e
of Qo(R).
intersection
then
: n A
of the m a x i m a l 9 we get
= (N Ae) c = e A and this
conditions
If A e C'(o)
implies pe
= P.
:
that
g
: (n A) e c N A e
Hence,
for an e l e m e n t
A
in C'(o)
to
is such
that
[A : R] e C(o)
then
A
is
of R. Let o be a s y m m e t r i c
ring.
Suppose
In this
there
BC c A.
BR + [A
it f o l l o w s
A e C'(~).
o is a T - f u n c t o r ,
case,
every
T-functor left
and
ideal
suppose
that
A e C'(o)
is a
R
ideal.
C ~ A but
C + A and
is the
J(Qo(R))
sufficient
2.
left
that
ideal.
20.
left
prime
suppose
equivalent
Qo(R)
P = peC C
gives
left
PROPOSITION a prime
17.
: N A e = J(Qo(R)).
following
be a p r i m e
of P r o p o s i t i o n
are
P = peC
Conversely
and by c o n t r a c t i o n
to
hence
=
if P = n {A, A ~ C'(o)}.
functor.
radical
J(Qo(R))
Thus,
if and o n l y
exist
Therefore
: R] are that
left
(BR + [A
in T(o)
A e T(a)
ideals
B,C of : R])(C
and b e c a u s e
contradictory
R
s~ch
that
+ A) C A,
T(o)
B { A
but
since
is m u l t i p l i c a t i v e l y
to the h y p o t h e s i s
that
26
2.
Suppose
Now
we h a v e
left
(BR) e = Q o ( R ) ( B R )
Theorem
9, C o r o l l a r y
and thus
ideals
B,C
is an ideal 2, y i e l d s
C e ~ A e yields
such
that
of Q~(R)
that
BC c A,
and this
A e is a m a x i m a l
C e + A e = Qo(R)
then
(BR)C
entails left
and we d e r i v e
c A.
(BR)eC
ideal
c A e.
of Qo(R)
from
(BR) e : (BR)eC e + ( B R ) e A e C A e,
by c o n t r a c t i o n ,
COROLLARY
1.
is a p r i m e
COROLLARY C'(Op) not
that
2.
If Op
: (xm)y
DEFINITION. dule
1.
S
is s y m m e t r i c
exactly
An R - m o d u l e
the
G(P);
M
tor
:
every
A e C'(o)
then
maximal
the e l e m e n t s
in the
set of
of
ideals
6.
if
M
is a r i g h t
R-module
such
that
:
x,y e R.
idempotent
kernel
for o if
functor.
An R - b i m o -
:
R-module.
sub-bimodule
If
S
S' c S, the q u o t i e n t
is a q u a s i - s u p p o r t
extension
2. E v e r y
nonzero
3.
is a b i m o d u l e T
R
see P r o p o s i t i o n
free
1. S is an e s s e n t i a l
then
then
S/S'
is a o-
R-module.
21.
T
of
to be a q u a s i - s u p p o r t
nonzero
PROPOSITION
If
ideals
Let o be an a r b i t r a r y
2. For e v e r y
functor
and r e s t r i c t e d
is an R - b i m o d u l e
S is a o - t o r s i o n
o, t h e n
left
for all m E M and
is said
torsion
kernel
ideal.
intersecting
x(my)
(BR) ° or C o is in A o : A.
If o is a r e s t r i c t e d
left
are
either
sub-bimodule
is a l s o
sueh
of e v e r y of
that
a quasi-support
S
for an i d e m p o t e n t
nonzero
sub-bimodule
is a q u a s i - s u p p o r t
T D S with for o.
o(T)
kernel
of
func-
S.
for o.
= 0 and o(T/S)
= T/S,
27
4.
If ~S is the k e r n e l support
5.
If
S
contains
proofs
the
corresponding
22.
assertions
properties
Let
a •
P
that
R/P
properly,
of R/P.
P.
~(R/P)
Therefore,
P the
quotient
R/P has port
= 0 we m a y
the
module
property
follow
be
same
supporting
then
idempotent
rise
and
S
is a q u a s i -
for a t h e n
S
I • T(a)
only
P • C(a).
derive
for
that
that
is
of
[12].
be an ideal
in
R.
if P • C(~).
then
every
module I
ideal
ideal
properly
Hence,
I
any
is G - t o r s i o n ;
I
of con-
R/I b e c a u s e
Conversely,
every
is G - t o r s i o n . (R/P)/M
P
for e v e r y
= 0 entails
as the p r o o f s cf.
let
to a G - t o r s i o n
Hence,
R/I
lines
modules,
is a q u a s i - s u p p o r t ,
from
properly
i/P
containing P • C(a) containing
sub-bimodule
thus,
R/P
is
M
of
is a q u a s i - s u p -
for a.
DEFINITION.
A symmetric
there
a quasi-support
exists
Proposition
4 together
ciated
a prime
with
kernel
with
ideal
functor
S
for
the P
~ is said
~ such
foregoing,
of
R
R
is a left
Noetherian
ring
P of
every
essential
left
of R/P
there
is a o n e - t o - o n e
modules
I
responding
and
to the
coincides
to-one
prime
prime
with
correspondence
ideal
correspondence ideals
ap
of
ideal (cf.
Let
that
every
a R _ p asso-
ideals
P
of
injective
theory 0 a R _ p = ~p, R
prime
a nonzero
idecomposable
Ip be the
Since
for each
contains
The t o r s i o n
[21]).
between
such
between
R. P.
that
if
is a q u a s i - p r i m e .
If
R,
that
to be a q u a s i - p r i m e 0 a = ~S"
yield
Remark.
M(R)
S,
is a s u p p o r t
the
for a if and
gives
a sub-bimodule
and a(R/P)
S' w h i c h
for
F(R)
is a q u a s i - s u p p o r t
Suppose
taining
a sub-bimodule
of t h e s e
PROPOSITION
PROOF.
with
for a.
The
R/P
associated
for ~S"
a support
Then
functor
ideal
ideal,
injective module
induced
by
we also
get
and q u a s i - p r i m e
then R-
cor-
Ip in a one-
kernel
28
functors
associated
PROPOSITION
23.
to one a n o t h e r
PROOF. a'
Let
with
If t h e then
A • T ( o ' ) - T(o)
elements
of
i.e.,
kernel
R/P
Moreover,
o'-torsion.
Therefore,
since
every nel fact
Qo(R/P)
P • C(o)
functor that
different
in some
module.
P • C(o)}
the e x a c t
for
f r o m o and
element
Qo(R/P)/(R/P)
0 ~ R/P ~ Qa(R/P)
that
{Qo(R/P),
is a q u a s i - s u p p o r t
functor
is c o n t a i n e d
o'-torsion.
yields
injective
are
isomorphic
o is a q u a s i - p r i m e .
P • C(o),
is a s y m m e t r i c
an i n d e e o m p o s a b l e
P
o.
o' ~ o.
of C(o),
ig G-torsion
sequence
Suppose
while
~ Qo(R/P)/(R/P)
~ 0
P • C(o)
with
by t h e h y p o t h e s i s .
Hence,
o is the
largest
Qo(R/P)/(R/P)
is t o r s i o n
is G - t o r s i o n
is
:
for the
E m Qo(R/P)
R/P
ideal
it is c e r t a i n l y
is o ' - t o r s i o n
for w h i c h
An
that
yields
P n A, h e n c e
0 o = T E.
free, that
symmetric
i.e., 0 o = TR/P,
for ker-
The
concluding
the
proof.
PROPOSITION the
24.
inf b e i n g
PROOF.
As
we h a v e
that
the
other
COROLLARY o'(R/P)
A symmetric
taken
before
# R/P
since
If o,o' f o r all
Let
the e l e m e n t s
is e q u a l
COROLLARY
2.
S
different
f r o m o such
= R / P and t h u s o(R/P)
are
o'
P • C(o),
then
kernel
o'(S)
0
inf ~ R / P
= (inf T R / P
)0
from o then be
smaller
for
,
some P • C(o) 0 t h a n TR/P. On
follows.
functors
such
that
o' ~ o and
for e v e r y
o' >i o, o'
o = o'
be a q u a s i - s u p p o r t that
cannot
0 = 0, o < T R / P
symmetric
to
P • C(o).
if o' ~ o and o' d i f f e r e n t
o'(R/P)
hand,
1.
over
o • F(R)
for o, t h e n
# 0 we h a v e
that
o'(S)
= S.
29
PROOF. tiori
o'(S)
is a s u b - b i m o d u l e
o'-torsion,
thus
of
S,
hence
S/o'(S)
is o - t o r s i o n
~ for-
S = o'(S).
C O R O L L A R Y 3. If for all P • C(o), the i n d u c e d s y m m e t r i c k e r n e l f u n c t o r s 0 0 TR/P c o i n c i d e , then : a = TR/p, each R/P is a q u a s i - s u p p o r t for o and o is q u a s i - p r i m e .
PROPOSITION such
that
25. [A : R]
PROOF.
Let
that I ~
= 0.
and this torsion for
Let a be a s y m m e t r i c
0 # ~ e R/P and Then,
some
I(x)
and
ideal
suppose
c P for
0 A e T(TR/A)
yields free
= P is a p r i m e
then
there
functor. If A • C'(a) 0 0 ~ R / A = TR/P"
0 I e T(~R/A)
is an ideal
some x ~ P r e p r e s e n t i n g
c T(TR/A) , c o n t r a d i c t i o n .
0 ~ 0 TR/A TR/P.
Now
0 J • T (TR/p) , i.e.,
ideal
kernel
let J Ryc
Thus,
0 # ~ e R / A and A and
J(A
~, h e n c e
+ Ry)
such I C P c A 0 is ~ R / A -
R/P
suppose
is
that J y
C A.
Since
= 0
o
0 is s y m m e t r i c and o(R/P) = 0 it is e a s i l y seen that A + Ry 6 T ( T R / p ) . 0 0 Now, T R / P b e i n g s y m m e t r i c , we get that J ( A + Ry) e T ( T R / P ) and 0 A e T(T /p). The l a t t e r y i e l d s P = [A : R] • T ( T R / p ) , c o n t r a d i c t i o n . 0
Thus
T R / P ~ ~R/A"
COROLLARY
1.
From
Proposition
tor ~, the e l e m e n t s 0 0 fore T R / A = rR/P.
COROLLARY symmetric functors
2.
A of Ct(a)
20 it f o l l o w s
that
are
[A : R]
Let a be a r e s t r i c t e d
for e v e r y A e C'(a), 0 T R / P c o i n c i d e for all
then
such
that
kernel the
P e C(o)
functor
fact
that
implies
for a r e s t r i c t e d = p e C(a).
such the
that
that
funcThere-
TR/A
is
quasi-prime
kernel
~ is a p r i m e
kernel
functor.
PROPOSITION
26.
prime
of
ideal
Let R.
R
be a left N o e t h e r i a n
Suppose
that
prime
ring,
a R _ p is a T - f u n c t o r
let
such
P
that
be a the
30
elements of G(P) are units in Q R - p ( R ) '
then
:
1. ~P : ~ R - P 2. R satisfies the left Ore condition with respect to G(P). 3. ~ R - P is a r e s t r i c t e d quasi-prime. 4. a R _ p is a prime kernel functor if and only if the induced TR/A are symmetric for all A • C'(~). 5. The elements of C'(~ R _ p) are prime left ideals, they are the left ideals maximal with the property of being disjoint from G(P).
PROOF. cause
1. If g e G(P) then Rg is killed under e x t e n s i o n to Q R - p(R) beg
is a unit in Q R - p(R).
Rg e T(a R _ p) and T(Op)
Property
(T) for a R _ p then implies that
= T(~ R _ p) follows because every A e T(Op)
con-
tains some g e G(P). 2. The left Ore condition for
R
with respect to G(P)
is equi-
valent to the fact that left ideals Rg, g e G(P), are in T(ap).
The
latter is a consequence of 1. above. 3. Proposition
17 may be applied; we get P = n {A, A e C'(ap)}
and thus o R _ p = ap is restricted. 4. If ~ R - P is prime then all TR/A, A • C'(o), coincide and coincide with a R _ p because o R _ p : inf{~R/A, A • C'(a R _ p)}. let TR/A be symmetric for every A • C'(~ R _ p). ted, Proposition 25
Corollary
Conversely,
Since o R _ p is restric-
2 apllies, yielding directly that a R _ p
is a prime kernel funetor. 5. This is Proposition
20
Corollary 2.
The foregoing p r o p o s i t i o n also holds
in case
R
is not prime; the proof
then uses reduction techniques thus disposing of "torsion problems".
We m e n t i o n the following
:
1. If for some A e C'(a R _ p) it happens
0 that P ~ T ( ~ R / A ) then
31
0 : ~R/A"
OR-P
The c o n v e r s e
is o b v i o u s l y
0 0 Since a R _ p ~ T R / A and T R / A ( R / P ) 0 lary 1, e n t a i l s that a R _ p = ~R/A"
PROOF.
2. If a R _ p is prime
then P ~ T(~R/A)
is true w h e n T R / A is s y m m e t r i c
II.
a l s o true.
# R/P,
Proposition
for e v e r y A E C'(o),
24
Corol-
the c o n v e r s e
for all A • C'(o).
3. R e d u c t i o n s .
In this tors
s e c t i o n we h a v e put t o g e t h e r
and l o c a l i z a t i o n ,
the l o c a l i z e r ' s R/a(R),
point
therefore,
applications
with respect of view,
although
some p r o p e r t i e s
to c h a n g e m e n t
it is m o s t
we h a v e
of the r e d u c t i o n - t h e o r y
natural
chosen
of k e r n e l
func-
of g r o u n d ring.
From
to t r a n s f o r m
a more general
R
into
set up, m a n y
deal w i t h the s p e c i a l
case
R ~ R/a(R).
Let R 1 and R 2 be rings gories
of R l - m O d u l e s
in R.l is a s s o c i a t e d , tinuous
i = 1,2.
c T2,
via
and let M ( R 1) and M(R 2) be the cateresp.
A ring homomorphism A continuous
i.e.,
if
f
a filter T i
f : R 1 ~ R 1 is conand s u r j e c t i v e
is an open m a p of t o p o l o g i c a l
homomorphism f
and a2(M)
t h e n an M e M(R2)
A m a p g : M 1 ~ M 2 is c a l l e d a r e d u c t i o n of M1,
if the f o l l o w i n g
1. g is R l - l i n e a r .
Hence
g
spaces.
m a y be c o n s i d e r e d
c a l ( M ) for all M • M(R2).
f : R1,T 1 ~ R2,T 2 is onto and c o n t i n u o u s
reduction
To a a i e F(Ri)
f : RI~ T 1 ~ R2, T 2 is said to be a final m o r p h i s m
a continuous module
and R 2 - m o d u l e s
if and only if f-l(T 2) C T 1.
homomorphism f(T1)
with u n i t
ring
if If
f
is
as an R 1-
Suppose
and take M 1 • M(R1) , M 2 • M(R2). (over f), or M 2 is said to be a
two p r o p e r t i e s
is c o n t i n u o u s
hold
:
for the a i - t o p o l o g i e s
i = 1,2. 2. a 2 ( M 2) c g ( a l ( M 1 ) ). In case M i is an R i - r i n g , i = 1,2, g i v e n by r i n g h o m o m o r p h i s m s
in M i,
32
Ji : Ri ~ Mi' then a zing-reduction gJl
g : M 1 ~ M 2 is a reduction
such that
= J2 f"
Examples. 1. If M = M 1 = M2, then the identity and only if ~2(M) M e M(R2)
C ~I(M).
g
Hence,
of
M
1 M is a reduction
near map g : M 1 ~ M 2 is a reduction g
is onto and Ker g c ~(M1)
terion
: ~ is idempotent
tive R-linear Surjective torsion
maps
reductions
free module
tions yields
over
over
f
f
if
for all
if and only if a 2 ~ ~1 on M(R2).
2. Let R 1 = R 2 : R, ~1 = ~2 = ~ and f = 1 R.
If
is a reduction
then we may deduce the following
if and only if for any M e M(R),
are called
are reductions
epireduetions.
is a2-torsion
An R-li-
if and only if a(M 2) = g a(M1).
g with Ker g c a(M)
a reduction
Let M1,M 2 e M(R).
free.
over the composition
all surjec-
over 1 R.
A reduction
Obviously,
cri-
of a a 1-
composition
of the underlying
of reducring ho-
momorphisms.
PROPOSITION homomorphism.
27.
Let f : R1, T 1 ~ R2, T 2 be a surjective Let ~1 be idempotent,
and Ker ft c al(R 1) while ~l(Ker
PROOF.
Since al(Ker
of ring homomorphisms
then f = f0 o f t
f) is an ideal we may consider
ft is final
the following
diagram
:
0 ~ Ker f/~l(Ker
f
R1
ft
follows
where
ring
f0) = 0.
0----~ Ker f .......
Thus al(Ker
continuous
-~ R 2
Ift
that ft is open and continuous
f)
f0
in R1/~l(Ker
while
0
]2
f) ~ R1/al(Ker
f0 ) = 0 and since the filter
~
R1/Ker f)
f
0
is Im T 1 it
f0 is continuous.
33
Proposition actually
28 w i l l
surjective
Ker f c ~1(R1)
f
it is s h o w n that ft
f : R1, T 1 ~ R2, T 2 such that
a torsion morphism.
is a t o r s i o n
reduction.
A reduction If
f
g : M1 ~ M2
is final,
reductions
are said to be final r e d u c t i o n s .
All kernel
functors
PROPOSITION
28.
are i d e m p o t e n t
reduction
PROOF.
Consider
of R l - m O d u l e s
the exact
Ker f c 01(R1 ) w h i l e
It is e a s i l y Noetherian,
c al(R1) checked
-1
otherwise
stated.
f : R1, T 1 ~ R2, T 2 is a f i n a l
f.
:
(~2(R2))__ ~ ~2(R2 ) ~ 0.
a 2 ( R 2) is a l s o 0 1 - t o r s i o n
entailing that
over
sequence
0 ~ Ker f ~ f
f-l(a2(R2))
unless
A final torsion morphism
torsion
Since
since there
ring h o m o m o r p h i s m
is c a l l e d
Ker g c ~1(M1)
over
the proof,
is a r e d u c t i o n .
A continuous
with
complete
that
f
it f o l l o w s
that
is a r e d u c t i o n .
if f : R1, T 1 ~ R2, T 2 is final and if ~1 is
t h e n 02 is also a N o e t h e r i a n
kernel
functor
(as d e f i n e d
in
[12]).
PROPOSITION If t h e r e
29.
exists
Let al, o 2 • F(R)
and s u p p o s e
a final homomorphism
that o I is i d e m p o t e n t .
f : R1, T 1 ~ R2, T2, t h e n ~2 is
idempotent.
PROOF.
Let A',
suppose
that A'/B'
A / B ~ A'/B'
B' be left ideals is a 2 - t o r s i o n .
and A • T 1.
Since A / B
of R 2 such that A' • T2, Put B = f - l ( B ' ) , is a 2 - t o r s i o n
and thus the fact that a I is i d e m p o t e n t follows,
proving
that a 2 is i d e m p o t e n t .
implies
B' c A' and
A = f-l(A').
Then
it is also a l - t o r s i o n B • T 1 and B'
= f(B) • T 2
34
PROPOSITION embedding equal
30.
A final m o r p h i s m
f : R1, T 1 ~ R2, T 2 gives
to an
of M(R 2) into M(R 1) so that the restriction of o I to M(R 2) is
to o~.
The proof
is easy.
localization
Before
functor
PROPOSITION
31.
investigating
Q1 to the e m b e d d e d
reduction-properties
of o - p r o j e c t i v e
Given
an exact
sequence
map h :
0
~ M 1'
M1
of the
Q2' we include
epireduction.
some
Then,
if
too.
M ~ M" ~ 0 of O l - t o r s i o n
: M 2 ~ M".
diagram
the r e s t r i c t i o n
M(R 2) equals
Let g : M 1 ~ M 2 be a t o r s i o n
and an R 2 - 1 i n e a r commutative
whether
modules.
M 1 is o l - p r o j e c t i v e , M 2 is o l - p r o j e c t i v e
PROOF.
rise
free
Since M 1 is o l - p r o j e c t i v e
R2-modules we obtain
a
~ 0
~ M1/M ~
M2
M
where
M"
M I / M ~ is Ol-torsion.
the latter
R2-module
Put M~
is ol-torsion.
M~ n Ker g c oI(M~) , and thus sion free g(Mi)
M.
Therefore
and we o b t a i n
~ 0
= g(M~).
~(Mi n m e r
g) is o l - t o r s i o n
~
commutative
½ h M
in the ol-tor-
M~ n Ker g c Ker 4, or 9 f a c t o r i z e s
the f o l l o w i n g
~ M"
onto M2/M ~
Now Ker g c oI(MI) , thus
0 i _ilg MIiM 0 0
Since M I / M ~ maps
~ 0
diagram
through
with exact
rows
:
35
Since
M 2 / M ~ is O l - t o r s i o n , we m a y
COROLLARY.
If in the
projeetivity
PROPOSITION jective.
situation
of M 1 y i e l d s
32. Let
Let ~1 be
g
ol-injective , then
PROOF.
Put A = f-l(A')
that
described
By o 1 - p r o j e e t i v i t y
of
0
above,
left
that
every
then
01 -
in T 1 is o 1 - p r o -
and
let M 1 be
faith-
o2-injective.
hence
A • T 1.
: A' ~ M 2 e x t e n d s
we h a v e
is final
ideal
epireduction
some A' • T2,
map h
mB
g
such
M 2 is f a i t h f u l l y
A
M 2 is e l - p r o j e c t i v e .
of M 2.
for
any R 2 - 1 i n e a r
that
c2-projectivity
: M 1 ~ M 2 be a f i n a l
fully
show
conclude
We h a v e
to an R 2 - 1 i n e a r
a commutative
diagram
to R 2 ~ M 2.
:
~A
A' >0
MI-~-~ M 2
where
A/B
Beeause
is ~ l - t o r s i o n .
M 1 is f a i t h f u l l y
: R 1 ~ M 1 and thus For any a • A,
h
proves
morphism
that
position
will
~ may
: gg(a) in M2,
= g(am) this
by g(m).
reduction,
yields map
Finally,
conditions ring.
under
to an R l - l i n e a r
fixed
Then,
= a'g(m),
a' • A'.
is d e f i n e d
M 2 is a 2 - t o r s i o n Since 28,
a final
one
for the However,
milder
m • M 1.
by d e f i n i t i o n
R 2 ~ M 2 which
see P r o p o s i t i o n
result
some
h(a')
o2-injective.
to d e d u c e
a similar
extended
= ag(m).
to an R 2 - 1 i n e a r
R 1 is a left N o e t h e r i a n yield
be
for all b • B and
M 2 is f a i t h f u l l y
31 C o r o l l a r y ,
(T) in ease
= b m
on the r i g h t
is a t o r s i o n
Proposition ty
structure
: A' ~ M 2 e x t e n d s
by m u l t i p l i c a t i o n this
~(b)
(h o f)(a)
of the R l - m O d u l e Hence
~1-injective,
could
descent the
free,
torsion use of p r o p e r -
following
hypotheses.
pro-
36
PROPOSITION
33.
Let f : R 1 ~ R 2 be a surjective
let ~1 be a T-functor free.
Equivalently
I. The Rl-mOdule
and
on M(R 1) such that R 1 and R 2 are both al-torsion :
structure
for QoI(R 2) via
ring homomorphism
of Q~I(R2)
defines
an R2-module
structure
f.
2. Define ~2 on M(R) by putting T(c 2) = {f(A), A e T(~I) , then Q~I(R2)
= Qo2(R2).
3. The unique extension ~ : Qol(R1 ) ~ Qal(R2)
of
f
is a ring homomor-
phism.
PROOF. functor,
The surjectivity
of
f
implies that a 2 is indeed an idempotent
and it is clear that a I and a 2 coincide
The Rl-linear
i' : R 2
and ~2-torsion.
~ Q~I(R2)
Thus i' extends
is R2-1inear
on R2-modules.
Suppose 1.
and Qal(R2)/i'(R 2) is a 1-
to a unique Rl-linear
map
: Q~2(R2) ~ Qol(R2), while i : R 2
~ Q~2(R2 ) extends to a unique R2-1inear
: Qal(R 2)
map
~ Qa2(R 2)
which is also Rl-linear
by definition
Qol(R2).
is easily seen to be g2-injective
Since Qol(R2)
follows that Qal(R2) the commutative
~ Qa2(R2).
.......
Qal(R 1 ) ....
Give gn : R1 ~ Qal(R1)'
f
structure
To prove that 2. implies
diagram of Rl-linear
R1
of the Rl-mOdule
morphisms
of
in M(R 2) it 3., consider
:
~R2
~ Qol (R 2) -~ Q a 2 ( R 2 ) .
by g~(r)
: rn, then g~ extends
to a unique
37
fn : Q~I(R1 ) ~ Qel(R1 )"
Now, ~fn(r)
on the other hand f~(n)f(r) f~(~)~ coincide
= ~(r)f(n)
on R1, therefore
If ~,n e QqI(R1),
then f(En)
if ~ is a ring homomorphism the latter equality, implies
extends
3.
the assumption
is onto.
more general Reduction Let
: f(r)~(n)
by Rl-linearity
f) of Q~I(R1)
that Ker f
is equivalent
(T).
: Q~I(R1)
~ Qal(R1/T(R1)) ,
(symmetric)
Corollary
T-functor
and therefore
T ~ a 1,
the fore-
a ring homomorphism
3 we turn to the following,
(T). ring and let T be a T-functor
If R/T(R)
Hence,
if
A
is a final torsion
epireduction.
that T is Noetherian
and moreover,
A G T(T) which
on M(R).
is equiped with the filter
is a final torsion morphism°
yields
33 implies
1 ~ 2 ~ 3 hold even without
Hence we obtain
of property
Consequently,
(T)
situation.
JT : R ~ R/T(R).
follows
: r~(n),
Since property
and this again
that Ker f is a ~l-ideal
Before deriving
Finally,
3 ~ 1 follows.
in Proposition
applies.
R be an arbitrary
A ~ jT(A)
of ~.
the implication
that o I has p~operty
going proposition
which
= ~(r)~(n)
Note that the implications
10 yields
on Qal(R1).
then ~(rn)
2. In case Ker f = T(R 1) for some
then Theorem
coincide
and
ffn and
: f(E)~(n).
1. Any of the conditions
to statement
COROLLARY
these maps
Hence,
= f~(n)~(~)
follows
to an ideal Q~l(Ker
= f(r)~(n).
= f(r)f(n),
= ffn(E)
that f is surjective,
COROLLARY
= ~(r n) : r ~(n)
jTT(T)
is a left ideal of
Consider then JT R
then
Since T has property every
B e T(T)
(T) it
contains
an
to Proposition
31
is z-projective. iT(T)
that every
is Noetherian
jT(T)-open
and the corollary
left ideal contains
ideal which also is in T(jT(T)).
Hence,
iT(T)
a jT(T)-projective
is a T-functor.
left
38
COROLLARY 3. We may restate Proposition
33 under milder assumptions.
Let f : R 1 ~ R 2 be a surjective ring homomorphism and let ~1 be a Tfunctor on M(R1) , then the following assertions 1. Q~I(R2)
are equivalent
:
is an R2-module via f
2. If ~2 = f(al) then Qal(R 2) = Qa2(R2) 3. The extension f : Qol(R1) ~ Qal(R 2) is a ring homomorphism. PROOF.
We reduce this to the proof of Proposition
sider the following commutative f
R1
L
33 as follows.
Con-
diagram of ring homomorphisms
~ R2
i
R1/~1(R1)
f~ • R2/a I (R 2 )
Equiping Zl(R1) and ~2(R2) with the residue filters of T(a 1) and T(~2) , we obtain a diagram of final morphisms with vertical arrows representing torsion morphisms. logous statements
The facts listed below reduce statement in the level of f~ and then Proposition
1,2,3 to ana-
33 applies.
1. The Rl-mOdule structure of Qal(R 2) is defined via ~1" 2. Qal (R2) : Qal(R2/al(R2)) 3. Commutativity
= Qal (R2/a2(R2))
of the diagram yields ~2 f T(al)
4. The extension f : Qal(R 1) ~ Qal(R2)
of
= ~2 T(a2)
f to Q~I(R1)
= f~lT(al )
is actually the
extension of f~, by definition. Another consequence
PROPOSITION
of the reduction of property
(T) is :
34. Let f : R1,T 1 ~ R2,T 2 be a final torsion morphism.
~1 be a T-functor,
let Ker f be a al-ideal of R 1.
Let
Then Qal (R2) = Q~2 (R2)"
For every M e M(R2) we have that Qal(M) ~ Qa2(M) , i.e., the localization functors Qa I and Qa 2 coincide on R2-modules.
39
PROOF.
By property
(T) for o I we obtain that Q~I(M) ~ Qal(R 1) ®R1 M and
descent of property From Proposition
(T) under
f yields that Qa2(M)
= Qa2(R2)
~2
M.
33 we retain that
Qal(R 2) = Qa2(R 2) and thus Qal(R 1) = Qa2(R2) (since Qal(R 2) = Qal(R 1) because Ker f is a ~l-ideal Rl-mOdules
via
f,
f
is torsion).
entails that both
M
and Q~I(R1)
thus the tensor products
g induces
a final torsion reduction
a final torsion epireduction consequence
are R2-modules
then g-1(a2(M2))
ga I : al(M 1) ~ a2(M2).
then M1/al(M 1) ~ M2/a2(M2).
g-l(a2(M2))
= ~1(M1 ) If
g
is
This is an easy"
c ~1(M1)
~ gg-l(a2(M 2) ~ 0
and g(al(M1))
c al(M2)
imply that
g~l : gI~l(M1 ) is a final torsion reduction
and if
isomorphism
To a final reduction
M1/~1(M1)
~ M2/a2(M2)
g : M I ~ M 2 there corresponds This map is not necessarily T-functor
and
of the fact that we have an exact sequence 0 ~ Ker g ~ g-l(a2(M2))
because
the fact that
are isomorphic.
If g : M 1 ~ M 2 is a final torsion reduction and
Finally,
follows.
a reduction
a reduction
while Ker f is a ~l-ideal
modules because
Qolg
g
is onto then the
: Q~I(M1)
~ Qo2(M2).
of Q~l(R1)-modules.
then Q~lg is a reduction
If o I is a of Qal(R1)-
in this case the Rl-linear map Q~I(R1) ~ Q~2(R2)
is onto
and a ring homomorphism.
LEMMA 35. metric
Let f : R1, T 1 ~ R2, T 2 be a final morphism.
if and only if ~2 is symmetric,
and if so then
Then o I is sym:
1. Taking inverse images under
f yields an injection
C'(~ 2)
2. Taking inverse
f yields an injection
C(a 2) ~ C(al).
PROOF.
images under
The first statement
is trivial.
~ C'(al).
To prove 1, let A' • C(~2).
40
Then A : f-l(A')
~ T 1 since
f
is open and
B D A with B q T 1 then B D Ker f yields Hence f(B)
= A' follows
and then
also, A • C'(o 1) because
B = f-lf(B)
if
where f(B) D A'
B has to coincide with A.
The proof
of 2, is similar.
PROPOSITION
36.
Let f : R1, T 1 ~ R2, T 2 be a final torsion m o r p h i s m and
let o I be a T-functor
such that Ker f extends
to an ideal Q~l(Ker f) of
Q~I(R1 ), then ~1 is a prime kernel
functor if and only if ~2 is.
PROOF.
is saturated because A D ~1(R1)
Note that every A e C'(~1)
Thus A ~ f(A) and C'(~2).
sets up a one-to-one
correspondence
The corollary to Proposition
and only if all Rl-mOdules one another.
We have
between the sets C'(o 1)
8 yields that ~1 is prime if
with A • C'(~ 1) are isomorphic
Let A,B • C'(~ 1) and write A'
A',B' e C'(~2). Proposition
Q~I(R1/A)
Qo1(R2/A')
= f(A),
~ Q~I(R1/A)
to
B' = f(B), then
~ Q~I(R1/B)
34 implies that Q~2(R2/A ') ~ Q~2(R2/B')
for arbitrary
D Ker f.
~ Q~I(R2/B'
and since this holds
B' • C'(o 2) we derive from this that o I is prime
if and
only if ~2 is prime.
Let f : R 1, T 1 ~ R2, T2, be a continuous phism.
and surjective
One easily checks the following elementary
of pure ring theoretic nature.
If
A
and
B
ring homomor-
properties
which are
are left ideals of R 1 such
that A D Ker f then, f[A B
: B] : If(A)
: f(B)] and
are left ideals of R 2 then,
[A : B] : f-l[f(A)
If
A
and
If
f
is final and if A • C' (~l) is such that A ~ Ker f then,
f(A) e C'(~2).
PROPOSITION
37.
[f-l(A)
: f-l(B)]
: f(B)].
= f-l[A
If P e C(~1) , p n Ker f then f(P) e C(~2).
Let f : R 1, T 1 ~ R2, T 2 be a final morphism.
: B].
41
1. If o I is restricted 2. If
f
then o 2 is restricted.
is a final torsion reduction
then o I is restricted
if and only
if o 2 is restricted.
PROOF.
1. Let A • C'(o 2) then f-l(A) • C'(o 1) and
Thus f[f-l(A ) : R 1] = [A : R 2] and
[f-l(A)
: R 1] • C(Ol).
[A : R 2] • C(o 2) by the foregoing re-
marks. 2. Since Ker f c Ol(R1) , every A • C'(o 1) contains
Ker f and hence
A • C'(o 1) then f(A) • C'(o2).
Moreover
[f(A)
f-l[f(A)
[f-lf(A)
: R 1] • C(Ol).
: R 2] • C(o 1) and thus
restricted
: R 2] • C(o 2) yields
that
Thus if o 2 is
then so is o 1.
CQn~lus~on.
If one considers
R ~ R/o(R)
o • F(R), then most properties analogous
if
properties
for an arbitrary
idempotent
of o and Qo on M(R) are equivalent
for the restriction ~ of o to the embedded
to the
category
of R/o(R)-modules.
Special references A.W.
GOLDIE
D.C.
MURDOCH,
for Section
[11]; A.G.
HEINICKE
F. VAN 0YSTAEYEN
F. VAN OYSTAEYEN
[38].
II. [13]; J. LAMBEK, [25],
[26],
G. MICHLER
[27]; S.K.
[21];
SIM [30],
[31];
III.
Ill.
1.
Spe c a n d
the
Zariski
R is a l e f t N o e t h e r i a n
LEMMA
be
2. F o r a set 3. F o r
X,
{As,
called
if
P
from
S.
of
e •
A,B
take
o f R}.
is a s s o c i a t e d , The
To a n y
this
following
the
if
R,
ideal
A
set o b v i o u s l y
of
R
the
depends
on-
is clear.
sets
of
topolo$y.
ideal
of
f o r an
V(A)
we have
to be the
of c l o s e d
irreducible
V([ A s)
V(A
=
).
u V(B).
open
P
A subset
n V(B).
that
= V(A)
A point
R.
the u n i o n
point
R
= V(AB)
X A = X - V(A)
is not
A generic
let A c B, t h e n
V ( A n B)
Zariski
S
and
I} of i d e a l s
of
the
R
is a m a x i m a l
irreducible
V(P)
ideals
r a d A of A.
ideals
ideals
So w e m a y
only
unit.
38.
1. Let A , B
on
prime
= {P e X, P ~ A}
ly on t h e r a d i c a l
Topology.
ring with
Put X : S p e c R : { p r o p e r set V(A)
SHEAVES
of
sets X
for a topology
is c l o s e d
if a n d
S c X is said t o be
sets w h i c h set
S
are d i f f e r e n t
is a P e X s u c h t h a t
: S.
PROPOSITION
39.
irreducible
closed
is the u n i q u e
PROOF.
such that
exist
C A.
if x e B n C t h e n Hence,
point
V(A)
for
Hence
to V(P)
for
Conversely, some
every
P e X and P
S.
W 1 a n d W 2 are W 1 : V(P).
= V ( r a d A) we a s s u m e
ideals
Put
is i r r e d u c i b l e .
S c X is e q u a l
P • W 1.
Since
there B'C'
V(P)
= W 1 u W2, w h e r e
sets,
irreducible. then
subset
generic
If V(P)
one of t h e s e
prime
If P • X, t h e n
B' a n d
B = A + B',
C' w h i c h
sets,
then
P
Conversely,
let V(A)
A : t a d A.
If
are
C : A + C'
x = a I + b = a 2 + c with
x R x C b R c + A c A and this
closed
not
A
contained
is be
is not in
T h e n A = B n C;
A indeed,
a l , a 2 e A a n d b e B, c e C.
yields
x • A since
A
is r a d i c a l .
43
Therefore,
V(A)
such
P
that
does
we get
that
bility
of V(A).
since As
= V(B)
V(A)
{As,
~ •
open
sets
compact
contain
the
ideal
~ % entailing is a n o t h e r
ideals
of
= X - V(A
but not
R
A
Hence,
exists
point
a P • V(A)
picking
a contradiction
with
for V(A)
an x • B - A ,
the
then
irreduciP n A but
= P.
ring
gives
it is e a s i l y
rise
) if and o n l y
necessarily
(x).
generic
A n p, w h e n c e
of a c o m m u t a t i v e
I} of X
for any x ~ A t h e r e
P
also
case
and
V(B) If
A • V(P)
in the
not
m V(C)
verified
to a c o v e r i n g
if 1 • [ A s,
Hausdorff;
for any
of
that
X
by m e a n s
It f o l l o w s
ideal
A
a set
of
that
R,
of X
the
is
open
set X A is compact.
Let A , B
be
ideals
of
then
positive
T(A)
= {left
ideals
L
of
R
containing
= {left
ideals
L
of
R
such
~A(M) Note
= {m • M, that
being the
T(A)
the
the
tients
and T(A)
for
= 0
"tops"
with
are d e n o t e d
by iA,
PROPOSITION
40.
in rad
defines
(0),
L D B mn and
expounded
i.e.
A not
an
in g e n e r a l
QA(R)
L • T(B).
rad L
of
A.
they
that
R-module be
the
M. upon
of X A.
in rad
(0),
defined
canonical
are not
every
L D A n and inclusion
to any
the r i n g because
of quo-
both
maps
as
Using
II, we a s s o c i a t e
The
B D A}
integer
looked
XA
R ~ QA(R)
injective.
ideal
A
rings
not
since T(A)
contained
on Spec
B c rad A and X B ~ % if and o n l y
proves
rad
positive
elements
of n o n c o m m u t a t i v e
then
a filter,
~A'
it m a y
is w e l l
to X A for
in T(A),
This
some
for e v e r y
in s e c t i o n
This
such
functor
contained
to a A.
a presheaf
(0).
Pick
i.e.
B
L n A n for
L•T(A)}
to A n c B for
we a s s o c i a t e
ideal
the m a x i m a l
Assigning
B { tad
an
R
of XA,
though
If X B c X A t h e n
of
kernel
on the r a d i c a l
PROOF.
A
of X A and
respect
only
some
B is e q u i v a l e n t
is a subset
technique set XA,
depend
that
Lm
set of
QA(R)
ideal
a symmetric
set C(a A)
open
To an
defines
localization
non-empty
n.
A • rad
some
Obviously,
integer
R,
R.
if
A D B m we o b t a i n
C T(B).
Thus
n}.
44
o A ~ O B and we get the canonical projection ~, ~(A,B) : iA(R) = R/OA(R) ~ R/OB(R) = iB(R). Now, consider the following diagram (all maps in the diagram are ring homomorphisms by Theorem 10)
iA(R)-R/ / ~ ( A '
,> QA(R)
B)1
~
I p(A'B)
iB(R)
,~ QB(R)
Since QA(R)/iA(R) is OA-torsion, a f o r t i o r i OB-torsion, the fact that QB(R) is faithfully oB-injective implies that ~(A,B) extends uniquely to p(A,B), and Ker ~ = Ker p(A,B) n iA(R ).
The uniqueness property al
so yields that p(A,A) is the identity on QA(R).
If ¢ ~ X C c X B c XA,
then the following diagram of ring homomorphisms results :
iA(R)
~
w(A,B)
QA(R) A,C)
(A,C) ic(R)
p(A~B)
--
:$c(R)
(B,C) IB(R)
• QB(R)
Since ~(B,C)~(A,B) = ~(A,C), it follows that p(A,C) and p(B,C)p(A,B) are extensions of ~(A,C) to QA(R) and as such they must coincide. Consequently, the diagram is commutative. _Remark.
Let {Ai, i e I} be a finite set of ideals of
R, then it is
readily verified that the radical of ~. A. is equal to the radical of i
~ r a d Ai . 3_
i
45
THEOREM i.e.,
41.
The presheaf
(XA,QA(R))
denoted R is a m o n o p r e s h e a f
if X A : u X i is a covering of X A by open sets X i = X - V(A i) then
g : 0 is the unique element of QA(R)
PROOF.
on X,
Since X A is compact,
covering of X A.
such that p(A,Ai)g
we may suppose
: 0 for all
that we are given a finite
Writing oj for the kernel functor corresponding
we obtain the following
commutative
i.
to Aj,
diagram of ring homomorphisms
R
(R) R{°A(R)"!
p(A,A.)
) Ri~j
Q (R)
~ Q~(R) ]
By definition
of QA(R)
So if g e Ker p(A,Aj)
there exists
then Bg c iA(R ) n Ker p(A,Aj)
Bg c oj(R)/OA(R ) for all Bg : 0 and g e OA(QA(R)) Conversely, C.] 6 T(Aj)
an ideal B e T(A)
j.
such that Bg C i A ( R ) .
= Ker ~(A,Aj),
If we prove that ©3 oj(R)/OA(R)
= 0 follows.
Obviously,
let x e O o.(R), then for each j ] 3 such that Cj x = 0. By definition
~ oj(R)
hence
= 0 then
contains
OA(R).
there is an ideal of o.] we have that
rad C. D A.. Take C = ~ C.. Then rad C is equal to rad(~ rad C.) and ] ] J ] ] ] contains rad(Z Aj). But then the fact that V(A) = n V(A.) implies that J J any P e X containing Z rad A hence containing rad A. for every j is j J' 3 ' in V(A). Hence rad A c rad(~ A.) and therefore C is in T(A). Finally
j
C x = 0 yields
x 60A(R).
might construct
Using classical
a sheaf of sections
bably this would lead to a theory a commutative
ring.
However,
with the close generalization further assumption
that
R
sheafifioation
methods we
associated with the presheaf.
Pro-
similar to the theory of schemes over
in the present of affine
context we contend ourselves
schemes.
Therefore,
we make the
is a prime ring, and prove that in this case
46
the monopresheaf
THEOREM
42.
actually
Let
R
is a sheaf of rings.
be a left Noetherian
on Spec R is a sheaf,
i.e.,
set XA, X A = ~ X , with X g~ e Q~(R)
for which
such that
then there exists
p(A, A )g = g~ for every ~.
to prove the theorem for a finite
Indeed,
if X A = u X. is a finite covering of X A for i i is true, then X is covered by the sets
which the statement X
= X - V(A ), such that there exist elements
Note first that is sufficient
covering of X A.
The monopresheaf
if we are given any covering of an open
p(A , A A6)g ~ = p(A6, A As)gB,
an element g • QA(R)
PROOF.
prime ring.
N X.z = X - V ( A
be the image of
Ai).
g
Let g e QA(R) map onto gi • qi (R) and let h
in Q (R).
Then ge and h
every map p(A , A A i) and the foregoing entailing the desired property
have the same image under
theorem yields
that h a
for the arbitrary covering.
g~ = 0,
So suppose
that X A = u X. is a finite covering for X A. If the gi • Qi (R) have the i z prescribed property then we may derive from the presheaf axioms that : P(A i, ~
Ai)g i = p(Aj, ~
zero ideal of
R
because
order of the factors only depending that,
Ai)g j for all the product
i and
is finite.
in TI] A. is not important i
on radicals.
j,
1
Denote [7] A by m i
where m
A i is a non-
Note that the exact because
everything
is
B.
We are going to prove
if gl • Q1 (R) and g2 • Q2 (R) are such that
P(A1,B)g I : P(A2,B)g2,
then there is an element g • Qc(R), where P(C'A1)g
= gl and p(C,A2)g
repeates
this process
= g2"
C = A 1 + A2,
such that
The theorem then follows easily
if one
a finite number of times, because ~A is exactly
the kernel functor associated with Z A.. Elements of QI(R) are defined i i to be equivalence classes [Ll,f 1] of pairs (Ll,f 1) with L 1 e T(A 1) and fl • H°mR(LI'R)
the equivalence
relation
is given by
: (Ll,f 1) ~ (L~,f~)
if and only if there is an L 1 • T(A 1) such that fl and fl coincide L~ C L~ N L1.
on
47
Put gl : [L1'f111
• Q1 (R)' g2 = [L2'f212
ring,
all the maps
shows
that
follows
every
~(Ai,B)
P(Ai,B)
reduce
• Q2 (R)"
to the
is injeetive.
Because
identity
on
These maps
R
R
is a prime
and thus
it
are then d e f i n e d
as
:
P ( A 1 , B ) [ L I , f l ] 1 : [Ll,fl] B P ( A 2 , B ) [ L 2 , f 2 ] 2 : [L2,f2] B, the right pairs
hand
sides
denoting
but for the T ( B ) - e q u i v a l e n e e
the same
element
(L2,f2),
i.e.,
the latter
fl(x)
of QB(R)
may be t a k e n
- f2(x)
of L 1 N L 2.
Then
Define
: 0.
DEFINITION. Zariski
that
Let
topology
R
This
- f2(x))
g
maps
to
onto gl,g 2 under
and the sheaf R, called
A a nonzero
ideal
functor
free
for every
~A, hence
nonzero
Assigning
QA(M)
to X A defines
+ f2(a2)
fllL1 N L 2 = f21L1 verification
N L 2.
of the to
p(C,A1) , p(C,A 2) resp.
prime
ring.
Spec R w i t h
the s t r u c t u r e
defined
the
sheaf on Spec R,
by C(~ ~) = {(0)},
i.e.,
of R}.
be a o ~ - t o r s i o n every
on the whole
[Ll,fl] 1 and T ( A 2 ) - e q u i v a l e n t
be a left N o e t h e r i a n
be the s y m m e t r i c
Pick
= 0 and thus
fl and f2 c o i n c i d e
Simple
kernel
proof
L'(fl(x)
to
L' • T(B),
~B is symmetric.
C = A 1 + A 2.
Let ~
M
because
because
scheme.
Let
gl and g2 m a p onto
(Ll,f 1) is T ( B ) - e q u i v a l e n t
is p o s s i b l e
is said to be an affine
= sup{~A,
Since
f : L 1 + L 2 ~ R by f(a I + a 2) = fl(al)
is T ( A 1 ) - e q u i v a l e n t
[L2,f 2] proves
by the r e s p e c t i v e
L' c L 1 n L 2 and with
Consequently
Take g = [L 1 + L2, f]c with g
for some
to be an ideal
with a I • L1, a 2 • L 2.
that
that
L'x • L' yields
• ~B(R)
defined
relation.
it follows
flIL ' = f21L'
an x • L 1 n L 2.
fact
the classes
R-module.
Then
submodule
of
M
is ~ A - t o r s i o n
M
is faithful.
a sheaf M of R - m o d u l e s
of the fact that R is a sheaf may e a s i l l y
on Spec R.
be m o d i f i e d
free
The
so as to
48
apply
III.
to the m o d u l e
2. A f f i n e
Throughout X = Spec
case.
Schemes.
this
section,
R be an a f f i n e
It is o b v i o u s
that,
ideals
T ( A I A 2)
= s u p { T ( A 1 ) , T(A2)}.
of Spec
R and
QA(R)
~ Qa~(R).
be c o n s i d e r e d
as b e i n g
, p(A,B),
stalks
of the
PROPOSITION
=
PROOF.
i.
latter
entails
for
some
Let
The
R,
we have
to call ideal
these
P • X, then
ring.
Let
Qo~(R)
A
of
the
R
that
of the d i r e c t e d
following
function
we h a v e
considerations
limit
that ring
injections
Qo~(R)
may
system
proposition
determines
the
:
P • X A}
lim
P~ XA
QA(R).
P ~ A,
o A ~ o R _ p.
not
prime
sheaf.
If P • X A t h e n
s
from
the d i r e c t
structure
43.
Q R - p(R)
We agree
0 ~ B c A}.
1. o R _ p : sup{oA, 2.
A 1 and A 2 of
o A ~ o ~ for e v e r y
It f o l l o w s
{QA(R)
is a left N o e t h e r i a n
scheme.
for
since
R
in
P.
hence
A D
Conversely,
Hence,
putting
(s)
for
some
s • R - P.
if B • T(o R _ p) t h e n A : (s) we h a v e
B D
The (s)
P • X A and
B • T(A). 2. This
is an easy
consequence
P • XAB.
It has
noted
rings QAB(R)
QA(R) maps
and
been QB(R)
is o b t a i n e d
as the
sup
{OA,
P • X A}
The
ring
QR - p(R)
that
are m a p p e d
injectively
{QA(R),
of
into
°AB
= sup{°A'°B}"
The
0 ~ B c A,
ring
if P • X A and
injectively
Qc(R).
p(A,B),
quotient
1, since
for the
into
direct
P • X B then
Furthermore, some
limit
Qc(R) of the
then
if the also
system
P • X B} symmetric
functor
= o R _ p. is c a l l e d
the
stalk
of R at
P.
The
stalks
of the
49
structure sheaf are mapped injectively into the function ring Qo~(R). Similar to P r o p o s i t i o n 43, we have that the stalks of the sheaf M are given by the modules Q R - p(M).
Mor~hisms of affine schemes are defined
to be m o r p h i s m s of the sheaf of rings, i.e., let R1,R 2 be prime left N o e t h e r i a n rings and let X = Spec R1, Y = Spec R 2 be the c o r r e s p o n d i n g affine schemes;
a m o r p h i s m from
X
to
Y
is then given by
:
1. A Zariski continuous map ~ : X ~ Y, such that 2. For every n o n - e m p t y open set
U
in
Y , write ~-I(u)
= V and let o U,
~V be the associated kernel functors on M(R 2) and M(R 1) respectively, then there exist ring h o m o m o r p h i s m s ~U
: Qu(R2 ~ ~ Qv(R1 ) which are
compatible with sheaf r e s t r i c t i o n s . A m o r p h i s m of affine schemes is said to be an i s o m o r p h i s m if and only if is a h o m e o m o r p h i s m of Zariski topological spaces and all induced maps CU are ring isomorphisms. The f u n c t o r i a l properties
Spec enjoys in the commutative
o b t a i n e d in full g e n e r a l i t y here.
case cannot be
R e s t r i c t i n g attention to special
open sets and the kernel functors a s s o c i a t e d to them, we obtain a satisfactory local theory. Conventions
and definitions
o A is a T-functor; (T).
: An open set X A in X = Spec R is a T-set if
the stalk Q R - p(R) is a T-stalk if o R _ p has property
If a T - f u n c t o r OA(T-set XA, T-stalk Q R - p(R))is such that
OA-perfect
(oA-perfect , o R _ p-perfect) In case
R
is
then it is called a geometric
funetor
(set, stalk).
R
is a c o m m u t a t i v e N o e t h e r i a n integral
domain,
all basic open sets X(f),
f • R, are T-sets and all stalks are
T-stalks. If
R
is not an h e r e d i t a r y ring however, not every open X A is a T-set.
For example if domain,
R
is Noetherian,
then, taking
M
integrally closed but not a D e d e k i n d
to be a n o n - i n v e r t i b l e maximal
find that o M is not a T - f u n c t o r
(cf.
ideal in
[12], example 2, p. 45).
sometimes be useful to have enough T-sets
in Spec R.
An
R
we
It may
affine scheme
50
Spec R is said to have a T-basis if there exists a basis for the Zariski topology•
consisting of T-sets.
Let
M
be a o ~ - t o r s i o n free R-module,
and denote by M the c o r r e s p o n d i n g sheaf of R-modules on Spec R.
A pre-
sheaf R ~ M may be defined by a s s i g n i n g QA(R) ® M to XA, we have
:
R
P R O P O S I T I O N 44.
If X = Spec R posseses a T-basis then M m F(R ® M);
F(R @ M) being the sheaf of sections of the presheaf R ® M.
PROOF.
Let X A be a T-set.
Then QA(M) m QA(R) ~ M.
A section
R
S e F(XA, R ~ M) may thus be identified with a section in F(XA,M).
Since
F(XA,M) m QA(M) and since both M and R ® M coincide on a basis for the t o p o l o g y in
X
it follows that M is isomorphic to the s h e a f i f i c a t i o n of
~ M, hence M ~ F(R ~ M). Note that,
since the modules considered are to be o~-torsion free,
it is
impossible to deduce from the foregoing that o R _ p is a T - f u n c t o r for all P • X.
This will be proved later, see P r o p o s i t i o n 49
P R O P O S I T I O N 45.
A geometric stalk is a "local" ring,
N o e t h e r i a n prime ring with a unique maximal
ideal.
i.e.,
corollary.
it is a left
The proof, using
P r o p o s i t i o n 11, is easy. If o ~ ±s " a T - f u n c t o r then Qo~(R) has
(0) for a maximal
ideal and hence
it is a simple ring, g e n e r a l i z i n g the function field of a variety over a commutative ring.
We return to a more general setting.
Let T ~ o be symmetric kernel
functors and suppose that o is a T-functor. e x t e n d e d left ideals A e• A e T(T). of an idempotent kernel functor e
LEMMA 46.
Let T(T e) be the set of
Then• T(T e) is o b v i o u s l y the filter on M(Qo(R)).
QTe(Qo(R)) m Q~(Qo(R)) m Q (R).
With these conventions
:
51
PROOF.
Consider
M e M(Qo(R))
Qo(R)-modules
then x e Te(M)
as R-modules
via R ~-~ Qo(R).
if and only if Aex
= O for some A e e T(~e),
if and only if Aox = 0, A ° e T(T),
or equivalently
and r e coincide
The Qa(R)-module
Te-torsion,
on Qo(R)-modules.
hence r-torsion
map ~ results, re-injective,
as an R-module
~ : Qre(Qo(R))~--~ while
Qr(Qa(R))
x e r(M).
Now,
is Te-torsion
Thus
QTe(Qo(R))/Qo(R)
and an R-linear
Qr(Qo(R)).
If
since
is
injeotive
Qre(Qo(R))
is
free and
QT(Qo(R))/QTe(Qo(R)) being ~e-torsion, follows.
of the fact that QTe(Qo(R))
let A e T(r);
quely to an R-linear is Qo(R)-linear
: Q e(Qo(R)) follows
is faithfully
m QT(Qa(R)) easily
A e ~ Q e(Qo(R))
by the o-injectivity
to R ~ QTe(Q~(R)),
by property
providing
extends
Since A e e T(~e),
Qo(R) ~ Q e(Qo(R))
the desired
uni-
(T) for ~, but this map
of Q e(Qo(R)).
to a unique
after
T-injeetive.
then any R-linear map A ~ QTe(Qa(R))
the latter map may be extended tricts
isomorphism
The fact that it is a ring isomorphism
verification Indeed,
a Qo(R)-module
R-linear
which res-
map extending
the initial A ~ Q e(Qo(R)). The isomorphism
THEOREM
47.
QT(Qo(R))
Let X A be a geometric
an affine scheme,
PROOF. position
m Qr(R)
ideals
11 implies
dence with proper
pect inclusions
the elements
ideals
ideals of QA(R).
it is some X C with C c A. it follows
10.
: X'
P ~ A are exactly
that these prime
prime
from Theorem
open set of X = Spec R, then X A is
in fact X A = Spec QA(R)
The prime
form XAB hence
follows
P ~ T(A).
are in one-to-one
An open subset Because
correspon-
of X A is of the
the operations
that X C is in one-to-one
Pro-
correspondence
c,e reswith
X' = {P' e X', P' ~ C e} and thus e defines a homeomorphism of the toCe pological spaces X A and X' (for the induced Zariski topologies).
52
We a s s o c i a t e d
Qc(R)
to X C and Q c e ( Q A ( R ) )
to X'Ce'
The
foregoing
yields
Q e(QA(R)) m Q c ( Q A ( R ) ) m Qc(R), p r o v i n g that X A w i t h the C Z a r i s k i t o p o l o g y and the r e s t r i c t e d s h e a f is an a f f i n e scheme.
THEOREM
48.
and s u p p o s e Speo QA(R)
Let X C and X A be d i f f e r e n t that X A is a g e o m e t r i c if and o n l y
correspondence
T-sets
set.
T-sets
properly
X
in Spec R. contained
induced
such that X C c X A
T h e n Xce is a T - s e t
if X C is a T - s e t
between
in
lemma
There
in
is a o n e - t o - o n e
in X A and p r o p e r
T-sets
in Spec QA(R).
PROOF.
First
suppose
that a C is a T - f u n c t o r .
ideal
L e T(C e) is a O c e ~ p r o j e c t i v e
perty
(T) b e c a u s e
R
a QA(R)-linear
QA(R)-m°dule
is left N o e t h e r i a n .
M' ~ M ~ 0 be an e x a c t
sequence
map h : L ~ M.
L c 6 T(C).
Restriction
R-modules,
M
a T-functor
it f o l l o w s
of
h
left
ideal
B 6 T(C),
maps
is c o m m u t a t i v e
and t h e n Oce has
Let L • T(C e) and
left pro-
let
of o e - t o r s i o n free Q A ( R ) - m o d u l e s . C F r o m L D (ce) n d e r i v e s L c D C n or
to L c y i e l d s
and M' are o c - t o r s i o n that
We show that e v e r y
an R - l i n e a r
Given
h c : L c ~ M.
As
free and f r o m the fact t h a t ~C is
L e is a C - p r o j e c t i v e
and thus t h e r e
B c L c such that the f o l l o w i n g
diagram
exists
a
of R - l i n e a r
:
B c
>L c
M'
-~M
f'
If
L
is p r o p e r
module,
it is f a i t h f u l l y
Therefore h ( q a)
then L c ~ T(A),
a QA(R)-linear
hence
0A-injective
the Q A ( R ) - l i n e a r
= qhc(a)
~0
map
h
B ~ T(A). (because
is d e f i n e d
w i t h a e L c, q • QA(R). map f : B e ~ M ' , f ( q b)
Because
M
is a Q A ( R ) -
°A is a T - f u n c t o r ) .
as f o l l o w s
:
In a s i m i l a r way f' e x t e n d s
= q f'(b)
for q e QA(R), b e B.
to
53
Now,
B • T(C)
lowing
- T(A),
diagram
proving
that
functor,
thus B e • T(C e) and B e ¢ QA(R);
commutes
~L
M'
~M
dule via QA(R)
Conversely,
sequence
assume
is a Qc(R)-module,
hence
it follows
that the inclusion
M ~ Qc(M)
Ker i
(because
= QA(Oc(M))
of QA(R)-modules
Qc(M)
an exact
sequence
is so because
Qc(M)/Im
Action
of Q e on C
i~ ~ 0
~ Q e(Qc(M)) C
~ 0
Qc(M)/i(M)
and also Oce-tOrsion
is Oc-torsion,
of Lemma 48 may be used to proof that Qc(M)
Oce-injective
QA(R)-module.
Thus
Qce(Qc(M))
thus
as a QA(R)-module.
The technique
m QC(M)
in-
:
Im i ~ D i(M) while
i ~ is Oc-torsion
Qce(QA(M))
extends
:
0 ~ Qce(QA(M)) This
a QA(R)-mo-
is oA-faithfully
c Oc(QA(M)).
0 ~ Ker i ~ ~ QA(M) ~ Qc(M) ~ Qc(M)/Im yields
that Oce is a T-
Qc(M)
i ~ : QA(M) ~ Qc(M),
Obviously,
the exact
Since
~ Qc(R),
to a QA(R)-linear jeetive).
~0
is Oce-projective.
let M • M(R).
the fol-
:
BeC
L
obviously
m Qc(M)
is a faithfully
and therefore
follows.
Now,
Q e(QA(M)) ~ Qce(QA(R)) @ QA(M) by (T) for . C °ce QA (R) Then, Q e(QA(M)) m Qc(R) @ [QA(R) @ M] m Qc(R) ® M. The isomorphism C QA(R) R R Qc(M) ~ Qc(R) ~ M is equivalent Noetherian.
Denote
The one-to-one X A n XC ' = X C. but P D L C'
to property
C ee by C', i.e.,
correspondence Indeed,
follows
(T) for o C because
C' = { x • R, L x C easily
is left
for some L • T ( A ) } .
from the fact that
X C C XC, and if P • XC,
for some L • T(A)
C
R
X C then P D C, P ~ C',
and thus P ~ L, entailing
P @ X A.
54
PROPOSITION
49.
If X A and X B are
T-sets
in Spec
R then
XAB
= XA n X B
is a T-set.
PROOF.
We have
t e r T(a) T(A)
has
a basis
u T(B).
contains
QA(R)
from
the
R~mark.
The
is a g a i n
a T-functor.
50.
some
entails
R has
T-set
X B.
that Since
by the
THEOREM
above
51.
If an i d e a l
PROOF. We m a y
T(a)
yields
set
that
in
X A is a u n i o n a B ~ aA,
A
of
has
R
A ~ T(a),
with
the
then
X
Knowing deduces
that
each
and
The in that
Qa(R)
that Thus,
a is a T - f u n c t o r .
stalk
let
is a T - s t a l k .
P e X A. hence
The
existence
P e X B c XA
that
p e XA,
X A is a T-set]
a R _ p is a T - f u n c t o r .
of f i n i t e hence
C.l 6 T(al)
T-functors,
a a2-ideal
products
then
let a be A
u T(a2 ).
sup{al,a2}.
is a a-ideal.
of ideals
in T(a 1) u T(a2).
A ~ T(a 1) and A ~ T(a2).
A C 1 . . . C n n C I' A C 2 . . . C n
Consider
Then,
n...n
' . "C'n A, C 1.
!
with ~
C i e T(a);
each
fil-
sup of a set of T - f u n c t o r s
of T-sets,
we h a v e
ideals
i = 1,...,r.
P e XA} = sup{aA,
is a a 1- and
a basis
of
easily
implies
a T-basis
open
one
= Qa(R),
Let a I and a 2 be s y m m e t r i c
suppose
[-~i C.I 6 T(a)
remark~
products
a product.
and this
argumentation
a R _ p = sup{aA, and
of f i n i t e
Qa(R)Ci
= Qa(R)
If Spec
a B} is a T - f u n c t o r .
as s u b r i n g s
that
Let X A be a Z a r i s k i
of a T - b a s i s for
Qa(R)C
same
QB(R)
fact
yields
PROOF.
a = sup{aA,
consisting
and
C e T(a)
PROPOSITION
that
Let C = C 1 ..... C r be such
both
1 e Qa(R)C
to p r o v e
inclusion
deriving
from
the
a 1- or a 2 - i d e a l
55
condition
for
A.
Theorem
14 yields
that
A
is a o-ideal
and Qo(R)A
is
an ideal of Qo(R).
COROLLARY
1.
If X A and X B are geometric
geometric
set.
COROLLARY
2.
If Spec R has a geometric
is a geometric For geometric
PROPOSITION
sets
basis
sets, we may prove
52.
an analogue
Let X C and X A be different that X A is a geometric
of Theorem
T-sets
set.
in Spee QA(R)
if and only if X C is geometric.
of Spec QA(R)
correspond
let
one-to-one
The correspondence
has already
been proven
is an ideal of
there
R
in Theorem
48.
an L' E T(OC)
to QA(R)
T(Oce)
is of the form L e for some L e T(OC)
ideal of
Conversely, R.
suppose
Then QA(R)B
Thus Qce(QA(R))B
finishes
from the foregoing
C' = {x • R, L x
(L')eB c B L e,
that QA(R)
c C
and Bc
for every L e T(OC)
Extension
it follows
of these
that and let
because
B
is OceB be an
X A is geometric.
The isomorphism
The one-to-one
correspondence
and the fact that XC, n XA = X C where
for some L • T(OA)}.
this
and since every ideal in
is an ideal of QA(R)
the proof.
because
Now B c ~ T(o C)
is Oce-perfeet
is an ideal of Qce(QA(R)).
Qc(R) ~ Qce(QA(R)) derives
that
of X A.
By contraction,
Bce = B.
such that L'B c c BCL.
subsets
that X C is geometric
and hence,
ideals
ideal.
yields
subsets
such that B ~ T(~ce).
is an ideal of Qc(R)
Xce is geometric
no problem
Suppose
since
48 :
Proper geometric
presents
and it is a ~A-ideal
that Qc(R)Be
exists
Q R - p(R)
in Spec R such that
Then,
to geometric
of T-sets
B be an ideal of QA(R),
implies
then each stalk
stalk.
X C c X A and assume
PROOF.
in Spec R then XAB is a
56
It has
been
to d e t e r m i n e For
such
and
so this
A ring
a class
a ring
R
of ideals Because ted
A,B
of
Qo(R/A)
~
that
every
the A r t i n - R e e s exists
II.
we
3.,
include
it is a m o s t
are o A - p e r f e c t
to the
consequences
Let and
1.
T-set
condition an
the
for e v e r y
problem
ideal
A.
R is g e o m e t r i c ,
case.
if and o n l y
integer for the
interesting
X A in Spec
commutative
n >0
such
theory
following
be a left N o e t h e r i a n
let o be a s y m m e t r i c
the R - l i n e a r
if for
that
any p a i r
B n A n C BA.
of r e d u c t i o n s
proposition
presen-
in its most
C'x c C and g(c') C n n A c A C.
C'x
g(c')
= f(C'x)
c C n.
2. P r o c e e d i n g induced
to show
ideal
given
T-functor.
ideal
A
that
and
may
[C,f]
B E T(o)
such
for e' • C',
C n • T(o)
x[O,f]
= x[C n
:
uniquely induced
to a r i n g
by the
struc-
canonical
we h a v e
fIC n]
structure
be r e p r e s e n t e d
= [C',f']
B c C n C'.
where
C' • T(o)
x • R.
as
Take
n
A
an-
[C,f]
if and o n l y
that
The
if
f
and
R-module
satisfies
such
that
that
: [C',g]
If x E A, C'x c C n m A c A C
and
thus
for c' E C',
C'x E A C.
as in P r o p o s i t i o n by R/o(R)
the A r t i n -
R.
in the R - m o d u l e
of Qo(R/A)
Since
= 0 because
Then
of
extends
by x[C,f] = [C',g]
= f(c'x)
satisfying
homomorphism.
f • H o m R ( C , R/A)
is t h e n
ring
: Qo (R) ~ Qo(R/A)
Elements
on a left
every
of Qo(R/A)
is a ring
Qo(R/A).
where
for
~o
It is s u f f i c i e n t
C • T(o),
structure
R
structure
: R ~ R/A
f' c o i n c i d e
also
closer
is an R / A - m o d u l e
and
nihilates with
us
R , there
R/A-module
ture
PROOF.
which
obvious
53.
condition
map
of rings
that
form.
PROPOSITION
2. The
out b e f o r e
it f o l l o w s
satisfies
in s e c t i o n
Rees
R
brings
of its
general
1.
pointed
~ R/A~,
33,
this
another
follows
proof
for
easily. 2. may
Since be g i v e n
~a
is
if one
57
modifies
1. as follows;
Qo(R/A)
is an R/Ao-module.
Then Proposition 33
applies directly because R/A o is o-torsion free.
References
for Section III.
D.C. MURDOCH,
F. VAN 0YSTAEYEN
[26], F. VAN OYSTAEYEN
[38].
IV. PRIMES IN A L G E B R A S OVER FIELDS
IV. 1. P s e u d o - p l a c e s of Algebras over Fields.
Let
K
be any field and let
A
be a K-ring.
K-ring
A
is given by a triple
(A',~,A1/K1) , where 9 : A' ~ A 1 is a
ring h o m o m o r p h i s m defined on a subring A' of v a l u a t i o n ring O K of is a place of
K,
A p s e u d o - p l a c e of the
A
such that A' n K is a
and such that the r e s t r i c t i o n of ~ to A' n K
K whith residue field K 1.
In the sequel, p s e u d o - p l a c e s
and places will always be assumed to be surjeetive, se specified,
K
and K 1 will be contained
respectively.
A pseudo-place
and unless otherwi-
in the center of
A
(A',~,A1/K 1) of the K-algebra
and A 1 A
will be
denoted by ~ when no confusion is possible.
If ~ is a p s e u d o - p l a c e of
A/K, and if {yl,...,ym}
A
sentatives
is K l - i n d e p e n d e n t
in
then any set of repre-
[Xl,...~x m} c A' with ~(x i) = Yi is clearly K - i n d e p e n d e n t
in
A. Examples of p s e u d o - p l a c e s
are
: places of fields, h o m o m o r p h i s m s
bras over fields and specializations The substantial part of K[A'] generated over
K
A
of alge-
of orders in central simple algebras.
with respect to (A',~,A1/K 1) is the subalgebra
by the elements of A'
A p s e u d o - p l a c e ~ is
special if for every x e A there is a ~ • OK, ~ ~ 0, such that ~x • A', i.e.
if and only if A : K[A'].
It is in general not really r e s t r i c t i n g
to c o n s i d e r special p s e u d o - p l a c e s only, however we do not s y s t e m a t i c a l l y impose this in the sequel. A p s e u d o - p l a c e ~ of A/K is said to be r e s t r i c t e d if for every n o n - z e r o x • A there is a ~ • K such that ~x • A' - P, where P = Ker ~. [A : K] < =
, then ~ is u n r a m i f i e d if [A 1 : K 1] = [A : K].
ties unramified,
restricted,
special, decrease
If
The proper-
in strength if listed in
this order.
PROPOSITION
54.
Let n = [A : K] < ~ and let ~ be a place of
K
with
59
valuation
ring
O K and m a x i m a l
unramified
pseudo-place
isomorphic
to ~.
PROOF.
Let { c i j , k l i , j , k
associated a cij,k sis
to a K - b a s i s
A
of
A
: 1,...,n}
of
A.
There
is t h e n o b v i o u s l y
{fl,...,fn}
is an ideal
a pseudo-place
in A'. of A/K.
Putting
1 • M.
left y i e l d s
Hence
of @ to
The O K - m O d u l e
Indeed,
A' g e n e r a t e d
is
a K-ba-
that
con-
by
M generated
epimorphism
observe
K
set of s t r u c t u r a l
a r i n g and the M K - m O d u l e The c a n o n i c a l
one
constants
fi : a el, we o b t a i n
z n e.f. = I • K - O K w i t h ~i e OK, then b e c a u s e i=1 i 1 that
at least
is an a • O K such that
such t h a t the c o r r e s p o n d i n g
{fl,...,fn}
exists
c K be a set of s t r u c t u r a l
{el,...,en}
{ c [ j , k } is a s u b s e t of O K .
defines
T h e n there
such that the r e s t r i c t i o n
= c i j , k • O K for all i,j,k.
{fl,...,fn}
stants
~ of
ideal M K.
by
~ :A' ~ A ' / M
=A 1
if
l-le i • M K it f o l l o w s
zn m i f i ' mi • MK M u l t i p l y i n g fj by 1 on the i=1 ~ k fk' and we o b t a i n 1 = i=1 zn mi cij,j • e MK, : f3• = k , izn = l mi cij,
a contradiction.
1 :
Moreover,
6 = i=1 zn m i f i • O K - M K
with m i • M K entails
1 =
zn ( m i 6 - 1 ) f i e M, again a c o n t r a d i c t i o n h e n c e M n K = M K. To proi=1 ve that ~ is u n r a m i f i e d , let i=1 zn ~i ~(fi ) = 0 w i t h ~i • KI" C h o o s e representatives
li • OK such that %(~i ) : ~i'
9(1 Zn1 lifi) "= entailing
Let
pseudo-place
~ to
A.
of A / K
k
zn l.f. • M, i:1 i i
be a s u b f i e l d
of
K
such that the r e s t r i c t i o n
is a k ~ p l a e e
of
K
t h e n ~ is c a l l e d
K
The d i m e n s i o n
cendence
or
degree
of a k - p s e u d o - p l a c e
of K 1 o v e r
k.
The r e s t r i c t i o n
of a p s e u d o ~ p l a c e
is s p e c i a l ,
~ is d e f i n e d
Let
B
restricted,if
a k-pseudo-place
pseudo-place
be a K - a l g e b r a ,
~ of A / K to ~ is such.
B
of the
to be the t r a n s -
The r a n k of an u n r a m i f i e d
is the r a n k of the p l a c e ~/K.
B/K w h i c h
Then
I i e M K and ~i = 0.
DEFINZTION.
of
= 0
i = 1,...,n.
K c B c A.
is a p s e u d o - p l a c e
of
60
PROPOSITION
55.
to subalgebra
PROOF.
of an unramified
B/K is an unramified
pseudo-place
Let {yl,...,yr } be a Kl-basis
to a Kl-basis tives
The restriction
of A1,
pseudo-place
of B/K.
of B 1 = ~(B n A') and complete
{yl,...,Yr+l,...,yn }.
then it may be completed down the q - r
relations
A
it
Choose a set of representa-
{Xl,...,x n} with @(x i) : Yi and with {Xl,...,x r} C B.
{Xl,...,x n} is a K-basis of
@ of A/K
The set
and if {Xl,...,x r} is not a K-basis of
to one,
{Xl,...,Xr,br+l,...,b q} say.
=
: br+t
B
We write
r ~n j=l atj xj + k:r+l ctk x k with atj and
Ctk in K. Put b'r+t = k:r+l En ctk xk"
Then {Xl,...,Xr,br+l,..., b'q} is still a K-ba-
sis of
t.
B.
Fix an index
There exists a c t • K such that e t c t k • O K
for all r + 1 4 k ~ n and ctCtl b r+t • : c b'zr+t • B n A' For suitable
~i • K1
= 1 for some r + 1 4 1 ~ n.
We obtain
En @(ctctk)Yk • ~(B') = B' and @(b r• +t ) = k=r+l i = 1,...,r,
= B 1.
we have k=r+lEn ~(ctctk)Y k = i=lEn ~iYi,
with ~(ctctl ) = 1, hence a contradiction. The following damental
theorem characterizes
for the application
simple algebras
THEOREM
56.
unramified
of pseudo-places
pseudo-places;
to the theory of central
(section V).
Let @ be an unramified
pseudo-place
of A/K then
1. A' is a free OK-mOdule
of rank n : [A : K].
2. There exists a K-basis
{el,...,e n} of A, generating
such that
PROOF.
it is fun-
:
A' over O K and
{~(el),...,9(en) } is a Kl-basis of A 1.
Pick a K-basis E = (el,...,e n) of A such that e i e A' for all
i = 1,...,n, nerated by
and such that ~(E) E
is a ring;
I.
of A 1.
The OK-mOdule
for, let e.e. ~n cij, k ek' there exist 1 ~ = k:l
cij e K such that cijcij,k cij,l = 1 for some
is a Kl-basis
= c[j,k is in O K for all k = 1...n, while
Hence,
~ n c ~lJ ,kek e A ' " cijeie ]. : k=l
ge-
61
For a couple cij,k
(i,j)
such that
c-lc ij ~ij,k • O K "
=
one hand
For a couple
: 9(cijeiej)
-1 cij • O K hence
cij ~ M K we have
(i,j) with cij
e M K we get on the
= 0 and on the other hand
@(cijeiej ) : k=lZ n ~(c[.],k ) ~(e k), contradicting (i,j)
the K l - l i n e a r
the p r o d u c t
any x • A', x = have
xjlx
=
idependency
eie j belongs
to the O K - m O d u l e
~n x.e. with x. • K. i=1 i i 1
all x i are
OK[E].
Furthermore,
xj I • M K yields
trivial
relation
: 0K[E] If
L
0
=
~(xjl)~(x)
and the proof
is a d i v i s i o n
such that
of
tion of a place
L
ring
tion
as follows.
A pre-place
an u n r a m i f i e d
case,
of a K - a l g e b r a
A
(cf.
[34]).
: ~(A')
P2
: If x,y e A and xy 6 A' then x ~ A' implies
i.e.
a ring
PROPOSITION
such that
57.
A 1 is a d i v i s i o n
Let
ring
then
to a non-
Thus
in its c e n t e r
and
(L', ~, L1/K 1) of ring.
The d e f i n i -
to extend
~ such that
defini-
this d e f i n i -
:
implies
y • P, similarly,
P = Ker 9.
to refer
to a ring w i t h o u t
(0) is a c o m p l e t e l y
prime
then ~ is a pre-place.
zero divisors,
ideal.
( A ' , ~ , A 1 / K 1) be a r e s t r i c t e d
ring,
that
ring.
x e P, w h e r e
We use the term d i v i s i o n
We wish
Pick
belongs
to the c o r r e s p o n d i n g
is a p s e u d o - p l a c e
= A 1 is a d i v i s i o n
K
if L 1 is a d i v i s i o n
P1
y ~ A'
x
because
pseudo-place
is a n a l o g o u s
E.
j, 1 ~ j ~ n, we
that
a contradiction
the field
couple
xj I ~ M K implies
in O K too, m e a n i n g
containing
of a s k e w - f i e l d
in the c o m m u t a t i v e
Now,
by
=
if and only
tion
generated
~n ~ ( x i ) ~ ( e i ) , follows. i=1 is complete.
[L : K] < = then
L/K is a place
So for every
For s o m e index
zn x? e. with x~ • OK, x~ = 1. i=1 1 1 3
xj is in O K and thus
A'
of ~(E).
pseudo-place
such that
62
PROOF.
Observe
• M K.
Thus
The
Ix • A ' -
~(Ix)9(~y)
xy • A',
if x ~ A'
if xy • A' w i t h
• O K so that entails
that
y ~ A'
following
P,
and
x ~ A'
shows
that
and y ~ P, t h e n
BY • A' - P.
= 0 and this
implies
if I • K is such
Hence
contradicts
Ix • A' - P t h e n
there
lBxy • A ' -
exist
I • MK,
P, but
I~ • M K
lx~y • A' - P.
Similarly,
x • P.
that
restricted
pre-places
do g e n e r a l i z e
L
be a field,
K c L c A.
A restricted
places
of
fields.
PROPOSITION
58.
Let
9 of A / K r e s t r i c t s
PROOF.
Let x e L and
I,~ • M K such
that
multiplicatively I~ • M K.
suppose
~x -1 and
closed,
Moreover
p n L is the A'
to a p l a c e
ideal
of
L.
that
Ix are
I~ •
xx -1 • A'
x ~ A'
n L and x -1 ~ A'
in
n p) n L.
x ~ A'
of n o n - i n v e r t i b l e
n L is a v a l u a t i o n
(A'
(A' - P) n L f o l l o w s , with
ring
of
that
an u n r a m i f i e d
such
: K] < ~ is an i s o m o r p h i s m
rem
56 and
sion
IV.
K/k
2.
Let that
@2
A~ n A½ that
the
fact
that
a k-place
is an i s o m o r p h i s m
Specialization
(A~,~I,A1/K1)
of
and
PROPOSITION
5g.
~IIK
Let
but
this
this
in A'
set
is
contradicts
x -1 E L n p .
Thus
n L, and t h e r e f o r e
k-pseudo-place of
of the
A. finite
This
of a K - a l g e b r a
follows
dimensional
from
of @1'
be p s e u d o - p l a c e s
field
and w r i t e
Obviously,
is a s p e c i a l i z a t i o n
@1 ~ ~2'
if and
the g i v e n
definition
(of places)
of ~21K.
( A ' , ~ , A 1 / K 1) be a p s e u d o - p l a c e
( A ~ , ~ , A 2 / K 2) be a p s e u d o - p l a c e
of A/K.
of A 1 / K 1 then
of A / K
exten-
~ o ~ defines
We
only
and
A
Theo-
K.
(A~,~2,A2/K2)
Ker 91 c Ker 92.
the p l a c e
Since
of P s e u d o - p l a c e s .
is a s p e c i a l i z a t i o n and
Take
L.
verified
[A
entails
elements
It is e a s i l y that
pre-place
say
if
implies
let
a pseudo-place
63
of A/K on the subring @-I(A[).
PROOF.
The only thing to establish
ring of
K which coincides
O K = A' n K contains then we distinguish ~ase_~.
x
-1
with 4-1(K1
4-1(K1 three
is that 4-1(A~) n A[) n K.
n A~) n K.
cases
n K is a valuation It is immediate
Now let x • K - 4-1(A~
that
n K1) ,
:
x ~ O K. Then x -1 e MK, thus 4(x -1) = 0 and this yields 1
• 4- (A~ n KI) n K.
~9§~_~"
x • O K - M K.
Then x -1 • O K and 4(x -1) • K 1.
~(x) ~ A 1 N K 1 it follows x - 1 • 4- I ( A ~ ~a~e_~. Thus,
From ¢(x)
of A/K.
= 0 immediately
x • 4-1(A~
n K1) n K is indeed a valuation
cides with 4-1(A~)
PROPOSITION
that ~(x -1) • A[ n K1, therefore
n KI) .
x e M K.
¢-1(A~
Since
60.
ring of
n K1) n K. K.
That
it coin-
n K is obvious.
Let
(AI,41,A1/K 1) and
(A~,42,A2/K 2) be pseudo-places
Then 41 ~ ~2 if and only if there exists
a pseudo-place
~ of
A1/K 1 such that 92 = ~ 0~1. The proof
is straightforward
41(A ~) ~ 41(A½)/41(Ker
Remark.
a K-basis
or special
in the situation ~Ul,...,Un~
This allows us to construct, K
pseudo-place
~. The
of
pseudo-places.
described A
in Proposition
such that
01 : A~ n K~ and A~ : 02[u I ..... Un] with
~1 and ¢2 of
that
if and only if 41 and ~ are unramified.
same is true for restricted
exists
in showing
42 ) m A 2 is the desired
42 is unramified
Furthermore,
and consists
: A~ = 01[Ul,...,Un]
with
02 = A½ N K.
for any algebra A/K,
for arbitrary
such that ~1 ~ ¢2 and for any unramified
(A~,42,A2/K 2) of A/K such that 42/K
60, there
= ¢2' an unramified
(A[,~I,A1/K 1) of A/K such that 41 ~ 42 and 41/K
= ¢1"
places
pseudo-place pseudo-place
64
With notations
as above; pseudo-places
phic if and only if there exists ~(K 1) = K 2 and ~2 : ~ 0 ~1"
~1 and ~2 are said to be isomor-
an isomorphism ~ : A 1 ~ A 2 such that
This definition
implies that
if ~1 m ~2 then
~1 ~ ~2 and 92 ~ ~1' hence both @1 and 92 are defined on the same subring of
A
and their kernels
If 91 ~ }2 are unramified then ~1 m ~2"
Moreover,
coincide.
pseudo-places
such that 91/K m ~2/K
if ~1 ~ ~2 are unramified
(as places)
k-pseudo-places
A/K such that dim ~l/k:dim ~2/k is finite then ~1 is isomorphic
Let
(A',@,A1/K 1) be an unramified
the place @ = @/K has finite rank specialization
chain
m.
of A/K
: 9m_1 ~...~ @1 ~ ~
ramified pseudo-place being isomorphic
chain of places
a chain
further properties
K-algebra, A/K.
(~)
There exists
A,
and let
chains of this type do ex-
justifies
our definition
of
~A/J
~A1/J 1
we mention that pseudo-pla-
ring homomorphisms.
Let
A
be a
(A',@,A1/K 1) be a pseudo-place
~ of A/J such that the following
is commutative,
w
and ~ not
if and only if the associated
of pseudo-places,
wi A1
j with @i/K = ¢i"
pseudo-place.
a pseudo-place
gram of pseudo-places
A
one; this
under surjective
J and ideal of
Maximal
is maximal
(~) is a maximal
ces are "reduceable"
(~)
imply
unramified
~ of A/K such that @m-i ~ ~ ~ 9m-i-1'
the rank of an unramified Amongst
made above,
if it is impossible to insert an un-
to @m-i or @m-i-l"
ist and moreover,
(~) ~ associating
chain of non-isomorphic
Such a chain is said to be maximal
m , i.e.
For such places there exists a
The observations
that we can find a specialization
to ~2"
of finite rank
: ¢m-1 ~ ¢m-2 ~'''~ ¢1 ~ ¢
a place of rank 1 to the given ¢.
pseudo-places
pseudo-place
of
i.e. ~
: ~1~ on A'
:
of
dia-
65
where J1 = ~(J e A'). fied too,
(proofs
Moreover,
in [34]).
if ~,~ are pseudo-places of
K
if ~ is unramified
Note also that if A,B are K-algebras,
of A,B respectively,
then the tensorproduct (A' 8 B' OK
then ~ is unrami-
inducing
~ @ ~ is defined
, ~ ~ ~, A 1 ~ B 1) K1
and
the same place
to be the pseudo-place
of
A ® B. K
We now focus on pre-places.
PROPOSITION
61.
Let
(A',~,A1/K 1) be a pre-plaee
(A~,~,A2/K 2) be a pre-place pre-place
PROOF.
of A1/K 1 then
Since A 2 is a division
ring and because to verify
that xy • ~-I(A~)
while x ~ ~-I(A~),
1. Both
are in A'.
x
yield
and
~(y)
x
y e
or
x resp.
y is
PROPOSITION
Ker ~,
thus
that
in
62.
defines
a pseudo-
P2"
Suppose
: = ~(xy) e A~ and ~(x) ~ A 1
Ker a o~.
Let @1 and ~2 be pre-places
c A' implies
that
y
or
of A/K then equivalently
:
of ~1" ~ on the residue
division
algebra
of ~1 such
= ~2"
The implication
implies
that a pseudo-place such that ~ o ~ 1
sion ring,
~ o~
Ker ~ c Ker ~ o ~.
is a pre-plaee ~o~1
then
Then ~(x)~(y) y •
PROOF.
exists
~ 0 ~, A2/K 2) is a
the condition
is not in A', then xy • ~-I(A~)
1. ~2 is a specialization 2. There
(~-I(A~),
of A/K.
place of A/K it will be sufficient
2. If
of A/K and let
2 = 1 is obvious.
= ~2"
~ of the residue The residue
so we are left to verify
Let (A~,~I,A1/K1)
and
Conversely, division
algebra
the condition
Proposition algebra
60
of ~1
of ~2 is also a diviP2"
(A~,~2,A2/K 2) be the pre-plaees.
Since ~1
66
specializes Write
B = 91(A~).
presentatives x ~ A~. larly,
e B with x,y e A 1 and with ~ ~ B, then for rewe obtain
may be restated
that ~ • ~l(Ker
by ~(~)
= ~ o 9.
Obviously, ramified stands
of A/K,
Im ~ contains
that
for the unramified
then ~ restricts of pre-places restrictions
Obviously,
of ~ are injective.
It is clear that elements and only if their
an injective
This entails
that ~(~)
In general
Given a field
k
D/k exists,
of generic
statements
PRK(A)
denotes
~ is injective,
Reformulating
map on the isomorphism
Proposition
hold
of A/K, the set
also the 62 yields
:
if
in PSK(A) , (or PRK(A)) , classes
of pre-places.
only depending
on A~ and
in IV. 4. difficult
a prescribed D.
Im ~ contains
to construct
If a field
K
and a K-algebra
9 of D/K with residue
a prescribed
subset
are given
in Section
D
algebra
S c PSK(D ) then
for D/k.
constructions
pseudo-places
set S c pSK(A ).
such that a pseudo-place
D/K is said to be S-generic Examples
in general
it may be quite
for which
(Similar
of PSKI(A1) , (or PRKI(A1)) , are isomorphic
and a k-algebra
can be constructed
of 9-
if and only if 9 is un-
If 9 is a pre-place
is, up to isomorphism,
Im ~ contains
defined
= PRK(A ) n Im ~.
Ker ~, this will be studied
such that
of A/K.
images under ~ are isomorphic
so ~ induces
for the set of
of all specializations
to ~ : PRKI(A 1) ~ PRK(A) , where
of A/K.
PSK(A)
to a map UPKI(A 1) ~ UPK(A) , where UPK(A)
pseudo-places).
~(PRKI(A1))
Remark.
Write
pseudo-places
pseudo-places
and special
92).
a map ~ : PSKI(A 1) ~ PSK(A)
Im ~ consists
unramified
an then ~ restricts
for restricted
of A/K.
then 9 induces
Observe
Simi-
as follows.
(A',9,A1/K 1) be a pseudo-place
pseudo-places
: x y • A~ with x,y • A~ but
y • Ker 92 and y • ~l(Ker ~2 ) or ~ • Ker ~ follows.
• B with y ~ B implies
These results Let
If ~
x,y of x,y resp.
Thus, x y
D A~ D Ker 92 D Ker 91 and thus ~ 1 9 1 ( A 5) = A 2.
to 42 we get A~
V.
67
Because a surjective place
K-algebra m o r p h i s m
if its Ker is completely prime)
jeetive K-algebra morphisms.
is a pseudo-place
(and a pre-
the above statements
However,
hold for sur-
we want to get rid of the surjee-
tivity hypothesis.
PROPOSITION
63.
To a K-algebra morphism
mapping ~ : PRK(B) ~ PRK(A),
PROOF.
Let
defined by ~(~)
(B',~ ,B1/K 1) be a pre-place
pseudo-place
x y
P2 has to be checked.
6 A' with x ~ A'.
of B/K.
Put A'
is a division ring,
= f-l(B'),
a
C B 1.
thus only the
Suppose x,y e A are given,
Then f(x y) = f(x)f(y)
f(y) e Ker ~ or y e f-l(Ker ~).
a
= ~ o f.
of A/K may be defined by ~ o f : A' ~ ~(ff-~(B'))
Now, as a subring of B1, ~(ff-l(B')) eondition
f : A ~ B there corresponds
such that
e B', but f(x) ~ B' yields
In a similar way it may be shown that,
if x y • A' and y • A', then x • f-l(Ker ~), proving that ~ 0 f is a preplace of A. Let A ~ K be the category of K-algebras be generalized
to a category of algebras using pseudo-places
ces for the morphisms). PSK(A)
with K-algebra morphisms (this may
It is possible
to put topologies
or pre-pla-
on the sets
such that we get a functor PS K : A ~ K ~ To~.
An interesting interest
variation
on this theme is subject of IV. 4., additional
is added there by the existence
of a fitting
localization
tech-
nique.
~E~"
If
Conditions
f
64.
injective.
in terms of the alge-
B of f(A).
3. Pseudo-places
PROPOSITION
f is not necessarily
for ~ to be injeetive may be expressed
bra-extension
IV.
is not surjective,
Let
of Simple Alsebras. A
be a K-algebra,
that A 1 is a Kl-central
simple algebra.
let
(A',@I,A1/K 1) e UPK(A)
Then
A
is a K-central
be so
simple
68
algebra.
PROOF. Then
Suppose
that
I = J n A'
A
is an
ideal
are two p o s s i b i l i t i e s
1. ~(I)
= A 1.
2. ~(I)
= 0.
56.
Then
Hence
Now,
= 0
but
K-basis
E;
may
be m u l t i p l i e d that
PROPOSITION
be a p r o p e r
is an ideal
ideal
in A 1.
of
There
contains
a K-basis
where
E
x e I exists, = 0 entails
for
and thus
is a K - b a s i s
then
for
x = i=1 ~n aiei
a i e M K for all
= i=1 Nn aiei • J
A
J = A. A
as
with
ai e
i = 1,...,n,
ai • e O K and
with
is an ideal•
in T h e o r e m
This
means
OK, and
for
a~] = 1. that
by an e l e m e n t
in
K
to y i e l d
an e l e m e n t
in
I
it
the
65.
to an u n r a m i f i e d 5S),
and
equality
Let
A
since
pseudo-place
the r e s i d u e
[A : K]
of the
algebra
= [A 1 : K 1] y i e l d s
be a f i n i t e
dimensional
center
of Z(A) that
K-algebra,
Z(A)
is
Z(A)
in the = K.
let
e UPK(A).
1.
If A 1 is s e m i - s i m p l e
2.
If A 1 is a K l - c e n t r a l sion
tent
The
ideals
yields
that
to c h e c k [34].
then
A
is s e m i - s i m p l e .
division
algebra
then
A
is a K - c e n t r a l
divi-
algebra.
PROOF.
first
statement
of
to n i l p o t e n t
A
that
A.
0 =
~ restricts
of A1,
and @(I)
J
J = (0).
(Proposition
center
let
Z n ~ ( a ~ ) ~ ( e i) is c o n t r a d i c t i n g the c h o i c e of i=1 I = 0 follows. Since e v e r y n o n - z e r o e l e m e n t of J
the
follows
and
:
1 ~ j ~ n, a ] l x
'
Finally,
of A',
= 0K[E],
~(x)
j,
simple
a [ l x e A' n j b e c a u s e ]
~(ajlx)
A
I
Let A'
an index
of
left
If a n o n - z e r o
e l• e E.
is not
A
is a c o n s e q u e n c e
is a K - c e n t r a l zero-divisors
in
ideals simple A
of A 1. algebra
reduce
of the
"descent"
Secondly, and
of n i l p o -
Proposition
64
it is s t r a i g h t f o r w a r d
to z e r o - d i v i s o r s
in A1,
of.
69
Let L/K be a f i e l d e x t e n s i o n
with Galois
group
G.
A crossed
a l g e b r a A = (G, L/K, {Co,T})
is d e f i n e d
to be the a l g e b r a
where
satisfying
the r e l a t i o n s
Uo, o • G, are s y m b o l s
O,T e G.
The
set {Co, T) d e f i n e s
Crossed product
algebras
a crossed product pseudo-place
if
a 2-eocycle,
are K - c e n t r a l
algebra,
t h e n ~ • UPK(A)
• H2(G,L~).
If A = ( G , L / K , { C a , T } )
w i l l be c a l l e d
in
L
With these notations
ponents Br(K),
PROOF.
66.
and c o n v e n t i o n s
of the c r o s s e d BA(K1) , t h e n
In Br(K)
exists
e
products
= $(Co,T)
for all O,T • G.
pseudo-place.
If e,e I are the ex-
A , A 1 in the r e s p e c t i v e
is a m u l t i p l e
Brauer groups
of e 1.
we h a v e A e = 1, this m e a n s Ce
G~T
:
that we can find e l e m e n t s fof~f~
•
o,T
•
(~)
G
"
an ~ • K such that ~fo • OL = A' n L for e v e r y o • G and
~fy ~ M L for some y • G. we d e r i v e
with Galois-
:
Let 9 be a g a l o i s i a n
fo E L, for e v e r y o • G, such that, There
a sa!oisia 9
and L 1.
G 2 : A 1 = ( G , L 1 / K I , { C o , ~} w h e r e CO,T
PROPOSITION
is
:
G 1 : ~ [ L • UPK(L) , G = G a I ( L 1 / K 1) and 9 is c o m p a t i b l e action
o • G]
U o U T : Co,~ U o T '
[{Co,T}]
simple.
L[Uo,
product
from
Suppose
(~) a r e l a t i o n
~ • M K.
Then
for the p a r t i c u l a r
y • G
:
ce ~f 2 = ~ f y ( e f y ) Y Y,Y ] Taking
images under @ yields
~(~fy)~(~fy)Y
Hence
e ~ M K and - 1
over,
if o is any e l e m e n t
a contradiction
= ~(~)~(C~,y)~(afy2)
e 0 K follows. of
:
G
Thus
foe
: 0.
0 L for all o e G.
t h e n t h e r e m a y be f o u n d
T O e G such that o T O = y , y the f i x e d e l e m e n t
as before.
More-
an e l e m e n t This y i e l d s
:
70
Ce
or
~fy
= f f~ f-l,
Ce o,Y
= ~f fo o Y
o
Since the left hand side is not in Ker 4, also ~ f
~ Ker 4 and
fT
if a runs through
~ Ker 4.
Hence fT
o
~ M L.
Now, y being fixed,
G
o
then t ° runs through ly, [eo,T = ~(C$,T) ~e
: ~(fa)4(f
G,
• 0 L - M L for every o • G.
= 4(fo)4(f~)~(f~$)
)o~(fo
being the smallest
hence f
Consequent-
and thus
)-1
implying that {~e } ~ 1. The exponent e 1 e1 integer such that {[ ,m} is equivalent to the trivial
factor set 1, it follows that e I divides e. A p s e u d o - p l a c e ~ of a K-algebra
A
such that ~/K is a place of
K
asso-
ciated to a N o e t h e r i a n valuation ring is said to be a N o e t h e r i a n pseudoplace.
There is an obvious link between N o e t h e r i a n p s e u d o - p l a c e s
central simple algebras and orders
in these algebras.
When
K
on K-
is more-
over complete with respect to the valuation c o r r e s p o n d i n g to O K and supposing that ~ is an u n r a m i f i e d pseudo-place
such that the defining ring
A' is a maximal order in A then A 1 is simple if
A
is simple
(a skew-field).
(a skew-field)
N o e t h e r i a n pseudo-places
to the theory of separable algebras,
cf.
[2],
[3].
if and only
also relate
We mention that,
in
case ~ is an u n r a m i f i e d N o e t h e r i a n p s e u d o - p l a c e of AIK such that A1/K 1 is a separable algebra, 1. A is a K - c e n t r a l
then
simple
:
algebra.
2. The residue algebras of u n r a m i f i e d p s e u d o - p l a c e s defined on m a x i m a l orders of
A
are separable Kl-algebras.
3. The subring A' of
A
where 4 is defined is itself a maximal order of
A.
Similar assertions are still true for n o n - N o e t h e r i a n u n r a m i f i e d pseudoplaces.
See Section VI on Azumaya algebras.
71
IV. 4. Primes in Algebras
Let
A
over Fields.
be a K - a l g e b r a and let ~ be a pre-place of A/K.
P = Ker ~ is a prime of A/K and ~
We say that
is said to r e p r e s e n t the prime
P.
Note that a prime may be r e p r e s e n t e d by several d i f f e r e n t pre-plaees. If ~ represents the prime is the place of
K
P,
then
induced by ~.
if for every y • A, y P c P
P
is called a ~-prime of A/K, where
A %-prime is said to be symmetric
is e q u i v a l e n t to P y
considered up to i s o m o r p h i s m unless otherwise the f o l l o w i n g notations
- PrimK(A) PrimK(A)
c P.
Places of
specified.
K
are
We agree to
:
= {P, P is a ~-prime of A/K}
= {P, P is a prime of A/K}.
Since the place ~ c o r r e s p o n d i n g to P we obtain that
is up to isomorphism determined by
II ~-PrimK(A), with p being the set of ~ep i s o m o r p h i s m classes of places of K, i.e., the set of v a l u a t i o n rings of
K.
: PrimK(A)
P
=
We introduce a Zariski topology in PrimK(A)
in %-PrimK(A)
is called the ~-topology.
A, and let D(F) subset of
A,
D(F 1 m F2)
PROPOSITION
PROOF.
= {P prime of A/K,
Let
F
p n F = ~}.
and the topology induced be any finite subset of The sets D(F),
F
finite
form a basis of the topology g e n e r a t e d by them because
= D(F1) m D(F2).
67.
Prim K and %-Prim K are c o n t r a v a r i a n t functors A I g K ~ Top.
Given a K-algebra m o r p h i s m f : A ~ B.
a pre-plaee of B/K r e p r e s e n t i n g a ~-prime p = f-l(Q) c f-l(B') bra f-I(B')/P,
Q
Suppose of B/K.
(B',~,B1/K 1) is Then,
defines a % - p s e u d o - p l a c e of A/K with residue alge-
which is a division ring because it is a subring of B 1.
If x,y e A are such that x y
• f-l(B'),
then f(x)f(y) • B', hence
x ~ f-l(B') would imply f(y) • Q or y • P; similarly, y ~ f-l(B') yields x • P and thus ~ 0 f defines a %-pre-place of A/K.
Consider
72
: PrimK(B) ~ PrimK(A) , given by ~(Q)
: f-l(Q).
This map ~ is continu-
ous, because
f-I(D(F))
= {Q • PrimK(B),f(Q)
• D(F)}
:
= {Q • PrimK(B) ' f-l(Q) n F = @} = {Q • PrimK(B), R e s t r i c t i n g f to %-PrimK(A)
Q n f(F)
= 4}
= D(f(F)).
yields maps f~ which are continuous by defi-
nition of the %-topology.
PROPOSITION
68.
Every prime
P
of A/K contains a prime ideal p0 of
p0 is the maximal element in the set of ideals of
PROOF.
Put p0 = {x e A , A x A
nor A b A
are not in a' A b '
Let
c AaAbA
Then
P
P
A.
implies a' • P.
but then a'A'b' c P contradicts
Obviously,
P.
Suppose that
and a b' • A b A
which
Since
Thus a' ~ A' implies
Therefore both a' and b' are in A'
the fact that A ' - P is m u l t i p l i e a t i v e l y
Observe that p0 is in general not completely prime.
A/K which is also an ideal of
in
contains neither
be r e p r e s e n t e d by (A',~,A1/K1).
c p • A', we have a'b' • A'.
b' • P while b' ~ A'
closed.
p0.
and thus we may find an a' • A a A
P.
contained
c p}, p0 is an ideal of
there exist a,b ~ p0 such that a A b c A aA
A
A;
A
for an arbitrary prime
A prime of
is a completely prime ideal of P
A.
of A/K, the set A - P is m u l t i p l i c a -
tively closed. The following c h a r a c t e r i z a t i o n of symmetric primes generalizes of I. Connell
[6].
P R O P O S I T I O N 69. mal ideal M K.
a result
Let % be a place of Then
P
K
with v a l u a t i o n ring O K and maxi-
is a proper symmetric ~-prime of A / K if and only
if :
1. P is an 0K-mOdule , such that P N K : M K2. P is m u l t i p l i c a t i v e l y closed and symmetric,
i.e., Py c P is
73
equivalent
to yP C P, for every y • A.
3. The complement
PROOF. then
of
P
is m u l t i p l i c a t i v e l y
The only if part is obvious.
closed.
Conversely
let A'
= {xeA,xPCP}
:
a) A' n K = 0K; since ~ • K and ~P c p entail - 1 b) A' is an 0K-algebra
containing
P
~ p, hence - 1
~ MK"
as an ideal.
c) Let x,y 6 A be such that x y • A' but y ~ A'. Since
P
is symmetric,
yP ~ P and Py ~ P.
the fact that x y P c P implies
Take p • P so that y p
x 6 p because A - P is m u l t i p l i c a t i v e l y x ~ A' yields y • P. is multiplicatively and thus
P
Finally, closed,
~ P.
that
P
x y • A' with
and since A - P
is completely
prime
in A'
is a ~-prime of A/K. following
PROPOSITION
If (A',~,A1/K 1) is a pre-place
70.
: PRKI(A 1) ~ PRK(A) is the place of
Proposition
62 we derive
K defined as the specialization
by ~, Prim
CA.) ~ PrimK(A ) K1 ' 1 topologies.
of the prime P = Ker @.
All mappings
where
of ~IK obtained by comagain denoted
are continuous
in the corres-
is the set of specializations
If P' e Im ~ then we write P ~ P'
A necessary
and sufficient
that there exist representing
pre-places
rings of definition A~, A~ resp., is an immediate
of A/K then
We also obtain a map,
The image Im ~ of ~ : PrimKl(A 1) ~ PrimK(A)
71.
:
induces maps %1-PrimK1(A 1) ~ ~-PrimK(A),
position with the place %1 of K 1.
PROPOSITION
• P implies
Similarly,
since A' is an 0K-algebra
it follows
c P while
Then x y p
closed.
From the remarks
ponding
Px y
consequence
condition
for P1 ~ P2 is
~1' ~2 for PI' P2 resp.
such that A~ c A[ and P2 D PI"
of Proposition
sented by ~ is called a minimal prime
62.
A prime
P
with This
of A/K repre-
if P e Im ~ with ~ e PRK(A)
74
implies ~ ~ 9. ExamD~e. prime.
A minimal prime ideal which is completely More examples may be constructed
tion rings,
see Proposition
over discrete
K,
such that if x y
72.
PROOF.
(A} ~,Af)
then
Let
If P • PrimK(Af)
'
represent
Let Af be a subalgebra of
• Af with x,y • A and x ~ Af, then
y • Ker f while y ~ Af yields x • Ker f, then
PROPOSITION
rank one valua-
77.
Consider a K-algebra m o r p h i s m g : A ~ B. A, containing
prime is a minimal
:
and P D Ker f then P e PrimK(A).
P.
If x,y E A and x y e A' but y ~ A} f
:
a) y ~ Af; then x y
• Af implies that x • Ker f c p.
b) y • Af; then x y • Af' with x ~ Af yields y • Ker f, contradiction Hence
let x • Af.
x • P since
P
But x,y • Af and x y • A} with y ~ A~ entails
is a prime of Af/K.
Similarly,
x y • A~ with x ~ AfT
yields y • P, hence P • PrimK(A). In case
f
is surjective we have that primes
Ker f yield primes : PrimK(B)
f(P) of B/K and it is easily verified that
~ PrimK(A)
defines
a homeomorphism
V(Ker f) = {p e PrimK(A),p m Ker f} which
PROPOSITION
73.
Let %1 ~ ~2 be places
rings 01 ~ 02 resp. (A~,gi,Ai/K i)
of PrimK(B)
K with associated
Let Pi be a ~i-prime
of A/K represented
i = 1,2, and suppose P2 ~ PI" of PI"
The situation may be summarized
onto
is Zariski-closed.
of
of A/K and a specialization
PROOF.
P of A/K which contain
in :
valuation by
Then P2 n A~ is a ~2-prime
75
PI C P2 n A: c A: n A l : A l c A. It is s u f f i c i e n t kernel
,21A~ defines
to show that
E A 1,
a) If y ~ A~ then x y
• A: yields
b) If y ~ A: then x y
• A 2' entails
tails
way it f o l l o w s
T
y • A 1 n P2" (A: n AS)
Since
- (A: n p2)
occur
then x y
Proposition
x • A 1' n P2"
• A 1' n A~ with x ~ A 1, n A2' implies
is m u l t i p l i c a t i v e l y
closed
that
and for this
is m u l t i p l i c a t i v e l y
it is
closed.
e A 1' n P2 and y ~ A 1, n P2"
e P2 n A~ forces
closed.
that
Hence
• A 1' en-
Again
two pos-
:
The a s s u m p t i o n
y e P1 C P, hence b) y ~ A~;
then
If x ~ A 1' then x y
x • P2"
A - (A~ n p2)
a) y ~ P2; then x y catively
A 2'
x E P1 c P2 n A:.
that x y
Let x,y • A be such that x y sibilities
n
(A~ n p2) n K = M 2 we are left to prove
to show that
The fact
A 2' and y ~ A~
n
y • P1 c A 2' c o n t r a d i c t i o n .
In a s i m i l a r
x e P2 b e c a u s e x ~ A~ with x y
A - P2 is m u l t i p l i e A~ w o u l d
lead to
x • A~ n P2" e A~ entails
t x • P1 c P2 n A1.
P2 n A~ is s p e c i a l i z a t i o n
of P1 is an easy c o r o l l a r y
of
72.
In the above full
on A~ n A~ with
P2 n A~.
if x,y • A are such that x y
enough
a pre-place
proof,
strength
the fact that
but m e r e l y
P1 is a c t u a l l y
a prime
that A~ has the p r o p e r t i e s
of Af
is not used
in
in P r o p o s i t i o n
71. With n o t a t i o n s
COROLLARY. prime
If
of Af. Taking
there
exists
as in P r o p o s i t i o n
P Note
is a ~ - p r i m e that
P
71 we have
of A/K,
the f o l l o w i n g
P n Ker f, then P n Af is a #-
is not n e c e s s a r i l y
a specialization
~ = ~1 = ~2' P = P1 = P2 in P r o p o s i t i o n a pre-place
of A / K r e p r e s e n t i n g
P
of P n Af.
73, we see that
such that
the ring where
76
the pre-place particular
is defined,
is minimal with respect to inclusion.
representative
for
P
will be referred
and if this pre-place
is special,
said to be absolutely
special,
PROPOSITION
restricted
P of A/K is special
2. A prime
P
P is absolutely
P
then
Let
representative
sentative for y e A{ yields
P.
P
P
is a restricted
P for
2.
A
However y y-1
Therefore A' and A
Then
and this pro-
(A',~,A1/K 1) be a restricted P.
In general A p c
A'.
Then certainly y e A' - P and thus y y-1 e A'
diction.
if y -1 were in
P
preSup-
implies
then also
e Ap with y ~ Ap entails y-1 e p, contra-
coincide. Assertion 3 follows from 4, P 65, 2, and the fact that a finite dimensional K-algebra which
is a division ring is a skew field
P
T (A 1,~1,A1/K1 ) be a
thus A~ c A[ follows
representing
e p.
then
and let
Let (A~,~2,A2/K 2) be any other repre-
To prove 4, let
1 = y y-1
PROPOSITION
(up to
P.
be restricted P.
prime of A/K if and on-
Then there is a unique
Moreover y -1 e A' - P because
Proposition
if and only if
If y e A~ - A~ then ~y e A~ - P with I e M K.
place of the skew field pose y e A' - Ap.
special.
if and only if every pre-
of A/K is unramified
~y e p, contradiction,
ves the assertion
y -1 ~ A' .
or unramified.
(A',9,A1/K 1) representing
1. is trivial.
restricted
is
is restricted.
is absolutely restricted.
isomorphism)
PROOF.
P
unramified.
is a skew field, P
then
if and only if it is absolutely
3. If [A : K] < =, then a prime
ly if
or unramified
of A/K is absolutely restricted
place representing
A
to by (Ap,~,A1/K 1)
74.
1. A prime
4. If
restricted
This
75.
(see Lemma 78 corollary).
Let ~ be a restricted
is symmetric
pre-place
and P = M K . (A'- P).
representing
a prime
P,
77
PROOF.
Suppose that y P • P, Py ~ P for some y • A.
and thus there
is a I • M K such that
we reach a contradiction.
ly • A' - P.
Let x • P.
Ix 6 A' - P, thus p0 = (0).
Moreover
There
Clearly y ~ A'
Since ly = y l • y
PC P
is a I e K such that
if y = ~x e A ' - P then I ~ O K fol-
lows, hence i-1 • MK and x = l-ly. RemaEk.
From Proposition
tersects
any subfield
58 follows
L/K of A/K in a valuation
that restricted primes generalize that PrimK(A)
PROPOSITION
that a restricted prime of A/K in-
PROOF.
This shows
the concept of a valuation ring and
76.
Let [A : K] < ~, and let P2 be an unramified by (A~,42,A2/K2).
~2-prime
Then for every place ~1 of
such that ¢1 ~ ¢2' there exists an unramified K
L.
is in a way related to the Riemann surface of a field.
of A/K represented
in
ring of
pre-place
K
41 inducing
¢1
such that P2 • Im ~1"
The remarks
unramified
following
pseudo-place
division ring.
Proposition
60 yield that there exists
(A~,41,A1/K 1) such that 41 ~ 42.
By Proposition
57, 41 is a pre-place
an
Hence A 1 is a
and since
Ker ~1 ~ P2 we have P2 6 Im ~.
COROLLARY.
(~)
To a specialization
chain of places of
:
1 ÷ ¢1 +'''÷ ~n = ~'
such that there exists an unramified ponds a chain (~)
K
(~)
of unramified
G-prime
P
primes of A/K
of A/K, there corres:
P = Pn n P n-1 n...n p1 n 0.
The chain
(~) is maximal
if and only if ( ~ )
is maximal.
Note that isomorphic unramified
primes are equal.
[A : K] < =, minimal unramified
primes
Noetherian
valuation
ring.
Thus,
are necessarily
In the absence
in case
defined over a
of the finiteness
condition
78
on
[A : K] we get,
PROPOSITION
77.
t i o n r i n g of
PROOF. i.e.,
Any absolutely
K
Let
P
restricted
a discrete
restricted
r a n k one v a l u a t i o n
A'I are s u b r i n g s
where
mal s u b r i n g
K
P
L E M M A 78.
Let
do-place
PROOF.
is r e s t r i c t e d
By T h e o r e m
into
~-prime
of A / K
pseudo-places
56, A' is a free
~(A')
of A / K d e f i n e d
( A ' , ~ , A I / K 1) is u n i q u e
COROLLARY.
0K-mOdule
~ dim0K(A')
Indeed
ideal of
A.
If
(A,1,A) A
If
there
is a i ~ O K
place
of
K
t h e n an
%-pseu-
Then ~ is the u n i q u e algebra
torsion
free
4-
is a p r i m e ring.
of d i m e n s i o n
[A : K] = n.
0K-algebras
: n, the sum r a n g i n g
on A' with p r i m e r e s i d u e
ring.
trans-
o v e r %Hence
as such.
If A / K is a f i n i t e
simple.
P 1 N K = M K.
( A ' , ~ , A 1 / K 1) be an u n r a m i f i e d
on A' such that the r e s i d u e
dimK1
O K is a m a x i -
is c a l l e d a site of A/K.
fact for f i n i t e d i m e n s i o n a l
: ~
Since
that
If ~ is a N o e t h e r i a n
[A : K] < ~ and let
defined
A well-known lates
P
of A / K such that A 1 is a p r i m e ring.
pseudo-place
that P1 is a n o n - z e r o
(ix) • M K A~ • PI'
Thus P : PI"
restricted
are defined.
F r o m i -1 • M K we d e r i v e
x = i -1
absolutely
Suppose
A 1' N K = O K and thus
x e p - P1 t h e n x e A 1' and since
aontradiction.
let P n K : O K be N o e t h e r i a n ,
0 # P1 C P c A' c A~, w h e r e A' and
and P1 resp.
we have that
ix e A' - P.
and
ring.
p r i m e of A / K such that P1 ~ P' i.e.
such t h a t
valua-
is m i n i m a l .
be a b s o l u t e l y
of
prime over a Noetherian
dimensional
is u n r a m i f i e d ,
is m o r e o v e r
prime
hence
a division
algebra
then
(0) is the u n i q u e
ring,
then
A
is
prime
it is a s k e w - f i e l d .
79
LEMMA
79.
If
is r e p r e s e n t e d
P
gebra
Let
then
PROOF.
P
P
by
that
restricted
let x e A' - A 1.
We h a v e
Pick
that
a relation
~x e A 1' - P1
lal(IX) n-1
Hence
relation
that
~x • A ' -
(Ix) n +...+
(Ix)n
thus
then
+'''+
since
Ix • A' - P
thus
a I • M K and
an_ 1 e M K and Let
The m a x i m a l fine
P
ideal
a function
PROPOSITION
al(IX)
+...+
so on.
Secondly,
A,
i.e.
ord
P
of ~.
First,
+ an = 0 with
Ix • A~
Ix • PI"
Since ai •
and
a i e OK .
Ima i • M K.
Therefore
suppose
A' C A 1
x • A 1' - A'
Ix • A' we h a v e
OK .
Pick a
Obviously
• PI'
e p
a I • P c P1 Finally
we a r r i v e
be a site
of a K - a l g e b r a
M K = P n K of : A ~ Z,
P
satisfies
:
Ix + an_ 1 • P C P1
Therefore, A,
A'
represented
O K is p r i n c i p a l ,
by o r d p ( X )
is a site
at
M K = (m)
= -n if and o n l y
of A / K t h e n
ord
= A 1' and by
P1
= P
(Ap,~,A1/K1).
say.
We de-
if m n x • A p -
is an o r d e r P
on
ring
i.e.,
We o b t a i n
contradiction.
contradiction.
If
an_lX
~-prime
ina n = 0,
Ix • P,
Ordp
80.
al-
we get
(ix) n-1
yielding
0K-integral
A.
of A/K.
Ix e p.
+ a 0 = 0 with
al(IX)
K-algebra
Further,
(~x) n +...+
but
+...+
75.
P, t h e n
al(IX)
a 0 E P1 n K = M K.
is an
restricted
x n +...+
ix • PI'
P c P1 by P r o p o s i t i o n
I • M K such
A'
+ "'" + Ina n • P1 b e c a u s e
(Ix) n • PI' y i e l d i n g
and then
that
O K is the v a l u a t i o n
(Ix) n + l a l ( I x ) n - 1
where
of an a r b i t r a r y
~-prime
P1 c A 1' is a n o t h e r
P1 n K = M K w h e r e
I e M K such
~-prime
( A ' , ~ , A 1 / K 1) such
is the u n i q u e
Suppose
A~ n K = OK,
be a r e s t r i c t e d
function
P.
80
1. o r d p ( x
+ y) ~ m i n ( o r d p ( x ) ,
ordp(y))
2. o r d p ( x y )
= ordp(x)
PROOF.
that P = {x • A, ordp(x)
Note
1. Put o r d p ( x
+ y)
+ Ordp(y).
= -n.
Then mn(x
at least one of the e l e m e n t s If -n -- ordp(X)
2. Let o r d p ( x y ) m
N
y m x • Ap
because
-- -n, _
P
being
The a s s u m p t i o n
d o w n to
P
ord(p,A,). to
let
a division
a prime.
that
P
being restricted.
the ring A' d e f i n i n g
section,
Thus o r d p ( x
+ (-N) or o r d p ( x y)
are r e l a t e d
is not
+ mny • A p - P
in
P,
the p r i m e
In case (maximal)
A
ordp(y)
thus only w h e n
A
If -n > ordp(x)
then
ordp(y)).
= -N. m
Then n-N
y • Ap
= N - n and
since
restricted
may be k e y e d
in that case the c h o i c e we o b t a i n o r d e r
dimensional
in skew fields. Sites
say.
+ ordp(y).
is a b s o l u t e l y
is a f i n i t e
orders
so -n ~> ordp(x)
Moreover
is not p r e s c r i b e d ,
[A : K] = n < ~.
ring,
= ordp(x)
However,
that
Put ordp(x)
Thus
~> 0).
yields
+ y) >~ m i n ( o r d p ( x ) ,
i.e., m n x y • A p - P .
Re_mark.
tions
mnx, m n y
= mnx
P and m n - N y ~ P b e c a u s e m N x • Ap.
m N x ~ P and
-n = ( N - n )
+ y)
= {x • A, ordp(X)
then mny • Ap y i e l d s - n ~ ordp(y).
m n y ~ A p or, - n > ordp(y).
n-N
> 0} and Ap
K-algebra,
of
func-
sites
F r o m n o w on in this
of A / K can only exist w h e n
is a skew f i e l d j b y
Lemma
A
is
78, C o r o l l a -
ry.
PROPOSITION A/K,
81.
t h e n A'
PROOF. tricted,
If ( A ' , ~ , A 1 / K 1) r e p r e s e n t s
is a v a l u a t i o n
r i n g of
a site
contradicts
~
COROLLARY.
If
to the v a l u a t i o n
Since a site
and ~x -1 • A' - P for some ~,~ • M K.
E MK, h e n c e
K
of a skew f i e l d
A.
S u p p o s e x,x -1 e A are such that x,x -1 ~ A'. ~x • A' - P
P
if x ~ A' then x
is c o m p l e t e ring
O K then,
with respect
-1
But ~
is res-
• A' - P
• A'.
to the v a l u a t i o n
if ( A ' , ~ , A 1 , K 1) r e p r e s e n t s
corresponding
a site of
81
A/K, A'
is an i n t e g r a l
PROPOSITION mified of
prime
P
Let
o r d e r of A/K.
( A ' , ~ , A 1 / K 1) be a r e p r e s e n t a t i v e
of A/K,
Since m i n i m a l
Noetherian Theorem
t h e n A'
is a m a x i m a l
A
valuation
56 y i e l d s
unramified r i n g of
that A'
tionary.
This y i e l d s
integral
order
of a m i n i m a l
and a m a x i m a l
associated tained
to O K .
there
unra-
subring
an i n t e g r a l
order,
for
M n K cannot
S. L o c a l i z a t i o n
P
be a p r i m e
(A',~,A1/K1).
valuation
(y~)-i
..., w h i c h
over
and
ring,
that
the p l a c e
every order
order
is an P
of
K
is con-
in A / K and A' c M .
~n Since y = i=1 Yiei ' w i t h
unless
By
Propo-
yj = y(y~)-i • M n K .
K c M, p r o v i n g A'
= M.
Sheaves.
is r e p r e s e n t e d
the f o l l o w i n g
kernel
is sta-
OK, thus A'
~ denotes
is in A', h e n c e
OK properly
of A / K w h i c h
symmetric
functor
by the p r e - p l a c e kernel
on M(A)
functors
:
correspoding
to the
corresponding
to the
set A ' - P.
2. a A _ p is the s y m m e t r i c
kernel
functor
on M(A)
set A - P .
is the s y m m e t r i c
plicative
that
at Primes,
1. aA, - p is the s y m m e t r i c
multiplicative
of A/K, w h e r e
to an a s c e n d i n g
L e m m a 79 e n t a i l s
so let M be a m a x i m a l
contain
Consider
multiplicative
A.
x
E = { e l , . . . , e n}
= y~ is in A' for some j, 1 ~ j < n.
81, y~ ~ P i m p l i e s
However,
3. OA,
relation
an e l e m e n t y • M - A ' .
Yi • K we get that y j l y
Let
A n y x e A' gives rise
: x O K c (x 0 K , X 2 0 K ) c
~-prime
over a
we have that A' n K = O K is N o e t h e r i a n .
Over a Noetherian
exists
defined
= 0 K [ e l , . . . , e n] for some K - b a s i s
in A'
restricted
in a m a x i m a l
Suppose
are n e c e s s a r i l y
0 K - a l g e b r a , and thus an o r d e r of
is the u n i q u e
sition
K,
primes
w i t h e l• e A' - P for all i.
c h a i n of 0 K - S U b m o d u l e s
IV.
and thus a m a x i m a l
A.
PROOF.
of
82.
0K-algebra
set A' - P.
kernel
functor
on M(A')
associated
to the m u l t i -
82
LEMMA
82.
metric
kernel
PROOF. being
C(OA_
p)
functor
p0 o b v i o u s l y disjoint
metric
this
OA, , w h e n
acting
83.
coincide
PROOF.
kernel
with
does
If ~ is special,
ideal
maximal = {p0}
p0 of
with
A
sym-
(as in I. 1.).
the p r o p e r t y
and b e c a u s e
a subcategory
of
o A _ p is sym-
of M(A').
not n e c e s s a r i l y
then
functor
If ~ r e p r e s e n t s
it is e a s i l y
on M(A),
a special
define checked
However
a kernel that
OA,
funcindu-
moreover,
prime
P
of A/K,
then
OA,
and OA, _ p
on M(A).
Take
M • M(A).
c O A , ( M ).
then
at
look
# 0 in
K
If, x •
s Ax. such
By d e f i n i t i o n
Suppose
that
~ y
y = 0, c o n t r a d i c t i o n .
PROPOSITION
84.
ring
and only
The
ideal,
C(o A _ p)
M(A)
on A - m o d u l e s
°A' - p(M)
a prime
to the p r i m e
o A _ p0 is the
o A _ p = O A - p0.
identify
an i d e m p o t e n t
LEMMA
associated
° A - P : O A - p0 w h e r e
is the u n i q u e
implies
tor on M(A).
hence
f r o m A - P, h e n c e
Via A 'C--~ A we
ces
= {p0},
proof
there
x
is O A , - t o r s i o n
A
= s(~ a)x
is also
Let ~ r e p r e s e n t if
i.e., s A ' x
= 0 for
is a n o n - z e r o
= ~ s a x
Thus
if and o n l y
if A'
~A,(M),
of the A ' - m o d u l e
is a p r i m e
some
y • s Ax.
on
M,
s • A' - P We m a y
find
is in s A' x = 0, h e n c e
an e l e m e n t
a special
structure
prime ring;
of °A' - p(M).
P
of A/K.
T h e n A'
is
A is O A , - t o r s i o n
free
representing
prime
if
free.
is s t r a i g h t f o r w a r d .
PROPOSITION
85.
If ~ is a r e s t r i c t e d
P, t h e n
OA, , OA, - p and
PROOF.
Since
P
is r e s t r i c t e d
p0
oA_ p coincide
is special, =
(0), h e n c e
OA,
pre-place with
o ~ on M(A).
and OA, - p c o i n c i d e
C(o A _ p)
the
= {(0)}
on M(A).
or o A _ p
=
o~
Since on
M(A).
P
83
We are
left
to p r o v e
I D
(s) w i t h
but
then
R~mark.
maps
OA,
rings.
y • Qo(A') - A ' have
Ly
tion
if Qo(A)
then
Qo(A)
if
on
s y
• A'
with
Qo(A')
Qo(A')
: A'
open
with
let
set.
E
A
kernel
of
functor
of A / K
X E c XF c XG , then
is a skew
subset
associate
E
containing
injective
if s • A' - P we
y ~ A.
of
E.
A,
To the
In this
and
situa-
[A : K] >
X E the
system
way we o b t a i n
QF'
o F < OE,
to OF,
o E resp.
(see T h e o r e m
p(F,E)
respecti-
corres-
= n { A - P, P • XE} , w h i c h
In this
a commutative
free
A',
field.
where
QF(A )
The
also
we h a v e
L D (s) w i t h
results,
we o b t a i n
hence
Indeed,
~ QE(A)
respect
A,
for w h i c h
if X E C X F t h e n [ c E and thus
: QF(A)
in T(OA, - p).
If @ is r e s t r i c t e d
o E.
~ s • A' - P,
o ~- and G - t o r s i o n
be a f i n i t e
A
that
then
s • A' - P y i e l d s
To X E we
subset
are
I • T(o A _ p), i.e.,
f r o m A' - P c A - P.
L • T(o),
= A'.
and
is also
A n Qo(A, ) = A'.
But
Indeed,
functors
then
ideal
: A,
p(F,E)
and A'
some
= A then
I
is r e s t r i c t e d ,
is any p r i m e
Qo(A),
X = P r i m K A and
X.
P
for
a symmetric
phism
P
A
that
Let
I • K such
directly
then
c A'.
basic
If
Moreover
is a m u l t i p l i c a t i v e ciate
be f o u n d
it f o l l o w s
by o.
A 'c--~ Q o ( A ' ) ~ - ~
ponding
may
OA, - p ~ o A _ p f o l l o w s
In g e n e r a l ,
Put
o A _ p : OA, - p on M(A).
There
(l s) c I
Denote
are p r i m e vely.
s • A - P.
from
inequality
that
QE are
diagram
a presheaf
a ring the
10).
Z we asso-
homomor-
localization
Furthermore,
of ring
if
homomorphisms
• QE(A )
QG(A) where ring
o(F,E)
is the u n i q u e
homomorphism
cation
of the
QF(A)
presheaf
extension
~ QE(A); yields
of the
similar
a sheaf
canonical
for
Q on
X.
p(G,F),
A ~ QE(A) p(G,E).
to a
Sheafifi-
:
84
THEOREM 86.
PROOF.
The stalk of Q at P • X is exactly Q A - p(A).
The direct
the direct
limit of the eofinal
the functors tems
E,
limit of the system {Qu(A),
a E are symmetric
system {QE(A),
and associated
the direct limit localization
of the symmetric corresponds
kernel functors
is exactly A - P, becuase
V c U} equals
p(E,F),
X F c XE}.
Since
to the m u l t i p l i e a t i v e
functor corresponds
involved.
to the multiplicative
p(U,V),
sys-
to the sup
The latter kernel functor
system generated
if s e A - P then P e X
by U{E,
P e XE} , which
and s
s e {s} c u{~,
p • XE}-
Let f : A 1 ~ A 2 be a K-algebra morphism. s E f(~),
let x • ~ be so that f(x)
that f-l(p,) f(~) c f(E). jectivity,
Then ~-I(x E) = Xf(E).
= s, then for P' e Xf(E) we have
= ~(p,) • XE and x ~ f-l(p,). If
f
is surjective
(for example tion).
: hereditary
In the commutative
neralizing
Spee and Gam K.
PrimK(A ) n Spec A and, plicative
QE(A1) ~ Qf(E)~A2).
are ring homomorphisms,
triction maps, more conditions
s ~ P' and by oE-in-
which composes with
QE f : QE(A1) ~ QE(A2 ) to give a morphism that these morphisms
This entails
then the foregoing yields,
a map QE(A2) ~ Qf(E)(A2)
If
compatible
To get
with sheaf res-
have to be imposed on the K-algebra
left Noetherian
rings with Artin-Rees
A
condi-
case, Prim K is a functor Alg K ~ Sheaves, This follows
ge-
from the fact that
if E = {el,...,e n} c A and <e> being the multi-
set generated by e = e I ... en, then <e> C ~ c A - U{Spee A, e ~ P}.
Hence,
the right hand set Deing the saturation
< e > -1 A m QE(A).
of <e>,
The rest of the proof is verification
bility with sheaf restrictions,
it follows
that
of the compati-
along the lines of [6], Proposition
24.
8S
Special
references
M. AUSLANDER, M. DEURING
for Section
0. GOLDMAN
[2],
[7], I.N. HERSTEIN
IV.
G. AZUMAYA
[3],
I.G. CONNELL
[14], F. VAN OYSTAEYEN
[34],
[6],
[35],
[37].
V. APPLICATION
V. 1. Generic
Central
: THE SYMMETRIC
Simple Algebras.
For finite abelian groups the category
PART OF THE BRAUER GROUP
G
we aim to construct
of fields with the morphisms
G-crossed product
"functors"
being the surjective
skew fields with galoisian pseudo-places
D(G) from places,
to
for the mor-
phisms. The skew fields ~K(G) thus obtained have a generic property on page 66.
Let ~ : ~K(G) ~ ~k(G) be the galoisian pseudo-place
ponding to a place ~ : K ~ k, then Im ~ contains lizations Dk(G)
as defined
of ~ obtained by composition
the set S of all specia-
with galoisian pseudo-places
such that every crossed product
(G,I/k,{Ca,T})
that DK(G) has to be constructed
that every crossed product
(G,I/k,{C
under a galoisian pseudo-place. the above described
LEMMA 87. Let
k
Let
G
H~ y m(G 'K~)-generic
be a field the characteristic
a purely transcendental
Then there exist fixed elements cocycle
i = 1,...,n,
f
of
G
and f(~,T)
K(tl,...,tn )~ generated
PROOF.
Ta-
in such a way
"generic"
stands
for
property.
n
and exponent
of which does not divide
Let
extension of
of S.
algebra of DK(G)
be a finite abelian group of order
taining the m-th roots of unity.
metric
,T}) is residue
In this section,
of
defined by a symme-
tric factor set [C~, T) is residue algebra under some element king ~ = IdK, it follows
corres-
G k
act trivially
and con-
in K = k(tl,...,tn) ,
of transcendence
(~l,T1),...,(~n,Tn)
n
m.
degree
n.
in G x G and a sym-
in K(tl,...,tn )~ such that f(~i,Ti) = ti, is in the free m u l t i p l i c a t i v e
subgroup
S of
by {tl,...,tn].
The lemma appears
in a slightly different
form
(and context)
in
[ls]. Since this provides
a universal
terms of the pre-seleeted
formula for the expression
f(~i,Ti) , it follows
of f(~,T)
in
that one easily generalizes
87
the p r e v i o u s {Co,T'
lemma to the case w h e r e
o,T • G} of the c o c y c l e
k
is arbitrary.
relations
Any
solution
:
Xo,T XOT,p -- XO.,Tp XT, p (~)
Xp,T
in k ~, yields hc : S
T,p
a map t i ~ Coi,T i w h i c h Co, T • he(S)
k ~ with
extends
to a h o m o m o r p h i s m
for all o,T • G "
From h c we derive
h~e : H2(G'S)
~ H2(G'k~)
with h~(f) : C, C : [{C~,T}],
xo, T denotes
a solution
of
one-to-one a method
correspondence
for f i n d i n g
all
(~) with xai,T i : t i. with
set-mappings
symmetric
factor
C e H2(G,I ~) where
i/k is any g a l o i s i a n
as follows.
specialization
cle d e f i n e d
Every
ameliorated
in the f o l l o w i n g
PROPOSITION
88.
C i being
cyclic
S be the free fixed
Let
abelian
such that
h d : S ~ k ~ exists,
PROOF.
{Co, r } d e f i n i n i n g extension
with
Gal(i/k) m G ,
a symmetric
every f a c t o r
in this way.
group
i = 1,...,r. generated
such that,
Let {Co, T } r e p r e s e n t
set r e p r e s e n -
The m e t h o d
of order Let
k
n,
R e=l
]
C
may be
G m C I × . . . × C r,
be any field
by {tl,...,tr}.
and let
Then there
under
d
the d e r i v e d
is a
element
f
homomorphism
is m a p p e d
onto
and let Cj be g e n e r a t e d
d.
by oj.
n°
=
coey-
for every ~ • H 2 ( G , k ~) a group h o m o m o r p h i s m
~ h 2 ( G , k ~) the fixed
a e"
]
j
aI dcr vI and de, T = d I . . r. if . a. = a I
v .ar r
dj
a class
way.
ni,
group
sets
vice-versa,
be an a b e l i a n
of o r d e r
f e H2(G,S)
h d : H2(G,S)
G
{tl,...,t n} ~ k ~, we o b t a i n
field
C e H 2 s y m ( G , l ~) is o b t a i n e d
where
Since H o m ( S , k ~) is in
t I• ~ ~.l • k ~ ' yields
by Coi,~ i : ~i' w h i l e
ting a class
f : [{xo,T}]
oj,
v i + w i > n i and e i = O otherwise.
=
'
1,...,r
~
:
wI .oWr °1 ""
The f a c t o r
sets
'
~i
=
1
if
{Co, T } and
Put
88
{da, T} are equivalent.
E1 tEr = tl "'" r with o,T,s i as before,
Define fo,T
then the map tj ~ dj, j = 1,...,r,
extends
to a h o m o m o r p h i s m
h d : S ~ k ~ and h d~ has the property h (f) = d.
Let
G
be a finite abelian group;
sion with Gal(i/k) unramified
m G.
place of
bally G-invariant,
I,
i/k an arbitrary
galoisian
Suppose that 01 is the valuation then, since 01 and its maximal
we get an injective h o m o m o r p h i s m
exten-
ring of an
ideal M I are glo-
:
H2(G,01) ~ H2(G,l~). The inclusion k* c I ~ yields and a commutative
a morhphism
the existence
T
of such "generic"
into the existence
cocycles
of generic
So we change from the level of cohomology
Let Gal(i/k) bles
{tl,...,tn}
by
= t to
i, G
of
l(t)[Ua,
G
t.
n
and add varia-
The equation
in the m u l t i p l i c a t i v e
t , such that x i,~ i = ti, the pairs
a crossed product Vl/k(G)
algebras.
to the level of simple algebras.
in k(t) ~ may be solved
This solution,
in Proposition
crossed product
acting trivially on
selected as in Lemma 87.
ing the algebra
as obtained
: G be a finite abelian group of order
with values ranging T generated
~ H2(G'I~)'
~ H2(G,I ~)
T 88 translates
: H2(G'k~)
diagram
H2(G,k ~)
Now,
~i' ~i
(~i,Ti)
aigain written
with l(t).
group
being pre-
{xa, T} defines
It may be looked upon as be-
a e G] with UaU T = xa, T UaT , UaU T = U T U ~ and
U a I : 1 ~ U a for all I e l(t).
For every crossed product
A = (G, i/k,
{Co,T})
with symmetric
pseudo-place
of DI/k with residue algebra
(~)
{Ca,T} , there exists A.
a galoisian
89
This is an easy consequence
of the fact that we may write
A = l[ua, o e G], the {ua, o e G} satisfying lations
for {U , a e G}, thus,
specializing
relations
similar to the re-
x ,T to Co, ~ and Us to u a we
get what we want. To an abelian L = k({Xa,
we associate
the field
a e G}) where X = {Xo, o 6 G} is a set of variables
such that
TX a = XTO ,
group
G
of order
a,T e G, represents
xed field for action of galoisian
G
field extension
due field of
in
n
and a field
the action of
i/k with Gal(i/k)
every crossed product A = (G, l/k,
{Ca,T))
i/k such that Gal(i/k)
place of
then a galoisian pseudo-place
Nk(G)
A crossed product A = (G, i/k,
L.
This easily yields
that
{Ca, T } and ar-
Now,
$(K,k),
K by
k
{Ca,T})
if $ : K ~ k is a ~(K,k) : NK(G) ~ N k ( G ) ,
in the construction is residue
of
algebra of
~k and ~K respec-
We have then that
~(K,k)
: PSkG(Nk(G))
maps ~k onto ~K' moreover, with as residue obtained
every galoisian pseudo-place
For more properties
88 allows us to construct
using less variables G = Gal(i/k)
clic decomposition {t~,...,t~}
~ PSKG(DK(G)) ~K of ~K(G)
algebra some crossed product A : (G, i/k,
in this way.
Proposition
Again,
Gal(L/k G) m G and every
and also of DK(G) under galoisian pseudo-places
tively.
Denote the fi-
with symmetric
stands for DL/kG(t)(G)).
may easily be derived from ~ (replacing
X.
m G, is residue algebra of
(Nk(G)
NK(G)).
on
m G may be found as the resi-
place of
Nk(G)/kG(t) K,
G
L by k G. Clearly,
L under an unramified
bitrary extension
k
= t' to
of NK(G)
cf.
another generic
{Co,T})
[35],
is
[17].
crossed product,
in the process. is abelian of order
of i
G.
n.
Let G = CI×...×C r be a cy-
Add
r
K-algebraic
independent
and let
G
act trivially
variables
on t'.
n.
Consider Sl/k(G)
= I(t')[V1,...,V r] with V. i = t! 1
C.. l
1 )
n. being the order of i
90
e~
The elements l(t')
i~l~r Vil,
0 ~ e I. ~ n.l are l(t')-independent
as it should~ Vi~= ~ i V i f ° r
of C i.
It is clear.that
tor set is obtained
all ~ • l(t'),
SI/k(G)
from t ~
is a crossed
" " " '
for SL/kG(G).
to be modified sition Di/~(G)
65 it follows
PROPOSITION
89.
galoisian
factor
in case L = k(~Xo,
of the generic
sets are chosen
a galoisian
algebra,
cf.
of SI/k(G)
~ • G~)
algebras proofs have so that Propo-
pseudo-place
of
[35], hence from Proposition
if SI/k(G)
is a skew field.
in Br(k(t'))
L = l(t,u).
pseudo-place
pseudo-place,
equals
the expo-
property
(cf.
[35])
divides
denoting
Since SL/K(G)
from k(u) ~ k(t')
crossed
notations
The crossed
introduced
product
skew field with center k(t).
a
a gaderive
and that
the exponent
~ k(t') while k(u,t)/k(u)
V. 2. Two theorems
on generic
e(SL/K(G))
~ SI/k(G)
= e(SL/K(G)).
90.
yields
DI/k(G)/K)" Since there is (DI/k(G)' ~t' Sl/k(G)/k(t')) we
e(Dl/k(G)) , e
Brauer groups.
We maintain
of SL/K(G)
(SL/K(G) , ~u'
66 that e(Dl/k(G))
divides
derives
The generic
dent, we have that e(Sl/k(G))
THEOREM
in
in Br(k(t)).
where
e(Sl/k(G))
morphism
was expressed
by S everywhere,
is a skew field
The exponent
from Proposition
ponding
as residue
its fac-
Put K = k(tl,...,t n) and let K(Ul,...,u r) = K(u) be the center
of SL/K(G)
loisian
Also,
The properties
There exists
that DI/k(G)
nent of DI/k(G)
PROOF.
88.
in that well-fitting
with SI/k(G)
such that
~,Y
fold if D is replaced
88 may be applied.
product
r
terms of ~dl,...,d r) in Proposition
Dk(G) , stated before
o.l is a generator
t' in th~ same way d
1'
we write Sk(G)
where
and V.l acts on
in the corres-
k(u,t),
this iso-
is purely
transeen-
products.
in V. 1..
algebra
SI/k(G)
= l(t)[Vo,o
e G] is a
91
PROOF• ring,
The proof
is b a s e d
o an a u t o m o r p h i s m
on the following.
of
S.
ring
S[T a] w i t h m u l t i p l i c a t i o n
that
S[T o] is a left
(and right)
f is i r r e d u c i b l e nerated
rule
(and right)
algorithm
S[T o] is an ideal
Consider
exists.
then
extensions
subfield
of
Gal(li/k)
i
: k
c
i
r-1
left fixed by H i .
S i ,i = 1,...,r,
S O = i c S 1 c...C is such that,
domain left
(f) is a d i v i s i o n
c.
with
if S i is a d i v i s i o n
ideal
a left
(f) of
if and only
C i cyclic
of order
We obtain
" .c 11 c I = i0, where
if
n i ge-
a chain
i.i is the
Gal(i/l i) = Hi, We are going
center
Sr_ 1 c S r = SI/k(G),
It is known
because
ring
i : 1,...,r..
Hence
m C i + l × . . . X C r , i = 0,...,r.
sion a l g e b r a s
ideal
Let G = CI×...×Cr,
ir
=
polynomial
= x°To, all x • S.
If the p r i n c i p a l
by oi, and put H i = Cl×...xCi,
of field
S be any d i v i s i o n
the n o n c o m m u t a t i v e
principal
S[T o] mod
in S[To].
ToX
Let
to c o n s t r u c t
l i ( t l , . . . , t i) such that
and the c o n s t r u c t i o n
algebra
divi-
then
S i + 1 is too,
of the
Si
i ~ r - 1.
Put t = { t l , . . . , t r } ; add t I • t to k and look at l(tl)[Tol] with aco1 tion defined by ~ Tol=TOl ~ fOrn all ~ • l(tl). Hence elements of k(t 1) commute
with To1 and since
left
ideal
thus
S 1 = l(tl)[V
1 ] where
starting
i < r.
The a l g e b r a
center
n1 by To1 - t I is an ideal.
P1 g e n e r a t e d
process
from
$1, and so on, we obtain
S'i-1 = Si-1
the skew p o l y n o m i a l
of l(tl)[Tol]
Put
V 1 is the P l - r e s i d u e
S i is then c o n s t r u c t e d
of Si_ 1 we get
We c o n s t r u c t
To1± - t I is in the center
S 1 = I(tl)[Tol]/P1,
of To1.
Repeating
the
I = S 0 , S 1 , . . . , S i _ 1,
as follows.
Extending
@ li-l(tl li_l(t I .... ,ti_ 1) ring
the
the
"'''ti)-
SI_I[Tol ] such that O.
ToiV ~
= V j
To. for i ~ j ~ i-1, oj i
and T
~ = ~ iT oi
for all °i n.
I • l(tl,.. • ,t i)
•
The
left
ideal
Pi of S'i_l[To
] generated •
by T O .1
i
is an ideal;
we put
S i : S~-I[To.]/Pi"
t.1
i
Thus
i
S.I = l(tl,. . ,ti)[Vol, . . . . 1 ~ i < r, is a d i v i s i o n
. ,Voi] alsebra
Obviously then
S r m SI/k(G).
S i is a d i v i s i o n
Now,
al~ebra.
if S i _ l ± Since
92
Si_ 1 is li_l(tl,...,ti_l)-oentral extension
it contains
l(tl,...,ti_ 1) of li_l(tl,...,ti_l) , it follows
a crossed product, where
and because
the galoisian that Si_ 1 is
Si_ 1 = (Hi_l, l(tl,...,ti_l)/li_l(tl,...,ti_l),
{Co, T } is the factor
set defined
{Co;T})
by the l(tl,...,ti_l)-basis
e.
for Si-l' hence { ~ V o Z , 0 ~ e i < n.} 1
{C O,T } is symmetric
and we have
1 n.
V° 3
= tj, j <
i - 1 while VopVoq
ferred to by {t}i_ 1. duct
(Hi_l,
= 1.V o o p q
This factor
Then S~_ 1 is a division
l(tl,...,ti)/li_l(tl,...,ti)
algebra
set will be re-
and a crossed
pro-
, {t}i_l} , if i = 1 then S~ is n.
that f : To.m _ t.z is irreducible 1 in S' [To.] because then S. is a division algebra. Moreover, since i-1 l i V . commutes with li(tl,...,ti) , Gal(i/l i) = Hi, we have that S i contains
simply
l(tl).
It is sufficient
to prove
i
l(tl,...,ti)/li(tl,...,t crossed
product
i) as a maximal
commutative
(Hi, l(tl,...,ti)/li(tl,...,ti)
g,h • S~._I[T .]' both non-trivial
subfield,
, {t}i).
and such that f = gh.
so S i is a
Assume The coefficients
i
of g,h are rational division Choose
algebra
functions
of t i with coefficients
ranging
over the
Si_ 1.
a • S~ such that i-1
af
=
:
in.
[ m aj(ti)TJ j=O Oi
where aj(ti) , bs(t i) e Si_l[ti]
jim.-1 [ m
s:O
b (ti)T s s i.
}
'
(this can be done with a e li_l(tl,...,ti)).
n.
Modulo
(ToZ-ti) this yields 1
(~)
0
=
[nil j=O
}I n l
aj(ti)V]~i
s:O[i
}
bs(ti)V$i
"
n.
Replacing
t i in the coefficients
by Va.m yields m
of (~) is the product g ( V ) h ' ( V o . ) 1
Si_l[Vo.]. 1
that the right-hand
of n o n - t r i v i a l polynomials in
1
If we show that Si_l[Vo.] i
~ Si_I[To.] i
then
(~) is a
side
93 contradiction.
Therefore
we show that the ideal
(f) n Si_l[Vo. ] of i
Si _ l [ T O i ] i s t h e z e r o i d e a l , isomorphism
i.e.
S!l_l_[Tc~] -* S~--l[V°']l r e s t r i c t s
S.i_l[To. ] ~ Si-l[Vo. ] " i
no r e l a t i o n
This is equivalent
t o an
to showing that
i
of the form : n.
(~)
g(ti,Tq.)(Tol. - t i) = h(To.), 1
1
g(ti,To.)
e S!i-i [Tq . ]' 1
1
exists.
0 # h(T ° . ) • Si_l[To . ]' 1 1
The above relation reduces,
polynomial
with
l
up to multiplication
by a suitable
C(t i) • Si_l[ti] , to n.
(~)
g~(ti'To.)( T o.l _ ti ) = C(ti)h(T ° ), with 1
g(ti,Toi)
1
1
e Si_l[ti,To. ]. l
We write, h(To.)
g~(ti,To l ) = j ~
a~(t i) T oi, j with a (t i ) • Si-I[T°']I and
m+ni b. T~ , with b. e = j~0 3 . ] Si-l"
I
i
Distinguish
two cases.
1. m < n i.
Then ( ~ ) ,
m
with g(ti,Tq
-
) = j[0 aj(ti)T~, i
ao(t i)
= bn.,...,bn.+m 1
However also 2. m ~ n i.
[
Thus a j ( t i ) • Si_ 1 f o r j = O , . . . , m .
: t i aj(t i) = bj, j ~ m, but this contradicts
Comparing
coefficients
occuring
in ( ~ )
t i ~ Si_ 1.
we get three sys-
:
= C(ti) b 0
an~a l(ti)ti
:
1
tems of equations
I I~ao(ti)ti
= am(ti).
yields
l
: C(ti)bn I 1
II
l
arli(ti)t i + ao(t i) = C(t i) bni ...............................
|a~(t.)t.
i
m
1
1
+ a~
m-n i
(t.) = C(t.) b 1
1
m
94
III
m_ni+l(t i) = C(t i) bm+ 1 . . . . . . . . . . . . . . . . . . . . . . . .
la~(t.) m l
= C(t.) b + l m ni
Let r > 0 be the t.-degreel of C(ti). IIl i m p l i e s we get
deg a~(t i)
= r.
: deg am_n.(ti)
contradiction
:
Combining
= r + 1.
am~-n.(ti)ti
As bm+ n" # 0, the last equation of i this
the
last
equation
of
II
Now if @ ~ m-n i ~ n.-ll then I yields a
= C(ti)
bm-n."
1
the equation
with
Otherwise
we d r a w f r o m
II
1
:
am_2ni(ti)
+ a m-ni ~ (ti)t i = C(t i) bm_n.i ;
then deg am_2ni(t i) has to be equal to r + 2 and~ either we keep increasing t h e
degree,
o r we d e r i v e
from
I a contradiction.
Because
the
t.1
degree of a~(ti) , j = 0 ...,m, is limited, the latter thing takes place. ]
COROLLARY.
THEOREM 91.
The generic algebra NI/k(G)
The exponent of Sl/k(G)
is a skew-field.
in its Brauer group 8r(k(t))
equal to the exponent of the abelian group
PROOF.
Let
G.
sI s G be of order n = Pl "''Pqq' Pi prime.
k(t) of SI/k(G)
ponent G 3• of
K.
Denote the center
theorem, ef. [14] , $. SI/k(G) ~ S 1 ® . . . ® Sq, where Sj is of degree pj3 Starting from the eyK clic decomposition G ~ i=lF -It Z / n i ~ , we compute the pj-component of G. If n l. = p~il .
by
is
By a well-known
Siq ."Pq ., then, . for j = 1, G
.,q, sj
=
~r i=1 sij and the pj-eom-
is Gj = i:l c T r
with nl3 "" = ni/P 13 and o.l being a
generate for Z/n i Z . n..
Let lj be the pj-extension
of
i,
then Sj may be
s..
written as l j ( t ) [ V 13, i = 1,...,r] and the pjl3-th power of V nij equals i ~i t i. Hence Sj is an algebra precisely of the type earlier constructed, i.e., Sj ~ SI/k(G j) 4
kj(u) where kj is the center of SI/k(G j) while
u
95
stands
for the
subset
{Ul,...,u m}
of t = {tl,...,t r} c o n s i s t i n g n.
variables
which
do not occur
s.. (Vol]). exp pjl3
as
of the
•
(pj does
not n e c e s s a r i l y
I
divide
all n.). 1 n..
Write
V ~ for V 13. A d d i n g u to the center of S I / k ( G j) does not affect °i oi the e x p o n e n t in the c o r r e s p o n d i n g Brauer groups. T h e r e f o r e , since e(Sl/k(G))
= j__F~# ej,
is a p - g r o u p
ej = e(Sj),
from now on.
it follows
that we may assume that G ml mr) n : pm and G of type (p ,...,p ,
Assume
m.
i.e.,
n i : p i and m = m I +...+ m r .
of cyclic product
algebras,
algebra
= i=1 ®r Si ' where
SI/k(G)
(Z/ni~,
SI/k(G)
decomposes each
into the product
S.I is a cyclic
li(t)/k(t) , {t} i) w h i c h
is a d i v i s i o n
crossed algebra
m.
of index p l,
(li/k is the s u b e x t e n s i o n
Up to a p e r m u t a t i o n mk+ 1 # m k then because
for any
we assume
by G mod Z / n iZ)-
m I i>...9 m r .
the h y p o t h e s i s j,
of i/k left fixed
If m I =...= m k but mI = p yields e(Sl/k(G))
e( ®k Sj) j:1
m1 = p
mk+ 1 ~< j ~< m I we have Pj
SI/k(G)PJ
-~ ($1 ® ' ' ' ®
(S 1 ~ . . . @
P" Sk]+l ®. " .~ sPJ r
~ k(t)
Sk)
Sk )pj
®
MN(k(t)) ,
N >
1.
k(t) The p r o b l e m
is thus r e d u c e d
to the case w h e r e
m I :...= m r . Using obd vious n o t a t i o n s for the factor sets we get that sP with d < m I is d d i/k' d similar to (G,l(t)/k(t), {t p ... t p }) This means that S p is similar mi
pd)
'
to S'i = (%/P (V~)
~, li(t)/k(t), t
'
.
r
Write
n. tpd Oiv i : and V~.I : I ~. for all i i i
s = {Sl,... ,s r} and look at the S"i : li(s) [Vi,
i = 1...r],
"
1
S!l = li (t)[V~i ]' where
k e l.(t).
subalgebra
which
Put tP.d i = Si'
1
S" of S' l i'
is i s o m o r p h i c
to
m.
(Z/p and thus
isomorphic
~(v i) : v ~i' phism
iZ,
li(s)/k(s) ' {Sl,...,Sr})
to S i u n d e r
(~i defines
on the g r o u n d f i e l d ,
the map ~i d e f i n e d
an u n r a m i f i e d thus
pseudo-place
by %i(tj) which
%i is an isomorphism).
= sj,
is an isomor-
However,
S'7 i
96
being
a k(s)-central
S[ = S'7 @ i 1 k(s)
k(t)
simple
and
f r o m this
S~/k(G)
with
@ k(s)
~1 8 . . . 8
li(s)
If the k(t) p
then
like
lois
groups
is c y c l i c
G
G
is a p r i m e
of m a x i m a l a first
92.
which
~
that
:
S'{/k
8
k(t)
k(s)
maps
is in
power the
be
to SI/k(G)
l(t)
under
onto
= p
e(Sl/k(G))
rd
<
equal pm
is e q u a l
split
because
to its
d < m 1.
index.
SI/k(G)
is a p r i m a r y
algebra.
finite
abelian
groups
occur
k does
not
subfield the
SI/k(G) is
by
to
then
direction
than
completely
at least
: k(s)]
in this
If char
which
or Sk(G).
One
as Ga-
This
is un-
:
divide
the o r d e r
of
F/kG(t)
admitting
an a u t o m o r p h i s m
exponent
If
G,
then
Sk(G) of
G.
[17].
3. The M o d u l a r
be
then
would
within
step
greater
[k(t)
S$',, (G) were
subfields
an a b e l i a n
strictly
k
k(t))
is i s o m o r p h i c
~ S~',~ (G),±/K
: k(s)]
If
contain
Let
an i s o m o r p h i s m
(SU 8 i k(s)
SI/k''(G)
algebra
However,
PROPOSITION
The p r o o f
[k(t)
to c h a r a c t e r i z e
but
cannot
division
: k(s)].
of
would
solved
derive
S[ ~ 8 i k(t)
: SI/k(G)
the d e g r e e
order
order
Cr
k(s)-central
COROLLARIES.
V.
8 k(t)
in S[i' it f o l l o w s
= l(s).
= [l(s)
the
=
contained
we m a y
S['._(G)±IK = k(s)8 = S'7.1 Since
the map
m
algebra
Case.
a field
with
char
k = p ~ 0.
A purely
inseparable
exteni
sion
P
of
linearly purely there
k
is c a l l e d
disjoint
inseparable exist
over
a m o d u l a r e x t e n s i o n of k, if k i k n pP for all p o s i t i v e i n t e g e r s
extension
a k-basis
B
for
P/k P,
is said
to have
and a b i j e e t i v e
a basic map
and
PP
i. group
are
A finite G
G ~ B, ~ ~ u
if , such
97
that uou T : I(O,T)UoT with I(O,T) 6 k ~.
This d e f i n i t i o n may be generali-
zed to inseparable algebras over fields, el.
[36], but it is not necessa-
rily to go into that here. From the d e f i n i t i o n it follows that a basic group is an abelian p-group. Indeed, UTO = I-I(T,O)U UO = I - I ( T , O ) I ( o , T ) U o T , ly be k-dependent
if oT = TO.
deduce u I = 1(o,1) 6 k ~ and
e
From uou I
Furthermore,
:
but UTO and uoT can on-
l(o,1)u o with 1(o,1) 6 k ~
the exponent e(G) of
is the e x p o n e n t of the extension P/k;
indeed,
G
We
is pe
since o e(G)
= 1 for
e(G) e k • and every o • G we obtain that u e(G) o = ~u I with ~ • k ~, hence u o e k~ e pele(G) follows. Vice versa, u • for all o • G implies u p =~ uope with ~ • k ~, thus Uope • k ~ or Uope = ~ u I with I • k ~, e n t a i l i n g oP e = i and e ( G ) I p e.
PROPOSITION
93.
A finite purely inseparable extension P/k is m o d u l a r if
and only if it has a basic group.
PROOF.
It is possible to show that P/k has a basic group if and only if
it is a regular extension in the sense of [15].
Also,
P/k has a basic
group if and only if P/k is isomorphic to the tensor product over simple purely inseparable
subextensions
correspond to the cyclic subgroups of c o m p o s i t i o n of
T H E O R E M 94.
G,
cf.
of P/k. G
k
of
These simple factors
occuring in a fixed cyclic de-
[36].
Let P/k have basic group
G.
Then
G
is up to i s o m o r p h i s m
u n i q u e l y d e t e r m i n e d by the e x t e n s i o n P/k.
PROOF.
Induction on the exponent
then clearly,
of P/k.
If e = 1 and
every basic group of P/k has to be of type
Let e ~ 1 and let bases
e
G
[P : kl = pn,
(p,p,...,p).
and G' be basic groups for P/k with associated k-
{uo, o e g} and {vT, ~ e G'} resp.
It is immediate that
98
F = k[u~,
o • G]
and
F'
: k[v$,
T • G']
coincide
because
v p may
be ex-
T
pressed
in the
f : G p ~ F, sic g r o u p s
that
f(o p)
= u~ and f'
hypothesis
in the
both
groups
is the d i r e c t is a g r o u p
N
F/k.
of times
have
the
will
sum of
The
same
G
p
pn
the
leaving
of
implies
fields,
with
of F/k
occurs
a Galois
G
rise
to ba-
and G'
can o n l y
type;
the
fact
G' m G.
algebra
with
N : N 1 ~...~
fixed
The m a p s
is e - 1 > l a n d by the
in the
following
N.
versa.
= v p , give
the t y p e
order
be c a l l e d
of
and v i c e
exponent
Hence
isomorphic
element
k
a factor
of k - a u t o m o r p h i s m s
unique
in
: G 'p ~ F, f'(T p)
G p m G,P.
number
A k-algebra
1. The
coefficients
G p and G 'p for
induction differ
u p with
group
Nm,
G
such
if
N
that
G
properties.
is the unit
element.
1
2. G acts 3. A n y
transitive
x • N such
It is e a s i l y if t h e r e
seen
exists
shown
that,
where
P/k
k(X)
k(X)
that
N/k
a normal
as
{C~,T})
in the
left
case
for
nerated
over
The
{ S l , . . . , s n}
set
PROPOSITION G.
There
and
dim ~Ik
k
95.
exists
k-basis
some
factor
in k(X)
fixed
is a G a l o i s {b a
where
by the
then
~ X
The
action
of
symmetric
In
if and o n l y
[15]
it is
A
if and o n l y
with
crossed Let
G
group
G
product act
if
and
sym-
structure
is
in
= XT~ , T,~ e G. X a.
: k] 2
[P
The The
symmetric
subfield
S n is k S : k ( S l , . . . , S n ) , the f u n c t i o n s in the v a r i a b l e s
X
of
field
ge-
, o • G.
independent.
Let N / k be a G a l o i s a k-pseudo-place
G
of d i m e n s i o n
splits N
group
N
the v a r i a b l e s
is k - a l g e b r a i c
= 0, w h i c h
P
N is a field.
by p e r m u t i n g
with
algebra
algebra
{Co,T}.
as f o l l o w s ,
the
algebra
~ e G} for
Galois
set
~ • G is in k.
simple
extension,
for
, a • G})
S n acts
every
is a k - c e n t r a l
is a m o d u l a r
= k( { X
group
x a = x for
(k-rational)
defined
{N1,...,Nm}.
that
if A / k
A = (G, N/k, metric
on
algebra
with
finite
9 of k ( X ) / k S w i t h
is c o m p a t i b l e
with
Galois
abelian
residue
action.
group
algebra
N/k
99
PROOF.
Let k G be the subfield of k(X)
left fixed for the action of
G.
Consider f =
[-7 (X-X o) • kG[X]. This is an irreducible polynomial over ~6G k G but also over k S . The b ° • N, all ~ • G, b • N, are roots of a polynomial T =
F-] (X - b °) = n a e G i=0
The speeialization field
k.
si;
a .xn- i with a. • k. l l
; a i extends
Let 0 S be the valuation
= k(X)'
We have k(X)'
putting #(X o) : b °. tion, as follows.
to a k-place
ring of ~ and consider
n k S = 0 S.
This
0s[{X o, o e G } ]
Extend % to a map of k(X)'
onto
is easily seen to be a homomorphism,
First extend ~ to 0s[X 1];
is a h o m o m o r p h i s m
% of k S with residue
since X 1 satisfies
then ~1
N
by induc-
: 0s[X1] ~ k[bl]
f(X 1) = 0 and b I satisfies f(b 1) -- 0,
which is the equation obtained by reduction
of f(X 1) : 0 under %.
Let
~1'''''#i-1
be obtained this way. Then X i (i stands for o i) satisfies i-1 fi(Xi) = ( f / F ] ( X - X j ) ) ( X i) = 0, where fi is ks[X1,...,Xi_l]-irreducij:1 ble. This polynomial reduces to fi = 5/ F - ] i - l ( x - b j ) with coefficients j:1 in k[bl,...,bi_l] , (bj = b°J). So %i-1 extends to a h o m o m o r p h i s m of Os[X1,...,X i] onto k[bl,...,b i]. The following
theorem yields a generic description
of the p-component
of Br(k).
THEOREM
96.
For any class e in the p-component
representative 1. A ~ (G, N/k,
A 6 a and a finite abelian p-group {Co,T}) , where
{Co, T ) a symmetric factor
N
is a Galois
PROOF. hence
0 which is compatible
Any central
such that
:
algebra with group
under a k-pseudo-place
G
and
inseparable
be the modular closure of P'/k
of di-
with Galois action.
simple algebra representing
it has a purely
G
there exists a
set.
2. A/k is residue algebra of Dk(G)/ks(t) mension
of Ba(k)
splitting
(cf.
~ is a p-algebra
field P'/k,
[331) and let
G
say.
and
Let P/k
be the, up to
100
isomorphism, such that
unique p-group determined
[A : k] = [P : k] 2, hence,
that A ~ (G, N/k, symmetric
{Co,T})
by P/k.
since
P
There exists an A e splits
A
we may conclude
for some Galois algebra N/k with group
G
and
factor set {Co,z}.
Put A = N[uo,
o e G] with u0~ = ~ ° u
(k(X)',~,N/k)
constructed
pseudo-place
for all ~ • N.
The pseudo-place
in the foregoing proposition
~ of Dk(G)/ks(t)
in the obvious way.
field of ks(t) under ~ is exactly
k,
extends to a
Since the residue
we have that ~ has k-dimension
equal to zero. In general,
when
N
is not a field,
unramified
pseudo-place
References
for Section V.
A.A. ALBERT P. MULLENDER
[3s],
[36].
of Nk(G)
[i]; I.N. HERSTEIN [16]; W. KUYK
it is impossible
to extend ~ to an
over kG(t).
[14]; K. HOECHSMAN
[17]; M.E.
SWEEDLER
[15]; W. KUYK,
[33]; F. VAN OYSTAEYEN
VI. A P P E N D I X
VI.
1. T h e
If ring
R
Center
coinciding
tric als
of Qo(R)
kernel of
R
R
x = 0}.
kernel
is G - t o r s i o n
in T(o).
ting
it w i l l
that
an R-linear
momorphism. the
same
ments
PROPOSITION
R
such
of Qo(R)
by the
same
representing if ~ then
filter
that
o ~ G0,
consider
i.e.,
A
such
is an
map
represen-
It is e a s i l y
seen
to be t h e
zero h o -
B : B ~ R represent
B coincide following
on A n B.
Ele-
proposition.
~ • C o if a n d o n l y
canonical
may
an ~ • Qo(R)
an R - l i n e a r
has
ide-
section
and
: A ~ R and
Jo is the
symme-
= O with
: A ~ R where
0 • Qo(R)
then
o on M ( R ) ,
always
of Q o ( R ) ,
symbol.
in the
Ax
in t h i s
we will
C
T O = {left
~
~ and
If ~ • Qo(R) where
form,
by
Let o 0 be the
some results
map
functor
for which,
By d e f i n i t i o n
characterized
f o r all r e j o ( R ) ,
of
o o n M(R)
free.
o f Qo(R)
1.
A
be d e n o t e d
kernel
by the
generalized
Consequently,
97.
ALGEBRAS
is a c o m m u t a t i v e
it w i l l
by C o .
defined
Although
element
map
R,
idempotent
ideal
be d e n o t e d
of C o are
of
b y an R - l i n e a r
The
element
its c e n t r o i d
be d e n o t e d
functors
m a y be r e p r e s e n t e d ideal
an
in a s l i g h t l y
symmetric
then
center
on M(R)
containing
stated
that
the
will
functor
x • R implies be
with
unit
F o r an a r b i t r a r y
center
OF A Z U M A Y A
o f Qo(R).
is a r i n g w i t h
throughout. the
: LOCALIZATION
if r~ = er
ring homomorphism
R ~ Qo(R)2.
Let A e T(o)
If ~ is left 3.
be an ideal
and right
If e • C o t h e n
an i d e a l
in T(o)
and
R-linear
every is left
let ~ then
: A ~ R represent e e Co .
representative and right
~ • Qo(R).
~
: A ~ R such
R-linear.
that
A
is
102
PROOF. and
1.
thus
Right m
extends
ms
Since by
r~
s is
The
= sr left
s •
C
3.
Let
is
defined
that
B
for
by
that
R be
defined
sB
the
same
LEMMA
PROOF.
any
ja(R)
and
its
the
T-class
cal
ring
Then = m x.
: A ~ left •
it
: R ~
Qo(R),
a left
follows
is
R-linear
that
restriction that
sq
= qs
s(ax)
= e mx(a)
Hence
s(ax)
R is
left
R-linear
T(a)
and
and
left
to
R
for
all
R-linear map
multiplication
coincides q •
map
= m x s(a)
for
with
all
with
Qo(R)
m s-
and
because
= s(a)x,
6e(ab)
= B(~(a)b)
= s(a)8(b).
Bs
coincide of
on A B e
Qa(R).
o 0 ~> T i> a t h e n
: A ~
they
s e
B e
Then
s is
free,
Note
the
this
fact
consequence
QT(R). the
in
homomorphism
T-torsion
there
same
C T. Qa(R)
map
that of
is
~
QT(R)
T(a).
AB
b •
Ja
proving
~
(0)
a ring
C
and sB
the
B we
of
~
they
: B ~
Bs c o i n c i d e QT(R).
C
sB
have
because
This
restriction
maps
let
in
C T.
of
and
6s
that
:
represent T(a)
homomorphism
let
and
and Then
therefore
is
element
Hence,
B •
and
that
R represent
define
of
T(a)
Note
R-linear
a • A,
= s(a)B(b)
s
= m x s(a)
right
= ~(a6(b))
represent
hence
a direct
to
sB(ab)
If
and
that
s
element
Let
B e T(T)
ms
R-linear.
on AB
and
98.
R.
j~(x)
Suppose
Thus
way
r •
m s implies
x •
right
are
a unique
s,
.
is
: B ~
all
of
by
Qo(R).
R-linear
a • A,
s
in
: Qo(R)-~
uniqueness
thus
2.
multiplication
c TO.
C T"
Ca
R with on ABe proves the
T(T) that
canoni-
Since
R
is
injective.
Qa(R) Theorem
~
QT(R) 10 b u t
is it
a ring follows
homomorphism because
is
R/T(R)
not
103
and
R/o(R)
If
R
C o ~ C~, known are
coincide
with
R.
is a p r i m e
ring
then
and
that
C o = C~ n Q o ( R )
C
is a f i e l d ,
integral
o0 : o for
thus
and
every
if
R
then
we have
injections
a.
is w e l l -
symmetric
is p r i m e
then
It the
Zings
Co
domains.
PROPOSITION
99.
ro-divisor
of
sentatives
~
Let
Co.
R
be
Then
a semiprime
at
least
: A ~ R, w h e r e
A
ring
one
(and
is a n
and
let
thus
ideal
m • C o be
all)
of
in T ( o ) ,
its
is n o t
a ze-
reprea mono-
morphism.
PROOF. ding
Let
the
p(o,0)
identity
is o 0 - t o r s i o n
be
the
of
free
R,
unique i.e.,
it f o l l o w s
Lemma
that
sors
of
C a map
zero-divisor B
A,
c
such
~
that
~IB
el A n A' • T(o)
The
converse
or
for
arbitrary
tral
element.
p(o,0)(e)IB 0 = IB ideal
p(a,0)(e) of
R
then is
p(o,0)
there
= ~,IA
of
the
does
: A'
not
cf.
exten-
Since
inclusions
C ° into
is
Let exist
an
Qo(R)
:
an
ideal
and
proposition ideal
if p ( o , 0 ) ( ~ ) B
= 0 for
B commutes
: A ~ R represent
because
injective
zero-divi-
p(o,0)~
is a
representative
then
I e T O such entailing with
R
holds in
R
the
that IB~
B e
0 ~
= 0.
in Q 0 ( R ) ,
in c a s e
contains
some
and
a
B • TO,
fact
a contradiction.
above
ideal
C O and
injective
a ~ o 0 if e v e r y
some
e
[A.1],
~ R were
n A'
= p(o,0)(IB~),
because
maps
in C O .
injective,
If ~'
Indeed, = 0 for
that
c T O yields
Remark. also
~ Q0(R)
• Q0(R)
zero-divisors
in C o .
then A n A'
derive
onto
that
Qo(R)
p(o,0)
~ • Co,
zero-divisor for
we have
~ Qo(R)
98 w e
map
p ( a , 0 ) [ A , m ] ° = [A,m] 0.
Jo R
From
R-linear
a = a0 a cen-
C O then
IB c R. Now,
IB
thus,
Hence is a n
104
if y • C n I6, y # 0,
t h e n y~
: 0 yields
that
~ is a z e r o - d i v i s o r
in C . o Therefore, then
if s o m e
p(o,0)(~)
zero-divisor position
e
: A ~ R representing
is a l s o r e p r e s e n t e d
in C O .
The
foregoing
e 6 C ° is not
by ~
: A ~ R and
then
proves
injective
it is t h e n
the c o n v e r s e
a
to P r o -
99.
Generalizing we obtain
a result
of S.A.
Amitsur
concerning
o0,
ef.
[A.1],
:
PROPOSITION
100.
o is a T - f u n c t o r
If R C °
is s e m i s i m p l e
a n d an e l e m e n t
Artinian,
then
a • R is r e g u l a r
Qo(R)
: R C a,
Jf a n d o n l y
if
Ra • T(~).
PROOF. tent,
Let
A
be an i d e a l
t h e n A n eAe #
(0).
in T(o)
and e e R C o
Indeed,
pick
a non-zero
B e T(o)
such
that
idempoBe c R.
e = Z. r i c i w i t h r I• E R, c.1 E C [~ a n d p i c k D 6 T(o) s u c h t h a t I De i C R f o r all i; t h e n eD C R. Take I e T(o), I C B n D and take
Write
I to be a n ideal. xAx
= 0 for
too and Ax that
IeI
Then
any x e IeI. = 0 follows.
= 0.
tradiction.
Again, Let
ejaj
is r e g u l a r
= aj we g e t
a division
c eAe ~ A a n d
Since However
this
R Ca
is s e m i p r i m e ,
x • Qo(R)
implies
of A N ejAej.
in
that
algebra,
Ie
R,
for
ejRCob
thus
Consider
e j R C a b ej = 0
then
it is r e g u l a r
in R C
is G - t o r s i o n
We know
follows.
is r e g u l a r
as R C O
a = Zjaj
• A.
: 0.
hence
where
of i in R C a
ej
is s e m i p r i m e ,
torsion
x : 0 entailing
= 0 t h e n ba. = 0 a n d 3
aj
(0) t h e n
is s e m i p r i m e
if b a
R Ca
R
yields
R
were
: 0, e = 0, so we r e a c h
b e 3. = 0 b e c a u s e in
if e I A I e
1 = Zj ej be a d e c o m p o s i t i o n
aj # 0 be an e l e m e n t element
eIAIe
free.
b.1 o
that This
Therefore
and
let
This
since
ejRC
ej
entails
= 0 follows.
because
a con-
R C /R o
that
If a is o-
a regular
is
105
element in
R
is invertible in R C o.
regular element, thus has property
(T).
1 • RC oA
Further,
D • T(o), hence x • R C plies then that Qo(R)
Every A • T(o) contains a
and 1 • Q~(R)A, proving that
if x • Qo(R) then Dx C R for some
Dx = R C o x
= R C~.
but R C o D x c R C o R
Finally,
if
a
= R Co
is r e g u l a r in
then a -1 • Qo(R) and Da -1 c R for some D • T(a).
imR,
Hence D • Ra
and Ra • T(~).
Remark.
If in P r o p o s i t i o n 100, R is also a prime ring,
then Qo(R)
is a prime ring too and hence it is a simple A r t i n i a n ring. fore, for every 0 ~ A • T(~ ~) we have that Q~(R)A = Q~(R). ty (T) for ~ then yields A e T(~) and ~ = ~
follows.
ThereProper-
For prime
rings the assumptions made in P r o p o s i t i o n 100 force us into the case considered by S.A. A m i t s u r in [A.1]. for milder conditions on of
R.
R
This is the reason why we look
under which Qo(R)
is a central e x t e n s i o n
This search will lead to the c o n s i d e r a t i o n of rings satis-
fying non-trivial polynomial identities.
In the torsion free case
R and Qo(R) will satisfy the same identities and the nature of the identities does not interfer with the localization theory.
That
is why we present a general approach using A z u m a y a algebras.
Let
R
be a ring with unit,
the opposite ring of algebra of and only if
R
R.
Let R 0 be
R,
over C. R
C (in) the center of
then R e = R ® R 0 is called the e n v e l o p p i n g C The C-algebra R is said to be separable if
is a p r o j e c t i v e Re-module.
Let m : R e ~ HOmc(R,R)
be the ring h o m o m o r p h i s m defined by m(x @ y0)z : xzy for all x,y,z • R.
Recall from [A.2] the following
LEMMA 101. E q u i v a l e n t l y
:
1. R is a separable C-algebra.
:
106
2.
Put M = H o m R e ( R , R e ) , t h e n R e M
3. T h e m a p m
: R e ~ HOmc(R,R)
ly g e n e r a t e d 4. T h e m a p
m
projective
= Re
is an i s o m o r p h i s m
and
R
is a f i n i t e -
C-module.
is an i s o m o r p h i s m
and
C
is a d i r e c t
summand
of
R
as
a C-module.
DEFINITION.
If
R
to be an A z u m a y a
LEMMA
102.
The
is s e p a r a b l e
following
functors
its
center
statements
are
equivalent
and
finite
F o r the p r o o f tensive bras
and
may
Let
respondence
{An
cide
be f o u n d
is g i v e n
C n AB
for
and M(Re),
in
every
treatment
i.e.,
between
C-
over
of the
A
of
R
on R - m o d u l e s .
is c e n t r a l
ideal
m
[A.2],
theory
2..
Let
filter
via
then and
there
ideals
R A c = A.
of
sim-
C.
[A.3].
An
of A z u m a y a
ex-
alge-
is a o n e - t o - o n e A c o f C, t h e
This
o ~ o 0 be s y m m e t r i c of a s y m m e t r i c
: (A n C ) ( B n C) f o r all
as C - m o d u l e s
maximal
consult
C,
by A c : A n C +~
is the
and RImR
[A.5].
ideals
102,
type
one m a y
separable
between
from Lemma
R-modules
M(C)
of f i n i t e
details
contained
C, A e T ( ~ ) }
because
between
dimensional
be c e n t r a l
correspondence
diately
:
= r m f o r a l l r E R},
C-module
and m o r e
self
also
R
is s a i d
and R - b i m o d u l e s .
3. R is a f a i t h f u l ple
R
N ~ R ® N a n d M ~ M R, w h e r e C
an e q u i v a l e n c e
modules
then
algebra.
M R = {m e M, m r
define
C
al e b r a
1. R is an A z u m a y a 2. T h e
over
C ~ R.
Then
ideals the
follows
imme-
on M(R).
Then
functor A,B
of
functors
cor-
o' on M(C) R.
o' a n d
Consider ~ coin-
107
LEMMA
103.
If
R
is an A z u m a y a
homomorphism
go
PROOF.
an a • C o and
Take
representing maps
C
as
let a ideal
is e a s i l y
morphism
A e ~ C and as such,
All
is c l e a r y
this
correspondence
: A ~ R and
ments
of C ° then
is a n a t u r a l
ring
checked. c
defines
represents of the thus
is d e f i n e d
restriction
Therefore
a map
c
go
: A c~R
element
of Qo,(C).
representatives.
: Ca ~ Qo '(C)"
morphisms
on AB and
c
is a C - b i m o d u l e
a unique
choosen
B : B ~ R are b i m o d u l e ~B
The
homomorphism
The
Moreover,
representing
(~8) c c o i n c i d e s
ele-
with
~c6c
(AB) e = A C B c.
Obviously, its
center
an A z u m a y a
C
field
sional
K-central
of f r a c t i o n s , simple
We now t u r n
to the
PROPOSITION
104.
and
consider
Sticking
PROOF. tion
the
sheaf
Let
R
on
derive
: QA(R)
directly for
K say,
In that
and
ring case
if and o n l y C
if
is c o n t a i n e d
S = R ® K is a f i n i t e C
dimen-
theoretic
aspect.
left
C A of QA(R)
Noetherian
where
set X A C X = Spec
A
Azumaya
is an i d e a l
R defines
algebra of
a sheaf
R. ~(R)
of
X.
If ~B ~ aA t h e n
of p(A,B)
is a p r i m e
domain.
be a p r i m e
centers
rings
R
algebra.
C A to the o p e n
commutative
algebra
is an i n t e g r a l
in its
we
there
: A ~ R be a b i m o d u l e in T(o).
independent
~ ~ c
if a
on
then
: Co ~ Qo,(C).
it, A is and
A c into
algebra
that
the r i n g ~ QB(R) ~(R)
any c o v e r i n g
homomorphism and
the
fact
that
is a m o n o - p r e s h e a f .
prove
that
there
are g~ e C a for w h i c h
p(~
there
is a g e C A such
p(oA,O
that
from
C A ~ C B is the
XA = m XA , sup(~
(write
X
Q is a sheaf,
We are for X A
left
all
~.
to
) such
,~B))g e = p(o~, sup(o
)g = g~ for
restric-
Since
that
,~B))g6, Q is a
108
sheaf,
such a
g
exists
is mapped into C a under
in QA(R) by Theorem 42.
However,
since
g
p(oA,O ~) the injectivity
of p(oA,O ~) implies
that g • C A = ~ C a.
Remark.
In general
if
R
is an Azumaya algebra then there is a Noe-
therian subring C O of
C
such that R : R 0 C@O C, where R 0 is an Azu-
maya algebra with center C O . tions are concerned,
Thus, as far as purely algebraic
the left Noetherian
hypothesis
is not very res-
trictive because one may use general techniques
of descent,
[A.5].
is significant
cause,
Geometrically
speaking,
since Spec is not functorial
ting Spec R to a suitable
PROPOSITION with center There
105. Let C
in general,
"patching"
R be a left Noetherian
the problem of rela-
prime Azumaya
algebra sheaf.
and g is an isomorphism
is affine.
The one-to-one
topological
be-
of Spec R 0 and Spec C is open.
is a sheaf morphism g : Spec C ~ ~(R)
and ideals of
cf.
and let Spec C be equipped with its classical
if and only if ~(R)
PROOF.
this assumption
ques-
correspondence
A M
Ac
between
ideals of
R
C gives rise to a h o m e o m o r p h i s m g of the underlying
spaces of Spec C and C(R) defined by g(p)
p • Spec C = X'.
= Rp = P for
Since g-l(x A) = X~c we only have to check whether
the following diagram of ring homomorphisms
is commutative
:
gA
CA
QAc(C)
I
p(A,B)
p(AC,B c )
I gB
CB
QBc(C)
for any B c A, ideals of The action of the maps
R.
is given as follows
: take a representative
109
of the element on which the map has to act and then r e s t r i c t a suitable subset of the domain. ly.
Clearly,
Hence c o m m u t a t i v i t y
it to
follows easi-
if C(R) m Spee C' then C' has to be isomorphic to
C a = C and then g is a sheaf isomorphism.
To end this section, we mention that for an Azumaya algebra with center
C,
the zero-divisors
of
C
R
are c h a r a c t e r i z e d by Propo-
sition 99 and its converse, which holds because
ideals of
R
contain
central elements.
VI.
2. L o c a l i z a t i o n of Azumaya Algebras.
In this section we drop the a s s u m p t i o n a ~ 00 . is an A z u m a y a algebra with center Moreover,
C
In the sequel
R
and R = R/o(R), [ = C/o(R).
~ will always be a symmetric kernel functor and C a is the
center of Qa(R).
T H E O R E M 106. Qo(R)
PROOF.
is an Azumaya algebra and Qo(R)
Since the canonical R ~ R is surjective
= RC O : R @ C .
it follows that
is t - c e n t r a l separable. with eenter C o. multiplication
Therefore, R ~ C a is an A z u m a y a algebra C The ring h o m o m o r p h i s m ~ ® C a ~ R C a defined by
in Qa(R)
is onto and Co-linear.
A z u m a y a algebra with center C a .
We end up with R C °
both rings h a v i n g the same center. [A.3]) where
D
Hence,
commutes with R and since Proposition
dule we may use Lemma 102.2.
COROLLARIES.
Qa(R)
is the eommuting ring of R C a
~ 00, we get D = C a and R C °
Thus R C a
= RCa
in Qa(R).
Since Q~(R)
to conclude that Q~(R)
R
Q~(R) ~a
and
D (cf.
Since
D
97.1. obviously holds without
= Q~(R).
1. An A z u m a y a algebra
C
is an
is an R-bimo= R @ Ca .
is o-perfect for every
110
symmetric 2.
If
R
central Qa(R)
is p r i m e
the
R
morphisms
Consider
on M(R)
given
~
fact this
as g(o)
that
h
we may
C,
using
(T).
h.
The
then
whenever
that
inclusion because the
because
a'
C
derived
Proposition
106 e n t a i l s ,
R C
A = R C a A c = R C a , thus over
C a this
by p r o p e r t y
means
to a When and
with
h
that
this by
without
and put A = R A e .
Qa(R)A
that
free,
C ~ Q~,(C)
and
is s e p a r a b l e
= Qa(R)
yields
extends
to c o i n c i d e
been
of
is a T - f u n c t o r .
is a ' - t o r s i o n .
canonical
has
Moreover,
is G - t o r s i o n
C ~ Qa,(C) Ca/C
Let A e be an ideal
to Pro-
= Q~a~(~)
R
ga has
this
corollaries
T-functor.
and Qa,(C)
assume
that
The
functor
t h e n A e T(~)
R Ca
if
kernel
A c E T(a')
since
105
in
C a ~ Qa,(C).
symmetric
A e T(a)}.
it i n d u c e s
Note
of Spec
by P r o p o s i t i o n
T-functor
= Q~a(R)
ga is i n j e c t i v e of
exists
Let ~a be the
: C a ~ Qa,(C) to
by f u n c t o r i a l i t y
is a T - f u n c t o r
Qa(R)
theorem
the u n i q u e n e s s
and
of
~ ~(Qa(R))
~a is a s y m m e t r i c
a : ~a ~ a 0.
property
a sequence
resp..
= {~(A),
and ~a'
ga is r e s t r i c t e d that
algebra,
of C.
we o b t a i n
ca
: Ca ~ Qo(C)
: R ~ R.
that
C-linear
entails
being
g(a)
where
by T(~a)
~ C/a'(R)
unique
then
Qa(R),
simple
of f r a c t i o n s
Spec
If a is a s y m m e t r i c
33 imply
R : R,
and
is a C a - c e n t r a l
Noetherian
by Ca,Qa(R)
PROOF.
i.e.,
C a is a f i e l d
field
~ ga
ga
case,
107.
in p r o v i n g
m
from
THEOREM
the
is the left
Qa,(C)
C,R are r e p l a c e d
C/a(R)
a field,
K
and
then
:
ga d e r i v e s
position
a = a
over
is p r i m e
commutative
Thus,
and
separable
Spec
where
a on M(R).
~ R @ K where C
3. If sheaf
T-funetor
If
(T) for a.
R Ca(C a A c) = R C a C a A c = C a and a
iii
fortiori Qo,(C)A c = Qo,(C).
This implies that o' is a T - f u n c t o r
and because every A c e T(o') extends to C A c = C this entails that o o C o = Qo,(C).
Since go is the unique C-linear extension of
C ~ Qo,(C)
to Co, it is immediate that go is an isomorphism.
COROLLARY.
Let X A be open in X = Spec R, where
N o e t h e r i a n ring.
R
is a prime left
If X A is a T-set then we get a commutative dia-
gram of sheaf morphisms
:
~(R) IXA
m
, ~(QA(R))
Ac X' Ac
Spec C A
~
Spec Q A c ( C ) ~
where X' = Spec C. fact that
R
The isomorphisms
is oA-perfect.
gA is an isomorphism.
exist by Theorem 47 and the
The foregoing p r o p o s i t i o n yields that
Commutativity of the diagram is merely veri-
fication.
LEMMA 108. Let
P
PROOF.
If
R
is an A z u m a y a algebra then Spec R has a T-basis.
be an arbitrary prime ideal of
R,
then OR_ P is a T-functor.
For every r e R choose a(r) e RrR n C and put A(r)
Then A(r)
is an ideal of
Moreover,
since B contains a B' also in T(OA(r))
finitely generated, a T-basis
R
and if B e T(OA(r) ) then B n A n ( r ) = Ran(r).
it follows that OA(r)
buth with B' being
is a T-functor.
{XA(r) , r e R} is obtained for Spec R.
is a T - f u n c t o r one argues
= Ra(r).
in a similar way.
In this way
To prove that OR_ P
Note that in the
112
absence
of the left N o e t h e r i a n
as a t o p o l o g i c a l consequence
COROLLARY. rollary
space,
therefore
All
sheaf morphisms
X A t h e n XBC
is a T - s e t
CB(QA(R))
are i s o m o r p h i s m s
109.
R
Let
R
PI"
statement
P
: CB(QA(R))
is a sheaf
be a p r i m e
of G(P)
contained
in the T - s e t
the
~ QBc(CA) i.e.
defining
for a basis
isomorphism
g(A)
of the
and the c o m m u -
same for glX' . Ac
ideal of
R,
then
:
elements
in QR_p(R)
then
R
satisfies
the left Ore c o n d i t i o n
to G(P).
1. R e d u e t i o n
t h e n J. Lambek,
G. M i c h l e r ' s
of 1. and 2. and d e n o t e
the p r i m e
T h e n ~OR_ P = o I is e x a c t l y
the k e r n e l
functor
R1-P 1 and,
torsion
(T) and Op = OR_ P.
~ : R ~ R 1 = R/oR_p(R)
the m - s y s t e m
of the co-
R ~ QR_p(R).
Op has p r o p e r t y
Step
in the d i a g r a m
map onto i n v e r t i b l e
is left N o e t h e r i a n
108 y i e l d s
is not a
in XAC and we h a v e
X B is a T-set,
is left N o e t h e r i a n
theory
Consider
gB(A)
H e n c e g(A)
the c a n o n i c a l
with respect
PROOF.
contained
of the d i a g r a m y i e l d s
1. The e l e m e n t s
3. If
appearing
Let X B be a T - s e t
whenever
Zariski-topology.
2. If
the s e c o n d
Spec
m QBc(CA).
The r i n g h o m o m o r p h i s m s
under
lemma deals w i t h
107 are in fact s h e a f i s o m o r p h i s m s .
P R O O F of the C o r o l l a r y .
THEOREM
the
of the first.
to T h e o r e m
tativity
condition
OR_ P and o I c o i n c i d e
that OR_ P is a T - f u n c t o r
hence,
ideal ~(P)
associated
on R l - m O d u l e s .
by P r o p o s i t i o n
by
with Lemma
33, o I
113
is also a T-functor. gebra with center C 1. extension
Theorem
m Qol(R1)
have that Qol(M) be sufficient
consists
To prove
2. we will assume that
CR_ P m QC_p(C),
Consequently
lows that CR_ P is a local ring.
Since
QR_p(R/P)
of QR_p(R).
is regular,
being
simple Artinian.
hence
st • 1 + pe.
proves
that
s
thus
If s • G(P) invertible
Suppose
Rs • T(oR_p).
oR-P < Op implies
The latter is equivalent pect to G(P).
that
[Rs
separable
ideal such that
Therefore
pe is the Jas
it follows
in
the latter ring
is such that
invertible
(Rs) e = QR_p(R)
and this yields
is central
it fol-
1 - st e pe,
of units of QR_p(R)
Step 3. Proof of 2. From
ry s • G(P)
1. and
an isomorphism
in QR_p(R/P),
is (both left and right)
Moreover,
in proving
then the image of
t • QR_p(R)
Now 1 + pe consists
that the left Ore
is commutative
is a maximal
m QR_p(R)/P e is a simple algebra.
cobian radical QR_p(R/P)
C
Thus QR_p(R)
over a local ring and pe = QR_p(R)p
is left
free.
(T) for oR-P yields
where p = P n C.
R
state-
lifts to the left Ore condi-
is oR_p-torsion
Step 2. Proof of 1. Property
yields
In case
= G(P 1) entails
to G(P). R
in Qol(R1)
via ~.
for R 1 with respect to G(P1) R with respect
Indeed,
and thus the fact that G(P 1)
factorizes
the fact that ~G(P)
1. and 2. it will
for R1, P1 and 01 .
of elements which are invertible
Noetherian,
tion for
statements
~G(P) c G(P1)
ment 1. because R ~ QR_p(R)
condition
that for every M • M(R 1) we
to prove the analoges
c p, this yields
that Qol(R 1) is a central
to an ideal Qal(RI)P 1 of QoI(R1).
it follows
= QR_p(M).
Then R 1 is an Azumaya al-
106 entails
of R 1 and thus P1 extends
Since QR_p(R1)
aR_p(R)
Denote z(C) by C 1.
and this
in QR_p(R).
that
that Rs e T(Op)
for eve-
: r] n G(P) ~ ~ for all r e R .
to the left Ore condition
for
R with res-
114
Step 4. Proof of 3. Look at the following ring homomorphisms
commutative
diagram of
: ~p , R/~p(R)
-g
R/C~R_p(R) By Proposition
33. 3., ~
QR_p(R) Consequently,
to a ring h o m o m o r p h i s m
~ QR_p(R/Up(R)).
G(P) maps onto a set of invertible
QR_p(R/Up(R)). with respect
extends
Moreover, to G(~p(P)
R/~p(R)
in
the left Ore condition
= ~p(G(P)).
Thus Up has
{Rs, s • G(P)}
immediately
that Up has property
QR_p(R)s
satisfies
elements
for a filterbasis
= QR_p(R)
(T).
and thus it follows
Finally
since
for all s • G(P)
we get Rs • T(UR_ P) by property
(T) for UR_ P and ~R-P = Up follows.
Remark.
The canonical
QR_p(R) ~ Qup(R)
is always
a ring homomor-
phism.
The proof of this is only a slight extension
of Theorem
10. 2.
COROLLARY.
Let
R
be a prime left Noetherian Azumaya
1. Spee R has a T-basis,
all stalks are T-stalks
algebra,
then
and each stalk is
a local ring with simple Artinian residue ring. 2. If X A is a T-set of Spec R then X A ~ Spec QA(R) one-to-one
correspondence
between
sub T-sets
and there is a
in X A and T-sets
in
:
115
Spee
QA(R).
there sets
VI.
is a l s o
3. A z u m a y a
110.
suppose
rank
Alsebras
that
C
is a s e m i - l o c a l
for
R
that
that
g~
be the t o r s i o n
R
107,
that
go maps onto
B e T(a').
to Bx I @ . . . ~
Define
: A ~ R by 2
onto
e
(n 2)
go : Qo ~ Qo '(C)
functor
type The
use
rank
o on M(R).
and
of c o n s t a n t
first
part
of p r o p e r t y
homomorphism.
of the
(T) for
Let
fact
that
R
we are
in the
If
and R
let
is C - f r e e ~ = C/o(R)
torsion by ~
by ~IA c
free
yields
: C/o'(C).
case
R = R.
: A ~ R where : A c ~ C.
that
A
is an
To p r o v e
that
B : B ~ C be a r e p r e s e n t a t i v e
is w r i t t e n
Cx I @ . . . @
CXn2 , t h e n A : RB
BXn2.
2
~(~n b.x.) i:l i i
linear.
The
B e Qo,(C)
is e q u a l
b i e B.
kernel
not make
ec r e p r e s e n t e d
for
where
sub-
QA(R).
Then
C-module.
e e C o represented
take
~
R,
geometric
of c o n s t a n t
thus R m @ n2 ~ w h e r e
that
go is onto, with
of
Rings.
of f i n i t e
ring
A
= R/a(R)
and
assume
in T(o),
does
is an i n j e c t i v e
we may
B
which
ideal
between
in Spee
ring.
symmetric
is a free
reduction.
Ker ~ m @ n2 o'(C)
ideal
every
every
algebra
is C - p r o j e c t i v e
: R ~ R
Recall
Valuation
be an A z u m a y a
of T h e o r e m
Hence
sets
R
it f o l l o w s
o, y i e l d s
open
over
for
correspondence
Let
Because
proof
is o A - p e r f e c t
geometric
is an i s o m o r p h i s m
PROOF.
R
a one-to-one
of X A and
THEOREM and
Since
: E n 6(bi)x i i=l
It is r e a d i l y
Therefore = B under
verified
~ represents go"
that
an e l e m e n t
~ is left
and r i g h t
~ e C a which
R-
is m a p p e d
116
COROLLARY. tric
~.
C
Indeed
te type may
If
PROPOSITION there
functors every
The
ideal
~t
C,
between
of
R
and
contained
invertible by p r o p e r t y
These algebras let lows. with
C
and
Now
C
Theorem
ideals
of
C.
Q~(R)A
110
constant
because and
ideals
K
that
and
p
is a p r i m e (T) and
one-to-one
corres-
, and b e t w e e n
Every
ideal But
This
A
then
yields
of s
R is
A•T(~)
a = ~R-P"
pseudotplaces
algebras.
and d e n o t e let
and
Thus
~ is a T - f u n c t o r
s • C-p.
field
~ = ~R_p°
property
of C
follows.
of O K and
kernel
so C(a ') c o n s i s t s
that
P = Rp.
link b e t w e e n
O K and r e s i d u e
such
ordered
of the
entails
ring,
(T).
implies
Let ~ be a K - a l g e b r a
of ~ / K
of fini-
symmetric
i.e.,
at C-p has
of A z u m a y a of
R,
linearly
Put
this
the
ring
of
an e l e m e n t
Let M K be the r a d i c a l
pseudo-place
symme-
a valuation
between
property
= Q~(R)
~, thus
provide
ring
being
over
is c o m m u t a t i v e
of Q~(R)
be a v a l u a t i o n
module
the r a n k
algebra
P
localization
localization
112.
are
ideals
(T) for
valuation
has
is a T - f u n c t o r
and
results
PROPOSITION
Then,
say.
C
in P, c o n t a i n s
in C
about
ideals
funetor
is a T - f u n c t o r .
pondence
for every
projective
correspondence
prime
therefore
if ~v
every
is an A z u m a y a
p r i m e ideals of
if and o n l y
not
R
and
p
ring,
C O m Qo,(C)
case.
kernel
element, of
ideals
If
then
condition
is a o n e - t o - o n e
symmetric
of one
the
in that
111.
ring
a local
thus
o on M(R)
PROOF.
thus
over
is free,
be d r o p p e d
then
is a local
C
Let
K
of s i m p l e be a field,
by O K in w h a t
~ be the
place
of
folK
k.
let
(R,~,D1/k)
D 1 = R / R M K is k - c e n t r a l
be a ~-
simple.
117
1. For e v e r y residue
field
field
symmetric
i i
2. For e v e r y
me i d e a l
p
is c e n t r a l
i,
that
(T).
[R
[Qo(R)
k-central
separable
with
there
is a
simple
algebra.
is an u n r a m i f i e d
Hence
R
is
Choose
111
= p - l o K for the pri~ such that
(with P : Rp)
: Qo (R) ~ Qo (R/P)
0
has
is s i m p l e b e c a u s e = R o yields
the
of N/K. Q~(R)
: 0 o] ~
is an A z u m a y a
[D : K]
= [R
algebra with
center
0
: 0 ] and t h e n
IN : K] ~
If (R,~,D1/k) simple
~o
(P)
that
106 and P r o p o s i t i o n
over 0
to o.
: i] : [Q (R)
the e q u a l i t y
COROLLARY.
,Ro/I)
k,
K
a pseudo-place
and O K is local we h a v e
then Qo(R)/Q
pseudo-place
we d e d u c e
(Qo(R),~
of O K c o r r e s p o n d i n g
field
field
exists
o v e r O K and t h e n T h e o r e m
2. F r o m the fact that
proves
as in 1.,
: %1 of
of N/K.
Qo(R)
o has p r o p e r t y desired
I
separable
imply that
places
with residue
o such that t h e r e
1. Since D 1 is s i m p l e
central
i
exist
for w h i c h R o is an 1 - c e n t r a l
field
pseudo-place
residue
and ~2 of
T-functor
(Q~(R),~o,Ro/I)
PROOF.
such that t h e r e
[R ° : i] : i] and thus ~o is u n r a m i f i e d .
is a p s e u d o - p l a c e
then e q u i v a l e n t l y
of D/K such that D 1 is
:
1. ~ is u n r a m i f i e d . 2. R is an A z u m a y a
Indeed
if @ is u n r a m i f i e d
is s i m p l e OK .
algebra.
and k - c e n t r a l
The c o n v e r s e
yields
follows
i = k, o = °R-P w h e r e
then Ker ~ = RM K and the fact that @(R) that
R
is c e n t r a l
from Proposition
P : Ker 4.
112.
separable
over
2. if one takes
118
This has
corollary
consequences
(R,~,91/k) central
extends
for the t h e o r y
be an u n r a m i f i e d
of p r i m e s
pseudo-place
113. E a c h of the f o l l o w i n g
1. P is a c o m p l e t e l y 2. Qo(P)
prime
is a c o m p l e t e l y
64.
It a l s o
in s k e w - f i e l d s .
Let
of 9 / K such that 91 is a
ideal of
equivalent
statements
:
R
prime
ideal
of Qo(R).
of ~
as a p s e u d o - p l a c e
is a s k e w - f i e l d .
4. ~ is a s p e c i a l i z a t i o n
implies
that
~ is a p r i m e k e r n e l
PROOF.
1 ~
2.
P ~ T(o),
in
P
for some
w i t h x,y • Q~(R)
a n d 12Y are c o n t a i n e d resp.
(I l l 2 ) C x y
Then
in
R.
then
in
R
yields
l~x C p, e n t a i l i n g
obvious.
Since ~o is u n r a m i f i e d ,
suppose
in T(o). implies x ~ R
of P r o p o s i t i o n
implies
that Qo(P) Secondly,
R
= Qo(R)
if x • R t h e n
12 n O K
is c o m p l e t e l y
llX = R I ~ x c p and thus 2. and
3. is
4 ~ 3 is an imme65.
2.
Final-
Ix C P for some i d e a l
I
t h e n the fact that ~ is u n r a m i f i e d
Firstly,
contradicting
llX
c ~ P then If 12Y
of s t a t e m e n t s
Then
if
such that
62 and P r o p o s i t i o n
is a " v a l u a t i o n " x -1 • R.
P
the i m p l i c a t i o n
- Ker @.
If D 1 is a s k e w - f i e l d
= p.
Ix
I ¢ P because
12c for 11 n OK,
that
The e q u i v a l e n c e
that x • Qo(P)
that
11, c
and thus
Conversely,
the fact that
by L e m m a
consequence
1.
= l cl X l 2cY c R n Q~(p)
13.
Since
let 11, 12 • T(o)
x • Qo(P)
diate
I • T(o).
Write
cy ~ P for some c • 12c and thus prime
functor.
x or y has to be in P, so 2 ~
xy • Qo(P)
of 9/K.
If x,y • R and xy • P then xy • Q~(P)
or ly is c o n t a i n e d
ly,
56 and P r o p o s i t i o n
skew-field.
PROPOSITION
3. R
Theorem
r i n g of the s k e w - f i e l d
9, i.e.,
if x ~ R t h e n x -1 • R i m p l i e s P ~ T(~).
Ix c p y i e l d s
x • P contrary
to the
119
hypothesis
x ~ Ker ~ since P c Ker ~ = RM K.
Thus we have
Qo(R) D R D RM K D p = Qo(p) and this means that ~o ~ ~ by the theory of s p e c i a l i z a t i o n of pseudo-places. We are left to prove that any of the equivalent conditions that Op is a prime kernel functor.
implies
Combining T h e o r e m 109 with Pro-
p o s i t i o n 26. 3. we find that Op is a r e s t r i c t e d kernel functor, thus P c A for every A 6 C(Op). clear that Op < TR/A.
Since R/A is O p - t o r s i o n free it is
On the other hand,
if ~ e R/P is such that
Ix c p for some I e T(~R/A)
and a r e p r e s e n t a t i v e x e ~, then x ~ P
would entail
P
I c P because
is completely prime, but then A n p
would also imply that A e T(TR/A) which is a contradiction.
There-
fore R/P is ~ R / A - t o r s i o n free and Op = TR/P = TR/A follows
(for eve-
ry A e C'(o)).
COROLLARY.
Hence Op is a prime kernel functor by P r o p o s i t i o n 7.
To every prime Ker ~ of the K-algebra D, where ~ is a
pre-place of N/K defined on a subring R~, which specializes to the prime Ker ~, there corresponds a prime kernel functor ~ on M(R~) such that Q~(R~)
Vi.
4. Modules over A z u m a y a Algebras.
LEMMA 114. Let C.
= R~.
R
be a left N o e t h e r i a n A z u m a y a algebra with center
Let M,N be R-modules such that
has finite representation. such that fR-P
is of finite type where as N
If f : M ~ N is an R-module h o m o m o r p h i s m
: QR-P (M) ~ QR-P (N) is an isomorphism,
exists a T - f u n c t o r ~B such that fB for some ideal
M
B
of
R.
then there
: QB (M) ~ QB (N) is an i s o m o r p h i s m
120
PROOF.
Since
0 ~ f(M) ~ N ~ N/f(M)
QR_p(N/f(M))
= 0.
is of f i n i t e
type t h e r e
A(N/f(M))
= 0.
such that fa
Choose
that Qa(M)
Qa(N)
has
Since
°a < °R-P'
Taking
Then ORa
of f i n i t e
type and
fa ) = 0.
As b e f o r e
a
~ Qb(Qa(N))
(0) b e c a u s e
Since
is a Z a r i s k i
we o b t a i n
is an i s o m o r p h i s m .
Rab = 0 w o u l d
as in the p r e v i o u s
there
of Qa im-
type.
that fba is an i s o m o r p h i s m
because
Exactness
Ker fa is of f i n i t e
and a B = s u p { O a , a b } .
In a s i t u a t i o n
N/f(M)
Therefore
: Q b ( Q a (M))
Rb, t h e n B ~ P,
and b e c a u s e
that
= o a is a T - f u n c t e r
are Q a ( R ) - m o d u l e s
QR_p(Ker
o b such that fba
isomorphism QB(M)
we h a v e
(T) y i e l d s
Remark.
a • A n C.
and Qa(N)
B = Ran
it f o l l o w s
is an ideal A • T(OR_ P) for w h i c h
finite representation.
Ra or Rb is in perty
is o R _ p - t o r s i o n
: Qa (M) ~ Qa (N) is s u r j e c t i v e .
plies
T-functor
H e n c e N/f(M)
~ 0 is e x a c t
QB(R)
QB(M)
i m p l y that
= Qb(Qa(R))
pro-
~ QB(N).
lemma,
f
is c a l l e d
a local
o p e n set X B such that
m QB(N).
THEOREM
115.
Let
be a p r o j e c t i v e
R
be a left N o e t h e r i a n
R-module
R and put p = P n C.
of f i n i t e
T h e n M'
type.
= QR_p(M)
Azumaya Let
P
algebra
and let M
be a p r i m e
ideal
is a free R ' - m o d u l e ,
of
where
R' = QR_p(R).
PROOF.
From Theorem
son r a d i c a l
of R'
10g we d e r i v e By p r o p e r t y
that pe = Q R _ p ( p ) is the J a c o b -
(T) for oR_p, M'
= R' @ M and thus R
M' is a p r o j e c t i v e presentation. pe m R' @ C'
of f i n i t e
The e q u i v a l e n c e
pC' but b e c a u s e
of C' on R' and pC' the right,
R'-module
we h a v e pe ® R'
between
that pe m pC' @ C'
C'
M(R e) and M(C)
C' is c o m m u t a t i v e
is the same w h e t h e r
M' m pC' ®
type, h e n c e M' has
M'.
R'.
entails
and b e c a u s e
considered
f i n i t e rethat
the a c t i o n
on the left or on
Consequently,
121
Now,
R'
type, that pC'
is C ' - f r e e
therefore ~
of f i n i t e
M'
rank
and M'
is C ' - p r o j e c t i v e
is p r o j e c t i v e
of f i n i t e
: pC'
® M' ~ C' ® M' m M' d e r i v i n g C' C' is also i n j e c t i v e . Furthermore,
~ C'
M'/peM ' ~ QR_p(R/P)
®
type
from
of f i n i t e
and this
the
entails
injection
M'
R'
is a p r o j e c t i v e QR_p(R/P)
QR_p(R/P)-module
is s i m p l e
this
implies
So we end up w i t h
the
1 ° . M' has
presentation
finite
2 ° . The
R'-linear
3 ° . The
R'/pe-module
and
since
a free
pe
is the
116.
free.
If
of c o n s t a n t
rank.
PROOF.
F(R)
F(R)
Let
~ M ~
T-funetor
OB such
corresponds
given
by r(P)
open
stant
: pe ® R'
because
M ' / p e M ' is Q R _ p ( R / P ) - f r e e .
situation
as a left
M' ~ M'
and
:
R'-module,
is i n j e c t i v e ,
M ' / p e M ' is free,
Jacobson
If R
M
radical
of R',
is R - p r o j e c t i v e
is m o r e o v e r
be a free
0 be exact.
there
two
following
that
type
this
implies
that
M'
is
is
lo-
R'-module.
PROPOSITION cally
~
of f i n i t e
in that
is an
that
QB(F(R))
thus
a locally
= [QR_p(M)
subsets
a prime
R-module
There
of Spec
case.
of f i n i t e ring
B
m F(QB(R)) constant
: QR_p(R)]. R intersect
then
of f i n i t e ideal
If
type M
rank of
R
R
free
and
let
and
an a s s o c i a t e d
To an M e M(R) r
: Spec
is a p r i m e
non-trivially
M
is l o c a l l y
m QB(M). function
then
ring
and thus
R ~ then r
any
is con-
REFERENCES
[A.1] S.A. AMITSUR, vol.
On Rings of Quotients,
8, p. 149-164.
[A.2] M. AUSLANDER,
FOR THE APPENDIX
Academic Press, London 1972
O. GOLDMAN, The Brauer Group of a Commutative
Ring, Trans. Americ. Math. [A.3] G. AZUMAYA,
Symposia Mathematiea,
Soc., vol.
97 (1960), p. 367
On Maximally Central Algebras, Nagoya Math. J.,
vol. 2 (1951), p. 119-150 [A.4] I.N. HERSTEiN, Noncommutative Rings, The Carus Mathematical Monographs,
number 15, Math. Assoc.
[A.5] M. KNUS, M. OJANGUREN, Azumaya,
of America,
1968.
Th~orie de la Descente et Alg~bres d'
Lecture Notes in Math.
389, Springer-Verlag,
1974
[A.6] L.W. SMALL, Orders in Artinian Rings, J. of Algebra, vol. 4 (1966), p. 18.
REFERENCES
[i]
A.A. ALBERT,
New Results on Associative
Division Algebras,
J. of Algebra ~ (1967), pp. 110-132. [2]
M. AUSLANDER,
O. GOLDMAN,
Trans. Amer. Math. [3]
G. AZYMAYA,
Maximal
Orders,
Soe. 97 (1960), pp. 1-24.
On Maximally
Central Algebras,
Nagoya Math. J. ! (1951), pp. 119-160. [4]
A.W. CHATTERS,
S.M. GINN, Localization
in Hereditary
Rings,
J. of Algebra 22 (1972), pp. 82-88. IS]
A.W. CHATTERS, Hereditary Proc.
[8]
A.G. HEiNICKE,
Noetherian
London Math.
I.G.CONNELL,
Localization
at a Torsion Theory in
Rings,
Soe. XXVII,
1973.
A Natural Transform
of the Spee Functor,
J. of Algebra 10 (1968), pp. 69-91. [7]
M. DEURING, biete,
[8]
Algebren,
Ergebnisse
der Mathematik
und ihrer Grenzge-
vol. 41, Springer Verlag 1968.
V. DLAB, Rank Theory of Modules, Fund. Math. 64 (1969), pp. 313-324.
[9]
P. GABRIEL, Bull.
Des Categories
Soc. Math.
Abeliennes,
France 90 (1962), pp. 323-448.
[10] A.W. GOLDIE, A Note on Non-commutative
Localization,
J. of Algebra 8 (1968), pp. 41-44. [11] A.W. GOLDiE,
Localization
in Non-commutative
J. of Algebra ~ (1967), pp. 89-105. [12] O. GOLDMAN,
Rings and Modules
of Quotients,
J. of Algebra 13 (1969), pp. 10-47.
Noetherian
Rings,
124
[13] A.G.
HEINICKE,
On the Ring of Quotients at a Prime Ideal of a Right
N o e t h e r i a n Ring, Canad.
J. Math.
[14] I.N. HERSTEIN,
24 (1972), pp.
Noncommutative
703-712.
Rings,
The Carus Math. Monographs 15, Math. Assoc. Amer.
[15] K. HOECHSMAN, Algebras Proc. Amer.
Math.
Split by a Given Purely Inseparable Field,
Soc. 14 (1963), pp.
[16] W. KUYK, P. MULLENDER, Indag. Math. 25, Nr.
1968.
768-776.
On the Invariants of Finite Abelian Groups,
2, 1963.
[17] W. KUYK, Generic C o n s t r u c t i o n of a Class of Non Cyclic Division Algebras, J. Pure and A p p l i e d Algebra
[18] J. KUZMANOVITCH, J. of A l g e b r a 2 1
[19] J. LAMBEK, Waltham,
2 (1972), pp.
Localization of D e d e k i n d Prime Rings, (1972), pp.
378-393.
Lectures on Rings and Modules,
Toronto,
London,
1966.
[20] J. LAMBEK, Torsion Theories, A d d i t i v e Quotients,
121-131.
Semantics and Rings of
Lecture Notes in M a t h e m a t i c s
177
(1971).
[21] J. LAMBEK, G. MICHLER, The Torsion Theory at a Prime Ideal of a Right N o e t h e r i a n Ring, J. of A l g e b r a 25,
(1973), pp.
364-389.
[22] L. LESIEUR, R. CR01SOT, Alg~bre N o e t h 6 r i e n n e N o n - c o m m u t a t i v e , M~mor.
Sci. Math.
[23] E. MATLIS,
154 (1963).
Injective Modules over N o e t h e r i a n Rings,
Pacific J. Math. ~ (1958), pp. 611-528.
[24] D.C. MURDOCH, Canad.
Contributions
J. of Math., vol.
[25] D.C. MURDOCH,
F. VAN
and Kernel Functors,
to N o n c o m m u t a t i v e
6, Nr.
Ideal Theory,
1, 1952.
0YSTAEYEN, A Note on Reductions of Modules Bull. Math.
Soc.
Belg., to appear.
125
[26] D.C. MURDOCH, Sheaves,
F. VAN OYSTAEYEN,
J. of Algebra,
[27] D.C. MURDOCH, Quasi-primes,
F. VAN OYSTAEYEN, Indag.
Math.,
[28] M. NAGATA,
Local Rings,
[29] L. SILVER,
Noncommutative
J. of Algebra [30] S.K.
Math.
Math.
and
Symmetric
Kernel Functors
and
Localization
pp.
1962.
and Applications,
44-76. Kernel Functors with Property
T,
Soc., to appear.
SIM, Noncommutative
Proc. Amer.
Localization
to appear.
Wiley-lnterscience
SIM, Prime Idempotent
Proc. Amer. [31] S.K.
7 (1967),
Noncommutative
to appear.
Localizations
and Prime
Ideals,
Soe.
[3~] H. STORRER, On Goldman's
Primary Decomposition,
Lectures
on Rings
and Modules, Lect. [33] M.E. Ann.
Notes
in Mathematics
SWEEDLER, of Math.
Structure
Soc. Math.
[35] F. VAN OYSTAEYEN, Bull.
Soc. Math.
[36] F. VAN OYSTAEYEN, Characteristic Indag.
On Pseudo-places Belg., XXV,
1972.
Extensions,
of Algebras,
1973, pp.
139.
Generic Division Algebras, Belg., XXV,
1973, pp.
The p-component
259-285.
of the Brauer Group of a Field of
p ~ 0,
Math. 36, Nr.
[37] F. VAN OYSTAEYEN,
1, 1974, pp.
Primes
J. Pure Applied Algebra, [38] F. VAN OYSTAEYEN, to appear.
of Inseparable
Berlin
87 (1968), p. 401.
[34] F. VAN OYSTAEYEN, Bull.
246, Springer-Verlag,
67-76.
in Algebras
over Fields,
to appear.
Extension
of Ideals under Symmetric
Localization,
SUBdECT
Absolutely
restricted,
Absolutely
special,
Absolutely
unramified,
Affine
scheme,
Artin-Rees Atom,
76
76 76
47
condition,
56
14
Azumaya algebra, Basic group,
106.
96
Bilateral kernel functor, Bimodule, C(o),
9
26.
20
Closed ideal,
11
o-Closed
ideal,
Critical
left ideal,
11 11
Crossed product algebra, Dimension
of a k-pseudo-place,
Envelloping
algebra,
Epireduction, Faithfully Filter,
69.
105
32
o-injective,
5
4
Final morphism,
31
Final reduction, Galoisian
33.
pseudo-place,
Generic algebras, Generic point,
66
42
Geometric
functor,
Geometric
set, 49
Geometric
stalk,
49
49.
69
59
INDEX
127
Ideal,
4
o-Ideal,
21
Idempotent
kernel functor,
Inf of kernel functors, o-lnjective,
9
S
Irreducible Isomorphic
5
set, 42 pseudo-places,
64
Isomorphism of affine schemes, Jacobson radical,
14.
Kernel functor,
4
K-pseudo-place,
59.
Left Artinian
condition,
Left Noetherian
ring,
Left Ore condition, Left P-primary
15
ideal,
Local isomorphism,
20
120.
Maximal pseudo-place, Minimal prime,
64
73
Modular extension, Module,
15
8
96
4
Morphism of affine schemes, M-system, o-Perfect,
18
Pre-place,
62
Prime
(of an algebra),
G-Prime,
71
71
Prime ideal,
8
Prime kernel functor, Projective
module,
~-Projective Property
49
10.
13
7
module,
7
(T), 11
Pseudo-place, Quasi-prime
58.
kernel functor,
Quasi-support,
26.
27
49.
128
Rank of an unramified pseudo-place, Reduction of modules, Reduction of rings,
31
32
Restricted kernel functor, Restricted pseudo-place, Ring of quotients,
7.
Separable algebra,
105
Site of an algebra,
24
58
78
Specialization of a prime,
73
Specialization of pseudo-places, Special pseudo-place,
62
58
Stalk, 48 Structure sheaf, 47 Sup (of kernel functors), Support,
9
13
Symmetric kernel functor,
9
Symmetric prime of an algebra, T-basis,
50
Tensorproduct of pseudo-places, T-functor,
7
v-Topology, ~-Torsion~
5 4
u-Torsion free, 4 Torsion morphism, Torsion reduction,
33 33
T-set, 49 T-stalk,
71.
49.
Unramified pseudo-place, Zariski topology,
42.
58.
65
59
