7V
Higher algebraic
K-theory: I
Daniel Quillen*
The purpose of this paper is to develop a higher
K-theory for additive categories
with exact sequences which extends the existing theory of the Grothendieck group in a
natural way.
To describe the approach taken here, let
M
embedded as a full subcategory of an abelian category
extensions in
A . Then one can form a new category
but in which a morphism from
subquotient
MI/M °
of
M,
M'
to
where
M
M .
=
category
has a classifying space
Q(M)
M=
is closed under
Q(M) having the same objects as
is taken to be an isomorphis~ of
Mo~M
are objects of
be an additive category
A= , and assume
I
are subobjects of
M
such that
Assuming the isomorphiom classes of objects of
BQ(~)
M'
M=
M ,
with a
M°
and
M/M I
form a set, the
determined up to homotopy equivalence.
One can show that the fundamental group of this classifying space is canonically isomorphic to the Grothendieck group of
~ , which motivates defining a sequence of
K-groups by
the formula
El(~)
=
~i+I(BQ(M),O) •
It is the goal of the present paper to show t~mt this definition leads to an interesting
theory.
The first part of the paper is concerned with the general theory of these
K-groups.
Section I contains various tools for working with the classifying space of a small
category.
It concludes with an important result which identifies the homotopy-theoretic
fibre of the map of classifying spaces induced by a functor.
to obtain long exact sequences of
One notes that the category
short sequences
category.
K-theory this is used
K-groups from the exact homotopy sequence of a map.
Section 2 is devoted to the definition of the
ties.
In
Q(M)
O - @ M ' -* M - , M " --~O
K-groups and their elementary proper-
depends only on
in
M=
M
and the family of those
which are exact in the ambient abelian
In order to have an intrinsic object of study, it is convenient to introduce
the notion of an exact category, which is an additive category equipped with a family of
short sequences satisfying some standard conditions (essentially those axiomatized in
[Heller]).
of
For an exact category
K-groups
Ki(M)
M
with a set of isomorphism classes one has a sequence
varying functorially with respect to exact functors.
contains the proof that
Ko(M)= is isomorphic to the Grothendieck group of
Section 2
also
M=. It should
be mentioned, however, that there are examples due to Gersten and Murthy showing that in
general
KI(M)
is not the same as the universal determinant group of Bass.
The next three sections contain four basic results which might be called the
exactness, resolution, devissage, and localization theorems.
a well-known result for the Grothendieck group
Each of these generalizes
([Bass, Ch. VIII]), and, as will be
apparent from the rest of the paper, they enable one to do a lot of
K-theory.
The second part of the paper is concerned with applications of the general theory to
rings and schemes.
Given a ring (resp. a noetherian ring) A , one defines the groups
• Supported in part by the Rational Science Foundation.
85
78
Ki(A)
(resp. K~(A) )
tive
2
to be the K-groups of the category of finitely generated projec-
A-modules (resp. the abelian category of finitely generated
canonical map
theorem.
Ki(A ) --> K~(A)
which is an isomorphism for A
There is a
regular by the resolution
Because the devissage and localization theorems apply only to abelian categories,
the interesting results concern the groups
K~(A)
for A
A-modules).
=
K:(A[t])
,
K~(A) . In section 6
K:(A[t,t-11)
=
we prove the formulas
K:(A)~K:_,(A)
noetherian, which entail the corresponding results for K-groups when
regular.
A
is
The first formula is proved more generally for a class of rings with increasing
filtration, including some interesting non-co,,,utative rings such as universal enveloping
algebras.
To illustrate the generality, the K-groups of certain skew fields are computed.
For a scheme (resp. noetherian) scheme X, the groups
Ki(X )
(resp. K~(X) ) are
defined using the category of vector bundles (resp. coherent sheaves) on X, and there is
a canonical map Ki(X) -*K~(X)
devoted to the K'-theory.
which is an isomorphism for X
regular.
Section 7
is
Especially interesting is a spectral sequence
cod(x) = p
obtained by filtering the category of coherent sheaves according to the codimension of the
support.
In the case where
X
program proposed by Gersten
is regular and of finite type over a field, we carry out a
at this conference ([Gersten 3]), which leads to a proof of
Bloch's formula
AP(x)
proved by Bloch in particular
p
=
HP(x, Kp(_Ox))
([Blech]), where
cases
cycles modulo linear equivalence.
AP(x)
is the group of codimension
One noteworthy feature of this formula is that the
right side is clearly contravariant in X, which suggests rather strongly that higher
K-theory might eventually provide a theory of the Chow ring for non-quasi-projective
regular varieties.
Section 8
contains the computation of the K-groups of the projective bundle
assOciated to a vector bundle over a scheme. This result generalizes the computation of
the Grothendieck groups given in ~SGA 63, and it may be viewed as a first step toward a
higher K-theory for schemes, as opposed to the K'-theory of the preceding section.
proof, different from the one in
The
[SGA 6], is based on the existence of canonical
resolutions for regular sheaves on projective space, which may be of some independent
interest.
The method also permits one to determine the K-groups of a Severi-Brauer
scheme in terms of the K-groups of the associated Azumaya algebra and its powers.
This paper contains proofs of all of the results announced in [Quillen I], except for
Theorem I
of that paper, which asserts that the groups K (A)
obtained by making BGL(A)
into an
H-space (see [Gerstenl5]).
here agree with those
From a logical point of
view, this theorem should have preceded the second part of the present paper, since it is
used there a few times. However, I recently discovered that the ideas involved its proof
could be applied to prove the expected generalization of the localization theorem and
86
79
fundamental theorem for non-regular rings
3
[Bass, p.494,663~.
These results will appear
in the next installment of this theory.
The proofs of Theorems A and B given in section I owe a great deal to conversations
with Graeme Segal, to whom I am very grateful.
One can derive these results in at least
two o t h e r ways, using cohomology and the Whitehead theorem as in
[FriedlanderJ , and also
by means of the t h e o r y of minimal f i b r a t i o n s of s i m p l i c i a l s e t s .
The p r e s e n t approach,
based on t h e Dold-Thom t h e o r y o f q u a s i - f i b r a t i o n s , i s q u i t e a b i t s h o r t e r than the o t h e r s ,
a l t h o u g h i t i s not a s c l e a r as I would have l i k e d , s i n c e t h e main p o i n t s a r e i n the
references.
Someday t h e s e i d e a s w i l l undoubtedly be i n c o r p o r a t e d i n t o a g e n e r a l homotopy
theory f o r topoi.
This paper was p r e p a r e d with the e d i t o r ' s encouragement during the f i r s t
o f 1973.
I mention t h i s because the r e s u l t s i n
~7
on G e r s t e n ' s c o n j e c t u r e and B l o c h ' s
formula, which were d i s c o v e r e d a t t h i s time, d i r e c t l y a f f e c t the papers
and
[Bloch]
i n t h i s p r o c e d i n g s , which were prepared e a r l i e r .
87
two months
[Qersten 3, 4 ]
80
CONTENTS
First part:
81.
Generaltheor ~
The classifyin~space of a small cate6ory
Coverings of BC
and the fundamental group
The homology of the classifying space
Properties of the classifying space functor
~2.
Theorem A:
Conditions for a functor to be a ho~otopy equivalence
Theorem B:
The exact homotopy sequence
The
K-~roups
of,,,,an exact
category
Exact categories
The category
Q(~)
and its universal property
The fundamental group of
Q(M)
and the Grothendieck group
Definition of Ki(~)
Elementary properties of the K-groups
§3.
Characteristic exact sequences and filtrations
~4.
Reduction b~Resolution
The resolution theorem
Transfer maps
§5.
Devissa~e and localization in abelian categories
Second part: Applications
~6.
Filtered rin~s and the homotopy property for regular rings
Graded rings
Filtered rings
Fundamental theorem for regular rings
K-groups of some skew fields
§7.
K'-theor~ for schemes
The groups
Ki(X ) and
K~(X)
Functorial behavior
Closed subschemes
Affine and projective space bundles
Filtration by support
Gersten's conjecture for regular local rings
Relation with the Chow ring (Bloch's formula)
Pro~ective fibre bundles
The canonical resolution of a regular sheaf on
PE
The projective bundle theorem
The projective line over a (not necessarily commutative) ring
Severi-Brauer schemes and Azumaya algebras
8S
81
~I.
5
The classifyin~ space of a small category
In the succeeding sections of this paper
K-groups will be defined as the homotopy
groups of the classifying space of a certain small category.
In this rather long section
we collect together the various facts about the classifying space functor we will need.
All of these are fairly well-known, except for the important Theorem
B
which identifies
the homotopy-fibre of the map of classifying spaces induced by a functor under suitable
conditions.
It will later be used to derive long exact sequences in
K-theory from the
homotopy exact sequence of a map.
Let
whose
C=
be a small category.
Its nerve, denoted
p-simplices are the diagrams in
C
NC= , is the (semi-)simplicial set
of the form
X ° - - , X I --~ ... ~ X p
The
i-th face (resp. degeneracy) of this simplex is obtained by deleting the object
X.
l
(resp. replacing
denoted
Xi
by
id : Xi--, X i) in the evident way.
BC,
is the geometric realization of
=
in one-one correspondence with the
NC.
=
For example, let
Then
BJ
elements of
J
CW
complex whose
C=,
p-cells are
p-simplices of the nerve which are nondegenerate, i.e.
such that none of the arrows is an identity map.
way.
It is a
The classifying space of
(See [Segal I] ,[Milnor I].)
be a (partially) ordered set regarded as a category in the usual
is the simplicial complex (with the weak topology) whose vertices are the
J
and whose simplices are the totally ordered non-empty finite subsets of
Conversely, if
K
is a simplicial complex and if
K, then the simplicial complex
BJ
J
J.
is the ordered set of simplices of
is the barycentric subdivision of
K.
Thus every
simplicial complex (with the weak topology) is homeomorphic to the classifying space of
some, and in fact many, ordered sets.
Furthermore, since it is known that any
CW
complex
is homotopy equivalent to a simplicial complex, it follows that any interesting homotopy
type is realized as the classifying space of an ordered set.
(I am grateful to Graeme
Segal for bringing these remarks to my attention.)
As another example, let a group
usual way.
sense.
Then
BG
G
be regarded as a category with one object in the
is a classifying space for the discrete group
It is an Eilenberg-MacLane space of type
G
in the traditional
K(G,I ), so few homotopy types occur in
this way.
Let
X
be an object of
C.
=
Using
BC=, we have a family of homotopy groups
groups of
C
with basepoint
X
X
to denote also the corresponding
O-cell of
~i(BC=,X), i ~ O , which will be called the homotopy
and denoted simply
Tri(C,X).
a group, but a pointed set, which can be described as the set
Of course,
Ir C
~To(C,X) is not
of components of the
O=
category
BC
C= pointed by the component containing
X.
are in one-one correspondence with components of
We will see below that
TrI(C=~)
In effect, connected components of
C.
and also the homology groups of
BC= can be defined
"algebraically" without the use of spaces or some closely related machine such as semi~dmplicial homotopy theory, or simplicial complexes and subdivision.
The existence of
similar descriptions of the higher homotopy groups seems to be unlikely, because so far
89
82
6
nobody has produced an "algebraic" definition of the homotopy groups of a simplicial
complex.
Coverings of BC= and the fundamental group.
Let
E
be a covering space of
fibre of
E
over ~ considered as a O-cell of BC.
determines a path from X
to X'
~-) E(X').
It is easy to see that
X ~-~ E(X)
from C
to
Sets
BC=. For any object X
in
of C,
let
=
If u : X -. X'
E(X)
denote the
is a map in C, it
BC=, and hence gives rise to a bijection
E(u): E(X)
E(fg) = E(f)E(g), hence in this way we obtain a functor
which is morphism-inverting, that is, it carries arrows into
isomorphisms.
Conversely, given
with X
in C_ and
such that
spaces
F : C --* Sets, let F \ C
x E F(X), in which a morphism
F(u)x = x'.
[Gabriel-Zisman, App.l, 3.2]
of
(X,x) --~ (X',x')
The forgetful functor F \ C -~ C
B(F\C) --~BC= having the fibre
ting, the map
denote the category of pairs
B(F\C)-~BC=
F(X)
over X
is a map
(X,x)
u : X--PX'
induces a map of classifying
for each object X.
it is not difficult to see that when
is locally trivial, and hence
F
Using
is morphism-inver-
B(F\C) is a covering space
BC=. It is clear that the two procedures just described are inverse to each other,
whence we have an equivalence of categories
(Coverings of BC=) ~
(Morph.-inv. F : C_ --~ Sets)
where the latter denotes the full subcategory of
from C=
to
let
G
=
Sets, consisting of the morphism-inverting functors.
= C_-- [(ARC)
-I]
=
denote the groupoid obtained from C=
the inverses of all the arrows
to G: m
Funct(C, Sets), the category of functors
by formally adjoining
[Gabriel-Zisman, I, 1.1] . The canonical functor from C
induces an equivalence of categories
Funct(_G, Sets)
(loc.clt., I, 1.2). Let X
--
(Morph.-inv. F : C --~ Sets)
be an object of C
and let
Gx
be the group of its auto-
morphisms as an object of =G" When C= is connected, the inclusion functor
Gx -, =G is
an equivalence of categories, hence one has an equivalence
Funct(=G, Sets) - ~
Funct(Gx, Sets)
=
(Gx-sets).
Therefore by combining the above equivalences, we obtain an equivalence of categories of
the category of coverings of
E ~
E(X).
morphism:
BC= with the category of
Gx-sets given by the functor
By the theory of covering spaces this implies that there is a canonical iso~I(C=,X) ~
GX.
The same conclusion holds when C= is not connected, as both
groups depend only on the component of C=
containing X.
Thus we have established the
following.
Proposition l.
The category of covering s~aces of
the category of mor~hism-inverting functors
thing, the categor~
Funct(G, Sets), where
BC= is canonically equivalent to
F : C --b Sets, or what amounts to the same
__G =
C [(ArC=)-~
is the groupoid obtained by
formall[ inverting the arrows of C=. The fundamental group ~TI (C,X)
isomorphic to the group of automorphisms of X
is canonically
as an object of the grou~oid
It follows in particular that a local coefficient system
L
G.
of abelian groups on BC
=
may be identified with the morphism-inverting functor X ~-~ L(X) from C= to abelian groups.
90
83
The homology of
7
BC
=
It is well-known that the homology and cohomology of the classifying space of a discrete group coincide with the homology and cohomology of the group in the sense of homological algebra.
We now describe the generalization of this fact for an arbitrary small
category.
Let
be a functor from C=
A
to
Ab, the category of abelian groups, and let
Hp(C,A)_ denote the homology of the simplicial abelian group
Cp(C,A)
of chains on
gC
=
II
with coefficients in
associated normalized chain complex. )
C
Hp(C,A} =
where
I~ ~
Ftmct(C,Ab)
A(X o)
A.
(By the homology we mean the homology of the
Then there are canonical isomorphisms
lim_.~p(A)
denotes the l e f t derived f u n c t o r s of the r i g h t exact f u a c t o r
to
Ab.
This is proved by showing that
which coincides with
lim
i~
from
A ~-> H.(C=,A) is an exact ~-functor
in degree zero and is effaceable in positive degrees.
(See
[Gabriel-Zisaum, App.II, 3.3J • )
Let
H.(BC= ,L)
denote the s i t a r
coefficient system
where we i d e n t i f y
f i l t e r i n g the
L.
homology of
BC
=
with coefficients in a local
Then there are canonical isomorphisms
L with a morphlem-inverting f u n c t o r as shove.
CW complex
spectral sequence.
=
I
One has
E I = O for q ~ O and
Pq
C.(C=,L). (Compare [Segal I, 5.1].)
plex associated to
This may be proved by
BC by means of i t s s k e l e t a and c o n s i d e r i n g the a s s o c i a t e d
E.o = the normalized chain comThe spectral sequence degenerates
yielding the desired isomorphism.
Thus we have
C
and similarly we have a canonical isomorphism for cohomology
(2)
HP(~=,L)
= ~(L)
where l i ~
denotes the right derived functors of the left exact functor
~kuTct(C,Ab)
tO
lim
from
Ab.
Properties of the classifying space functor.
From now on we use the letters
f : C -,~C'
C, C'
etc. to denote small categories.
is a functor, it induces a cellular map
Bf : BC -+BC'.
If
In this way we
obtain a faithful functor from the category of small categories to the category of
complexes and cellular maps.
This functor is of course not fully faithful.
CW
As a particu-
larly interesting example, we note that there is an obvious canonical cellular homeomorphism
(3)
where
~c
C°
=
~c °
is the dual category, which is not realized by a functor from C
except in very special cases, e.g. groups.
91
to
C°
84
8
By the compatibility of geometric realization with products [Milnor I] , one knows
that the canonical map
(4)
---~ ~ ----. x ~ ='
B(C x C')
is a homeomorphism if either
BC
or
BC'
is a finite complex, and also if the
product is given the compactly generated topology.
As pointed out in
£Segal I], this
implies the following.
2.
A natural transformation
induces a homoto~y
BC= x I ~
In effect, the triple
is the ordered set
[O<I~
BC_' between
(f,g,@)
e : f -@ g
Bf
and
of functors from
can be viewed as a functor
, and
B|
C__.
to
C'
=
- -
Bg.
C x ! --~ C_ ', where
1
is the unit interval.
We will say that a functor is a h omoto~y e~uivalence if it induces a homotopy equivalence of classifying spaces, and that a category is contractible if its classifying
space is.
I.
If a functor
f
has either a left or a right ad~oint, the_._nn f
is a
homotopy equivalence.
For if
f~
f'f -@ id, id
is say left adjoint to
-~ ff', whence
Corollary 2.
Bf'
f, then there are natural transformations
is a homotopy inverse for
Bf.
A cate~or~ havin~ either an initial or a final obseot is contractible.
For then the functor from the category to the punctual category has an adjoint.
Let
I
and let
be a small category which is filtering (= non-empty + directed
i ~-~ C i
limit of the
be a functor from
Ci;
to small categories.
Let
[Bass, p.41])
C= be the inductive
because filtered inductive limits commute with finite projective limits,
we have 0~= = lira...,
ObCi,= ArC= = ~
0bC
I
~i'
~d
more ~,~erally ~C= = lira_.,.
NCi . Let X i
be a family of objects such that for every arrow
i --~ i'
in
I, the induced
=l
functor
C i --)C=i, carries
indexed by
Xi
to
Xi, , whence we have an inductive system
Proposition 3.
If
X
is the common ~ e
i_~ ,,(~i,xl)
Proof.
subset of
~rn(Ci,X i)
I.
Because
of the
Xi
in
C, then
= ,,(c=,x).
I is filtering and
NC= = lim~ NC i , it follows that any simplicial
NC= with a finite number of nondegenerate simplices lifts to
i, and moreover the lifting is unique up to enlarging the index
As every compact subset of a
every compact subset of
CW
i
NCi=
for some
in the evident sense.
complex is contained in a finite subcomplex, we see that
BC= lifts to
BC=i for some
i, uniquely up to enlarging
i.
The
proposition follows easily from this.
Coroll~
functor
I.
Suppose in addition that for every arrow
C i -~ C=i,
is a homotopy equivalence.
equivalence for each
Proof.
suppose
map of
i
CW
Replacing
Then the functor
~
I
the induced
C=i -9 C= is a homotopy
i.
I
by the cofinal category
is the initial object of
complexes
i -@ i'
I.
i~I
of objects under
i, we can
It then follows from the proposition that the
BC__i --, BC= induces isomorphisms on homotopy.
homotopy equivalence by a well-known theorem of Whitehead.
92
Hence it is a
85
Corollary 2.
Any filtering category is contractible.
In effect,
I/i
9
I
is the inductive limit of the functor
of objects over
i
i ~-~ I/i , and the category
has a final object, hence is contractible.
Sufficient
conditions
for ai functor I to be a . homotopy equivalence.
it
ii
Let
C'
_--
by
f : C= --*C' be a functor and denote objects of
Y, Y' P etc.
sisting of pairs
is a map
If
(X,v)
w : X --*X'
functor of
Y
is a fixed object of
with
f/Y
Theorem A.
the functor
f
f(w)v = v'.
consisting of pairs
If the category
denote the category con-
Y\f
Y\C'
of objects under
(X,u)
with
Example.
Y.
to
(X',v')
is the identity
Similarly one defines
u : fX --~Y.
is contractible for ever~ object
Y
of
(3), this result admits a dual formulation to the effect that
Let
g : K --, K'
f/Y
C', then
is homeomorphic to
f/~
Bf.
If
f
is a
are contractible.
he a simplicial map of simplicial complexes, and let
be the induced map of ordered sets of simplices in
K', then
(X,v)
f
is a homotop~ equivalence.
In view of
of
X, X', etc. and objectsof
In particular, when
homotopy equivalence when all of the categories
f : J -~ J'
by
Y\f
v : Y --~fX, in which a morphism from
such that
C', we obtain the category
the category
C
C',
let
=
~- denotes the element of
is the ordered set of simplices in
K
and
K', so that
g
J' corresponding to a simplex
g-1(o-).
In this situation the
theorem says that a simplicial map is a homotopy equivalence when the inverse image of
each (closed) simplex is contractible.
Before proving the theorem we derive a corollary.
fibred and cofibred categories
the fibre of
f
the identity of
C'
over
Y
[SGA I, Exp. VI]
Y, that is, the subcategory of
by
f.
It is easily seen that
First we recall the definition of
in a suitable form.
f
C: =
makes
~-1(y)
v : Y -. Y'
~ Y\f
Y
(X,v) ~
f
the functor
v'X, we obtain for any map
u'v* -. (vu)*
u,v
v.
The prefibred category
of composable arrows in
is an isomorphism.
C', the
We will call such
prefibred and fibred respectively.
Dually, f
f-1(y) ..,f/y
f-1(y,)
C'
-7 f-, (y)
C'
is a fibred
category if for every pair
=
,,
canonical morphism of functors
functors
of
a functor
v* • f-1 (y,)
over
denote
, x~.(x,i~)
Denoting the adjoint by
determined up to canonical isomorphism, called base-chsnw~,e by
C: =
f-1(y)
C__ a prefibred category over
in the sense of loc.cit, if and only if for every object
has a right adjcint.
Let
whose arrows are those mapped to
makes
induced by
category when
C=
into a precofibred cate~or~ over
have left adjoints
v : Y -~Y'
(vu). ~ v . u ,
(X,v) ~-~v.X.
is called c obase-chanse by
for all composable
C'
=
when the functors
In this case the functor
u,v.
v, and
Such functors
C
v.: f-1(y) -->
is a cofibred
f
will be called
precofibred and cofibred respectively.
Corollary.
Suppose that
contractible for every
This
follows from
Y.
f
is either prefibred or precofibred, and that
The.._~n f
is a homotopy equivalence.
Prop. 2, Cor. I.
93
f-1 (y)
is
86
Example.
Let
S(C)
which a morphism from
such that
be the category whose objects are the arrows of
u : X --~Y
u' = wuv.
10
(Thus
to
S(C)
fibres defined by the functor
CO
<
=
u': X' -+Y'
is a pair
is the cofibred category over
(X,Y) ~-~ Hom(X,Y).)
s
S(C)
_--
t
C , and in
=
v : X' --~X, w : Y -~Y'
COxC
with discrete
One has functors
~ C=
given by source and target, and it is easy to see ~hat these functors are cofibred.
categories
s-m(x) = X \ C
tible. Therefore
s
and
and
t
The
t-I(Y) = (C_/y)o have initial objects, hence are contracare homotopy equivalences by the corollary.
This construc-
tion provides the simplest way of realizing by means of functors the homotopy equivalence
(3).
We now turn to the proof of Theorem A.
We will need a standard fact about the
realization of bisimplicial spaces which we now derive.
Let
0rd
be the category of ordered sets
p ={0<I<
definition simplicial objects are functors with domain
from simplicial spaces to spaces
where
p6M, so that by
([Segal I]) may be defined as the functor left adjoint
to the functor which associates to a space
Horn denotes function space and
vertices.
..~p},
0rd °. The realization functor
Y
/~P
the simplicial space
is the simplex having
p ~-~ H o m ( A p, Y),
p
as its set of
In particular the realization functor commutes with inductive limits.
Let
T : ~, ~
to spaces.
~-~ Tpq
be a bisimplicial space, i.e. a functor from ~==0rd°x0rd
°===
Realizing with respect to
q
keeping
p
fixed, we obtain a simplicial space
P ~-, j q ~-k TpqJ which may then be realized with respect to p . Also, we may realize
first in the p-direction and then in the q-direction, or we may realize the diagonal
simplicial space
p ~-~ Tpp . It is well-known (e.g. [ Tornehave] ) that these three
procedures yield the same result:
Lemma.
There are h,,omeomerphisms
whi.ch a r e f u n c t o r i a l
Proof.
i n 'the s i m p l i c i a l
Suppose first that
T
h rs • S : (p,q) ~
where
S
space
T.
is of the form
Hom(p,r) • Hom(q,m) x S
is a ~ v e n space. Then
I • ~ ' * ~ o m ( , , ' ) • Horn(,,.) • s ] =
(This is the basic h o m e o m o ~
A r • A s ~ s.
used to prove that geometric realization commutes with
i
products
[Milnor I].)
1 P ~ Iq~,
On the other hand, we have
Hom(,,~-) • ~ o m ( q . . ) •
I
sli
. Ar•
s•
and similarly for the double realiza~iom taken in the other order. Thus the required
functorial homeomorphlsms exist on the full subcategory of bisimplicial spaces of this
form.
94
11
87
But any
T
canonical
has a
h
(r,=)-~(r'
presentation
x Trs
~
hrSx Trs
,l' )
.
(r,.)
which is exact in the sense that the right arrow is the cokernel of the pair of arrows.
Since the three functors from bisimplicial spaces to spaces uader consideration commute
with inductive limit8 , the lamina follows.
Proof of Theorem A.
with
X
(X,Y,v)
an object of
to
C
S(f)
and
v : Y --> fX
be the category whose objects are triples
(Thus
(X,Y) ~-~ Hem(Y,fX).)
S(f)
P2
Pl
S(f)
p~(X,Lv) = x, p2(X,Lv) =
T(f)
u : X --~X', w : Y' -->Y
is the cofibredcatagory
(X,Y,v)
C', and in which a morphism from
over
such that
C= x C= '° defined by the functor
We have functors
~,o <
given by
a map in
(X',Y',v') is a pair of arrows
v' = f(u)vw.
Let
Let
~C
Y.
be the bisimplicial set such that an element of
T(f)pq
is a pair of
diagrams
(Yp
in
C'
and
=
C
~ "'" -'~Yo --* fX o 0 X ° --~ ... --#Xq)
respectively, and such that the
=
deletes the object
i-th face in the
Yi (rasp Xi) in the obvious way.
p-(resp, q-)direction
Forgetting the first component gives
a map of bisimplicial sets
(-)
T(f)pq ......~
N%
where the latter is constant in the
T(f)
map
is the nerve of the category
BPl : BS(f) ~
BC= .
p-directlon.
Since the diagonal simplicial set of
S(f), it is clear that the realization of
(*)
is the
(By the realization of a bisimplicial set we mean the space
described in the above lemma, where the bisimplicial set is regarded as a bisimplicial
space in the obvious way.)
On the other hand, realizing
(*)
with respect to
p
gives
a map ug simplicial spaces
IL
B(~'/fXo)°
~
Xo-.,~...-~Xq
which is a homotopy equivalence for each
object.
pt = ~c
q
Pl
we see
the realization of
(*)
C='/fXo
has a final
([Tornehave, A.3] ), or the lemma
is a homotopy equivalence.
Thus the
is a homotopy equivalence.
Similarly there is a map of bisimplicial sets
is the map
=q
because the category
Applying a basic result of May and Tornehave
below (Th. B),
functor
~
Xo--~......~Xq
T(f)pq --~ N(C'°)p
BP2 : BS(f) .....~.. BC='° . Realizing with respect to
whose realization
q, we obtain a map of
simplicial spaces
Yo*-""~"Yp
Io"-" "*"Yp
which is a homotopy equivalence for each
tible by hypothesis,
p, because the categories
Thus we conclude that the functer
95
P2
Y\f
are contrac-
is a homotopy equivalence.
12
88
But we have a co~utative diagram of categories
P2
C'O ~
,
P2
c o ~_
P~
S(f)
Pl
s(i,~,)
, C
~ c,
=
where
f'(X,Y,v) = (fX,Y,v).
been proved, (note that
Thus
f
The horizontal arrows are homotopy equivalences by what has
Y\i~,
=
Y\C'
is contractible as it has an initial object).
is a homotopy equivalence, whence the theorem.
The exact homotopy sequence.
Let
g : E --~B
h_omotopy-fibre of
be a map of topological spaces and let
f
over
F(g,b)
consisting of pairs
For any
e
in
fum~tor sendir~E
with
e
a point of
E
and
~ ~ri(F(g,h), ~) -----~ =i(E,e)
denoting the constant path at
f : C --@C'
B.
The
is the space
p
a path joining
one has the exact homoto~y secuence of
~ = (e,~), ~
Let
be a point of
= E xBBI xB[b~
(e,p)
g'!(b)
---~ lri+,(B,b)
where
b
b
be a funmtor and
(X,v : Y ~
fX)
to
Y
Y
and
b.
e
~..
b.
C'.
If
j : Y~f
(X,v) ~-~ v : Y --~ fX
formation from the constant ~kmctor with value
g(e)
with basepoint
g* _~ Iri(B,b)
an object of
X, then
g
to
fj.
-,C
is the
is a natural trans-
Hence by Prop. 2 the composite
B(Y\f) --~ BC= -, BC=' contracts canonically to the constant map with image
Y, and so we
obtain a canonical map
B(Y\f) ~
F(Bf, Y).
We want to know when this map is a homotopy equivalence, for then we have an exact
sequence relating the homotopy groups of the categories
Y\f, C
and
C'
Since the
homotopy-fibres sf a map over points connected by a path are homotopy equivalent, it is
clearly necessary in order fsr the above map to be a homotopy equivalence for all Y, that
the functor
Y'\f--~Y\f
, (X,v) ~-k(X,vu)
equivalence for every map
u
in
C'.
induced by
u : Y ~-~.Y' be ahomotopy
We are going to show the converse is true.
Because homotopy-fibres are not classifying spaces of categories, and hence are somewhat removed from what we ultimately will work with, it is convenient to formulate things
in terms of ho~otopy-cartesian squares.
~t
. . .h'.
~E
B'
~ h
;B
Recall that a co~utative square of spaces
is called h~Otopy-cartesian if the map
E' ~
from
E'
B' x B B I x B E
to the homotopy-fibre-product of
96
,
e'
~-. (g'(e'), ~--q'6U, h'(e'))
h and g is a homotopy equivalence.
89
When
in
B'
is contractible, ~he map
B', hence one has a map
15
F(g',b') --~ E'
E' --kF(g,h(b'))
is a homotopy equivalence for any
unique up to homotopy.
square is easily seen to be homotopy-cartesian if and only if
b'
In this case the
E' --~ F(g,h(b'))
is a
homotopy equivalence.
A commutative square of categories will be called homotopy-cartesian if the corresponding square of classifying spaces is.
With this terminology we have the following
generalization of Theorem A.
Theorem B.
C'
=
Y
t
Let f : C ~
C'
=
be a functor such that for every arrow Y -* Y'
of C'
- -
in
,
the induced functor Y' \ f --~ Y \ f
is a hemotopy, eqtuivalence. Then for any object
...........
the cartesian square of categories
=
Y\f
J
,
C
j(X,v)
=
X
=
f,
if
Y\c,
is homotopy-cartesian.
--~
wher___Z
f'<x,v) --
~' ~ c_'
j,(Y,,~) : Y,
Consequently for any X
=i+i(c__,,y)
~ =i(Y\f,
....J*
[)
in
f-1 (y)
~ =i(c,x)
we have an exact,,,sequence
f* .> =i(c__..Y)--~
..
X = (X,i~).
As with Theore~ A, this result admits a dual formulation with the categories
Corollary.
Suppose
arrow u : Y --* Y'
f : C= -@ C=' is/Drefibred (resp. precofibred) and that for every
the base-chan~e functor u*: f-1(y,) --~ f-1(y), (resp. the cobase-
change functor u.: f-1(y) _.~f-l(y,)) is a homotop~ equivalence.
C__', the cate~or~
f/Y.
f-1(y)
Then for an[
is homotop~ equivalen ~ to the homotop[-fibre of
f
Y
over
in
Y.
(Precisely, the square
f-i(y)
i > C
pt
) C'
=
where
i
is the inclusion functor, is homotopy-cartesian.)
we
Consequently for any X
in
have an exact homotopy sequence
_+=i+l<c, y>
This is clear, since
~ ,~i(~-,(y),x)
f-1 (y) --~ y \ f
i. ~ %(c,x)
~" . =i(c,,Y) - - - .
is a homotopy equivalence for prefibred
f.
For the proof of the theorem we will need a lemma based on the theory of quasi-fibrations
[Dold-Lashof] , which is a special case of a general result about the realization
of a map of eimplicial spaces [Segal 2
spaces such that the canonical map
for all
e c~
in B.
A quasi-fibratlon is a map
E, B
are in the class
W
complex, one ~ o w s
from
LMilnor 2]
F(g.b>
=W, and g
that
97
of
of spaces having the homotopy type of
is in
W
Thus if
is a quasl-fibration, we have that g-1 (b) -e F(g,b)
equivalence, i.e. the square
g : E --~ B
induces isomorphisms on homotopy
When
also in
b
].
g-1 (b)--e F(g,b)
-l(b)
is a homotopy
is
14
90
-1
i
!t h,!~
is homotopy-cartesian.
Le0~ma. Let
and let
be a functor from a small c a t e ~ o r ~
i~-~X i
g : X I --~ BI
be the s ~ c e over
pl
)
~_
Proof.
to the
obtained by rea~zing the simplicial s~ace
xi
io-*..-~ip
If X i -~ Xi,
BI
o
is a homotouy equivalence for every arrow
It suffices by Lemma 1.5 of ~Dold-Lashof]
p-skeleton
F
P
of
BI
to topological space,s,
I
i --* i'
in
I, then
g
is a
to show thatthe restriction of
is a quasi-fibration for all
p.
g
We have a map of
cocartesian squares
~
Fp_I
C
g
Fp
C I (ep_I )
where the disjoint unions are taken over the nondegenerated
NI.
Let
and let
of
g
to U, V
and U n V
g
are
quasi-fibrations.
and Ug%V, since
to glU, assuming as we may by induction that
is a quasi-fibration, and using the evident fibre-preserving deformation
into glFp_|
we may, let
x
D
of
provided by the radial deformation of /kp minus barycenter onto 9 / ~ p.
g-| (x') induced by
/~P
This is clear for V
is a product map.
We have only to check that if
simplex
p-simplices
U be the open set of F
P
V = Fp - Fp_ I . It suffices by Lemma 1.4 of lo___~c,cir. to show the restrictions
We will apply Lemma 1.3 of loc. ~ .
,~U
g l (Fp)
i -~..--) i
of
o
p
obtained by removing the barycenters of the p-cells,
over each p-cell
glFp_1
~
D
D
carries x E U
into x'eFp_ I, then the map
induces isomorphisms of homotopy groups.
come from an interior point
z
of the copy of A p
s = (io-~..-~ip) , and let the radial deformation push
with vertices
Supposing
z
g-1(x) --~
x ~ Fp_ I as
corresponding to the
into the open face of
Then it is easy to see that g-1(x) = X i
and g'1(x')
o
X k , k = ijo, and that the map in question is the one Xio---~~
induced by the face
io--~ k
of
s.
jo < ..(jq.
As these induced maps are homotopy equivalences by hypothesis, the proof
of the lemma is complete.
Proof of Theorem B.
We retura to the proof of Theorem A.
The functor
is a homotopy equivalence as before, but not necessarily the functor
Bp2 : BS(f) -~ B(C='O) is the realization of the map
lemma to the functor Y ~-~ B(Y~f)
98
Pl : S(f)-~ C=
The map
(**). Thus applying the preceding
from C 'O to spaces, we see that
fibration, and hence the cartesian square
P2"
BP2
is a quasi-
91
Y\f ~
pt
15
s(f)
Y
> C '0
=
is homotopy-cartesian.
Consider now the diagram
Y\f
,s(f)
Y\c'
~s(i%,)
~
~c
~,~ •
c,
pt --V-~ c,o
in which the squares are cartesian, and in which the sign
equivalence.
Since the square
homotopy-cartesian,
hence
(I) + (3)
(I) + (2)
j2.
The
',,~ ' denotes a homotopy
is homotopy-cartesian,
(,)
it follows that
is
is also, whence the theorem.
K-~rou~s of an,,exact cate~or~
Exact categories. Let M= be an additive category which is embedded as a full subcategory of an abelian category A , and suppose that M is closed under extensions in A
in the sense that if an object
A
of
A=
has a subobject
A'
such that
are isomorphic to objects of ~, then
A
is isomorphic to an object of
A'
~.
and
Let
~A'
=E be
the class of sequences
(I)
in
o
M=
,,
~M'
i
,M
J
~M"
which are exact in the abelian category
~
A.
=
~o
We call a map in
M_-
monomorphism (resp. admissible epimorphism) if it occurs as the map
member
M')
(I)
#M
of
and
an admissible
(resp. j) of some
=E. Admissible monomorphisms and epimorphiems will sometimes be denoted
M ---k~M", respectively.
The class
a)
i
E
clearly enjoys the following properties;
Any sequence in M=
isomorphic to a sequence in
E
is in
=
E.
For any
=
M',M"
in
=M, the sequence
(2)
o
is in
i
E.
~-
,~'
(id.o)_M.e
For any sequence
(I)
in the additive category
M.
=
b)
in
~ ' ~ . "
E,
i
,o
is a kernel for
j
j
is a cokernel for
The class of admissible epimorphisms is closed under composition and under base-
change by arbitrary maps in
M__. Dually, the class of admissible monomorphisms is closed
under composition and under cobase-change by arbitrsly maps in
c)
and
Let
M -~M"
in M= such that
be a map possessing a kernel in
N --~ M -~ M"
admissible epimorphiem.
M.
=
If there exists a map
is an admissible epimorphism, then
M -~ M"
N -~M
is an
Dually for admissible monomorphiems.
For example, suppose given a sequence
Form the diagram in
M.
A
99
(I)
in
E
_--
and a map
f : N --~M"
in
M.
=
92
where
P
O
k M'
~ M
J .~ M"
~0
0
:~i'
;P
,N
rO
is a fibre product of
sions in
A, we can suppose
exists in
16
P
f
and
j
in
A .
is an object of
Because
M.
=
M
is closed under exten-
Hence the basechange of
Definition.
An exact category is an additive category
of sequences of the form
properties
by
f
a), b), c)
Examples.
M= equipped with a family
(I), called theishort)exact sequences of
hold.
An exact functor
F : _M_--, M'
an additive functor carrying exact sequences in
M=
between exact categories is
into exact sequences in
M'.
=
Any abelian category is an exact category in an evident way.
family of split exact sequences
(2).
=E
M__, such that the
category can be made into an exact category in at least one way by taking
[Holler]
j
M= and it is an admissible epimorphism.
Any additive
E
=
to be the
A category which is 'abelian' in the sense of
is an exact category which is Karoubian (i.e. every projector has an image), and
conversely.
Now suppose given an exact category
contravariant functors from
M
M.
Let
A
be the additive category of additive
to abelian groups which are left exact, i.e. carry
(I) to
an exact sequence
o
; ;(M")
~ F(M)
, F(M')
(Precisely, choose a universe containimg
M,
and let
A
be the category of left exact
functors whose values are abelian groups in the universe.)
(e.g.
[Gabriel I ), one can prove
embeds
M=
as a full subcategory of
sequence
(I)
is in
E
The categor~
M=
M
Following well-known ideas
is an abelian category, that the Yoneda functor
A=
if and only if
details will be omitted,
If
A
h
closed under extensions, and finally that a
h
carries it into an exact sequence in
A.
The
as they are not really important for the sequel.
QM .
is an exact category, we form a new category
but with morphiems defined in the following way.
QM
Let
M
having the same objects as
and
M'
be objects in
M=
and consider all diagrams
(3)
where
~
j
j
~>
i
,~,
is an admissible epimorphism and
i
is an admissible monomorphism.
is~orphisms of the~e diagrams which induce the identity on
being unique when they exist.
A morphlsm from
M
definition an isomorphism class o£ these diagrams.
to
M'
M
and
We consider
M', such isomorphisms
in the category
Given a morphism from
Q~= is by
M'
to
M"
represented by the diagram
M'~
j'
N' ~
i'
. M"
the composition of this morphism with the morphism from
is the morphism represented by the pair
j-pr I , i'-pr 2
100
M
to
M'
represented by
in the diagram
(3)
93
17
i'
N ;
4
M"
~M'
i
M
It is clear that composition is well-defined and associative.
classes of diagrams
then
~
(3)
form a set
Thus when the isomorphism
(e.g. if every object of
is a well-defined category.
M=
has a set of subobjects~
We assume this to be the case from now on.
It is useful to describe the preceding construction using ad~ssible sub- and
quotient objects.
By an admissible subob~ect of
admissible monomorphisms
over
M.
we will mean an isomorphism class of
Admissible subobjects are in one-one correspondence with admissible quotient
objects defined in the analogous way.
with the ordering:
morphism.
cokernel
M
M')---~ M, isomorphism being understood as isomorphism of objects
When
M ;I M_
MI~ M 2
The admissible subobjects of
if the unique map
MI~< M 2, we call
(MI ,M2)
M I --> M 2
over
an admissible subquotient of
isomorphism
of
((MI,M2), 8)
M
form an ordered set
is an admissible mono-
an admissible layer of
M, and we call the
M.
With this terminology, it is clear that a morphis~ from
identified with a pair
M
M
to
M'
in
consisting of an admissible layer in
QM= may be
M'
and an
@ : M - ~ M / M I . Composition is the obvious way of combining an isomorphism
~,i with an admissible subquotient of
sible subquotient of
M"
M'
and an isomorphism of
to get an isomorphism of
M
M'
with an admis-
with an admissible subquotient of
M tt•
For example, the morphisms from
0
to
M
in
the admissible subobjects of
M.
isomorphisms from
isomorphisms from
in
M.
=
If
from
M'
M
i : M' ~
to
M
to
M
in
M'
~
M
are in one-one correspondence with
to
M'
in
Q~= are the same as
is an admissible monomorphism, then it gives rise to a morphism
QM= which will be denoted
i! : M'---~ M.
Such morphisms will be called in~ective.
Similarly, an admissible epimorphism
j : M-~M"
gives rise to a morphism
J
: M" --~ M
and these ~rphisms will be called surjeetive.
By definition, any morphism
u
in
Q__M
t
can be factored
u = i!j', and this faetorization is unique up to unique isomorphism.
If we form the bicartesian square
N)
then
u =
i
~M'
j' !.t
x ! , and this injective-followed-by-surjective factorization is also unique
up to unique isomorphism.
A map which is both injective and surjective is an isomorphism,
101
18
94
and it is of the form
e! = (@-I)'
for a unique isomorphism
Injective and surjective maps in
~
epimorphisms in the categorical sense.
In fact, the category
missible layers in
QM=/M
@
in
M.=
should not be confused with monumorphisms and
Indeed, every morphism in
~
is a monomorphism.
is easily seen to be equivalent to the ordered set of ad-
M with the ordering"
(Mo,M1) ~ (Mo,M~) i f
Mo<.Mo~Ml~.M~.
!
We can use the operations
by a u n i v e r s a l property.
i ~+ i.r
and
j ~-~ j"
to c h a r a c t e r i z e the category
Q~=
F i r s t we note that these operations have the following
properties:
a)
If
Dually, if
(JJ')'
•
=
b)
i
and
j
and
j,'j'
If
i'
are composable admissible monomorphisms, then
j'
are composable admissible epimorphisms then
Also
•
(4)
(i%) t
--
(i~)'
" = i%1 •
is a bicartesian square in which the horizontal (resp. vertical)maps
!
are admissible mon~morphisms (resp. epimorphisms), than
~ow suppose given a category
C,
and for| each
-(reap.
(i'i)! = i'll , •
i : M' )--@M
C=
and for each object
(reap.
j : M--~M")
j" : hM" -~ hM) such that the properties
this data induces a unique functor
t
i!j" = j"i',
M
a map
a), b)
Q=M -, C , M ~-k hM
of
M_-
.
an
object
hM
of
i, ; hM' - e h M
bold.
Then it is clear that
compatible with the operations
|
i ~-~ i!
and
j ~-~ j"
in the two categories.
In particular, an exact functor
functor
QM--~QM=
=
',
F : M --~ M'
M~->FM , i ,.~ - * ( F i ) ,. ~ j! ~=
between exact categories induces a
(Fj)" .
We note also that if
_M_
O
is
the dual exact category, then we have an isomorphism of categories
(5)
Q(=Mo)
=
such that the imjective arrows in the former correspond to surjective arrows in the latter
and conversely.
The fundamental ~roup of
that the classifying space
Theorem ~.
Suppose now that
is defined.
Let
_M is a small exact category, so
0
The fundamental $,roup TrI(B(Q=M), O)
Grothendieck ~roup
Proof.
QM.
B(QM)
be a given zero object of
KOM= .
The Grothendieck group is by definition the abelian group with one generator
["I
for each object
(I)
in
M= .
M
of
~
and one relation
[M]
-- [M'][M'~ for each exact sequence
We note that it could also be defined as the not-necessarily-abelian group
with the same generators and relations, because the relations
LM"]tM']
F
of morphism-inverting functors
show the group
tot from
F
=
Let
to
Ko=M acts naturally on
K M - sets
M.
F : Q=M --~ Sets.
F(0)
for
F
in
B(QM)
is equivalent to the
It suffices therefore to
_F , and that the resulting func-
is an equivalence of categories.
o--
~. : O ) - - @ M
and
JM : M - - ~ 0
f u l l subcategory of _F_ consisZing of r
for all
LM'][M"] = [M' ~ M"o] =
force the group to be abelian.
Accordir~ to Prop. I, the category of covering spaces of
category
M.
is canonicall~ isomorphic to the
Clearly any
F
denote the obvious maps, and let
such that
F(M) = F(O) and
is isomorphic to an object of
102
F'
be the
F(iM!) = idF(o)
F', so it suffices to show
95
F' is equivalent t o
K M-
=
19
sets.
O=
Given a
Ko~ - set
S, let
F S • Q M --~ Sets
be the functor defined by
~s(~) = s , Fs(i !) = id s ,- Fs(j!) = multiplication by
using the universal property of
_F_'. On the other hand if
F(i! ) = idF(0).
QM= . Clearly
F6=F', then given
[Ker j] on S,
is a functor from
i : M'>-~M
we have
Ko=M - sets to
i.~, = ~
) hence
Given the exact sequence
0
> M'
!
i
~M
J
!
; M " ~ O
!
J~.~ = i!jM; , hence
we have
S ~-k F S
!
F(~) = F(jM;) 6 Aut(F(O)). Also
F(jM') = F(j jM.) = F(jM,)F(jM;,)
so by the universal property of
Aut(F(O))
any
F
in
F'.
=
KoM- , there is a unique group homomorphls,, from
~ F(j .
such that
~hus we ~ v e a naturnl action of KoM
In fact, it is clear that the resulting functor
K M - sets is an isomorphism of categories with inverse
o=
theorem is complete.
H~her
K-groups.
definition of
K.M
==i
(B(Q~),o)
=
I=
+I
Note first of all that the
O.
F~F(0)
from
for
F'
=
to
S ~-~ F S , so the proof of the
The above theorem offers some motivation for the following
K-groups for a small exact category
~.finition.
Ko=M to
on F(0)
M= .
*
E-groups are independent of the choice of the zero object
Indeed, given another zero object
there is a canonical path from
0
to
O'
there is a unique map
O'
in the classifying space.
0 --~ O'
in
QM , hence
Secondly we note that the precedin~ definition extends to exact cate6ories havin~ a
set of isomorphism classes of objects.
subcategory equivalent to
We define
M= , the choice of
KiM=
to be
KiM=', where
Mr' is a small
H'
being irrelevant by Prop. 2.
=
From now on
we wili only consider exact categories whose isomorphism classes form a set, except when
mentioned otherwise.
In addition, when we apply the results of
~I, it will be tacitly
assumed that we have replaced any large exact category by an equivalent small one.
Elementary properties of
K-groups.
QM__--~QM_', and hence a hemomorphism of
(6)
f : M= --~ M'= induces a functor
f . , KiM - - . Ki~'
In this way
groups.
(5)
An exact functor
K-groups which will be denoted
Ki
becomes a functor from exact categories and exact functors to abelian
Moreover, isomorphic functors induce the same map on
K-groups by Prop. 2.
From
we h a v e
(v)
Ki(~o)
The p r o d u c t
i s e x a c t when i t s
M x M'
K~.
=
o f two e x a c t c a t e g o r i e s i s an e x a c t c a t e g o r y i n which a s e q u e n c e
projections in
M
and
=
M'
are.
~
Clearly
Q(M
-- x M')
--
=
QM
x QM'.
=
=
the classifying space functor is compatible with products
(~I, (4)), we have
(e)
p r , . ( x ) • pr~.(x)
Ki(M= x M')
=
Ki~= e Ki~=' ,
10.3
x ~
.
Since
20
96
The functor 0 :
M x ~ - - ~ , (M,M') F - ~ M ~ M '
K.M ~ K.M
i=
i=
=
O~ _~ K M .
Ki(M x M)
--
is exact, so it induces a homomorphism
1=
--
This map coincides with the sum in the abelian group KiM
O~M
, M ~-~ M ~ 0
Let
j ~-~ Mj
because the functors
M ~-~
are isomorphic to the identity.
be a functor from a amall filtering category to exact categories and
functors, and let
lira
---, M.
=j be the inductive limit of the M.
=j in the sense of Prop. 3.
is an exact category in a natural way, and
Q ( l ~ Mj) = lira
Q=Mj ,
~
Then
lira
--* M=3
hence from
Prop. 3
(9)
we obtain an isomorphism
< ( l i r a ~.)
Example.
Let
A
= lira K.M..
be a ring with
I and let ~(A)
finitely generated projective (left) A-modules.
denote the additive category of
We regard
P(A)
as an exact category in
which the exact sequences are those sequences which are exact in the category of all
A-modules, and we define the K-groups of the ring A
KiA =
A ring homomorphism
A--~ A'
by
Ki(P(A)).
induces an exact functor
A' ~ A ? : P(A)--> =P(A') which is
defined up to canonical isomorphism, hence it induces a well-defined homomorphism
(I0)
( A ' @ A ?). : KiA .....~. K.A'.
1
making KiA
a covariant functor of A.
(11)
If
Ki(A x A')
j ~, Aj
KiA ~
(8) we have
KiA'.
is a filtered inductive system of rings, we have from
(12)
Ki(1~
(To apply
=
From
Aj)
(9), one replaces
=
P(Aj)
lim K.A..
---, I 3
by the equivalent category
P(AJ'
=I
the idempoten% matrices over A.3 ' so that =P(l~ Aj)' = ~
that P D-~ HomA(P,A)
A °p
is an equivalence of P(A)
is the opposed ring to A, hence from
(13)
Ki(A)
Remarks.
(7)
Finally we note
with the dual category to P(A°P), where
Ki(A°P ) .
1
into an
whose objects are
J
=P(Aj) . )
we get a canonical isomorphism
It can be proved that the groups K.A
defined by making BGL(A)
[Gersten 5]).
=
(9) an isomorphism
defined here agree with those
H-space and taking homotopy groups (see for example
In particular, they coincide for
i = I , 2
with the groups defined by
by Bass and Milnor, and with the K-groups computed for a finite field in [Quillen
On the other hand, for a general exact category
the universal determinant group defined in
M= , the group
[Bass, p.389].
K I(M)= is not the same as
There is a canonical homomor-
phism from the universal determinant group to K I (M=), but Gersten and Murthy have
produced examples showing that it is neither surjective nor injective in general.
104
2].
21
97
~3. C~racteristic
Let
M
exact sequences and filtrations
be an exact category and regard the family
as an additive category in the obvious way.
let
sE, ~E, qE
E
of short exact sequences in
We denote objects of
denote the sub-, total, and quotient objects of
E
by
M
E, E', etc. and
E, whence we have an
exact sequence
O
in
M
~ sE
~ tE
~ qE
associated to each object
E
of
E .
gives rise to three exact sequences in
exactness, it is clear that
functors from
E
to
Theorem 2.
Proof.
M
on applying
Q~
maps in
Let
QM .
u
(e,q) : Q =E - - ~
M,N
of objects of
QM= x QM:
Lemma.
M .
consisting of triples
C'
v
With this notion of
s, t, and q
are exact
is a homotopy equivalence.
Put
C"
(s,q)/(M,N)
is contractible
C = (s,q)/(M,N); it is the fibred
C
u : sE -->M, v : qE -~ R
consisti~
of the triples
are
(E,u,v)
be the full subcategory of triples such that
u
is injective.
The inclusion functors
C= ' - - ~ C=
Consider first the inclusion of
that there is a universal arrow
C' in
X -~ ~
in
!
u = j'i I•
will be called exact if it
s, t, and q.
(E,u,v), where
be the full subcategory of
is surjective, and let
is surjective and
E
is an exact category, and that
It suffices by Theorem A to show the category
for any given pair
Let
A sequence in
M .
The functor
category over
such that
E
~ O
where
i : sE~--~ M'
, j
and
C= " - - ~ C='
- -
C.
Let
C=
with
:
have left ad~oints.
X = (E,u,v)gC;
~
M ~M',
in
it suffices to show
C'.
=
and define the exact sequence
i.E
by 'pushout':
E
:
0
• sE
~
~qE
tE
,O
jl
i.E :
0
> M'
~qE
~ T
~ O .
t
Let
~ = (i.E,j',v); it belongs to
the evident injective map
0'
and there is a canonical arrow
X --~
given by
E -@ i.E .
!
~ow suppose given
by the pair
Ep-@
Eo,
X --bX'
with
E' :;3 E o
X' = (E',j",v')
sE >---> SEo<~.
represents
M ~
sE 0
uniquely
u, we can suppose
is
E ~
X --~ X -~ X'
j.
E°
sE'.~ j'
chosen so that
By the universal
C'.
=
Represent the map
sE >--> sE °
is the map
i.E b-~ E 0 , so it is clear that we have a map
is the given map
of
X -* X'
X ~-* X'
such that
and
E~
~ -~ X'
i, and
E0
factors
in
C_' such that
X --~ X'.
X"
is in
valent to the ordered set of admissible layers in
corresponding to
E--~E'
M
property of pushouts, the map
It remains to show the uniqueness of the map
X -+ X" -+ X'
in
Since
~ -@ X'.
C'
E'.
Consider factorizations
~ote that
Let
C/X'
(E° , E I)
(E"o ' E';) the layer corresponding to
105
=
QE/E'=
is equi-
be the layer
X" --~ X'
so that
98
(So 'zl)<(E"o ,z~,) and s~.i'= ~'.
22
Tbere is a least such laye. _(Z"o'q )
given by
tE O = tEO , tE~ = sE' + tEI , which is characterized by the fact that the map
E~'/Eo
is injective and induces an isomorphism on quotient objects.
factorizations
X -~ X" -~ X'
there is a least one~ unique up to canonical isomorphism,
and characterized by the condition that
morphism
qE ~
qE".
clear that the map
E -@ E"
Since the factorization
X -. X'
EI/E O --~
Thus among the
should be injective and induce an iso-
X -~ ~ -, X'
is uniquely determined.
Thus
has this property, it is
C' -~ ----.
C has the left adjoint
=
x~.
Next consider the inclusion of
v : qE--~ N
by the pair
C"
=
in
C' t and let
0
~ sE
b T
0
: sE ~
tE
....
N , and define
N' ~
..... •
(E,u,v)6C'
=
----
j : N' ~ ~ qE, i : N ' ~
•
qE -----*O .
•
is left adjoint to the inclusion of
By Prop. 2, Cot. I, the categories
(E,j',i,)6C", and let
!
"
=
(O, JM',~,)
JM : M --~O
!
to
C
and
and
~
in
C"
are homotopy equivalent.
: OP-~N
C'.
,
This finishes the lemma.
be the obvious maps.
may be identified with an admissible subobject
.
sE' = sE
object of
C"=. Thus
(E,u,v) ~ - ~
C"
Let
A map from
-~
(E,j',i,)
such that
by pull-back:
0
One verifies by an argument essentially dual to the preceding one that
(j*E,u,i!)
Represent
j*E
E'
of
E
!
and
qE' = O.
Clearly
C"=, and hence
E'
is unique, so
(O,jM" ,~!)
is an initial
=C is contractible, which finishes the proof of the
~heorem.
Corollary
Let
1.
0
and
M'
•
M be exact c a t e g o r i e s and l e t
F' ----* F
~
....
)
be an e x a c t 9e~uence of e x a c t f u n c t o r s f r o m
F"
M' to
F. - F . +F.. : Ki~,-~K.M~=
Proof.
~ O
M._ Then
.
It clearly suffices to treat the case of the exact sequence
~ s----~t
0
of functors from
E
to
Mo
Let
> q
,
.~0
f : M x M --~ E
be the exact functor sending
(M*,M")
to the split exact sequence
0
The functors
tf
, .~M'
t.f.
But
f.
Thus
t.
=
i s a s e c t i o n of
=
) M' ~ M "
and ~ ( s , q ) f
e.(a.,%)f.
(s.,q.)
, M"
~ 0 .
are isomorphic, hence
=
(s. + q.)f. : (KiM=)2--~ Ki_rJ_.
: KI=E --) (ELM_)2 which i s an isomorphism by the theorem.
s. + q., proving the corollary.
Note that the category of functors from a category
C=
to an exact category
M
=
an exact category in which a sequence of functors is exact if it is pcintwise exact.
thus have the notion of an admissible filtration
F.
This means that
F
I (X)--* Fp(X)
O = FoG F | C . . o F n = F
is an admissible monomorphism in
106
M
is
We
of a functor
for every
X
23
99
F/F
for q.( p, determined up
q
is an exact category, and if the
in C=, and it implies that there exist quotient functors
to canonical iscmorphl-m.
functore
It is easily seen that if C
F/Fp_ I are exact for
Co rolla~ 2.
|~< p <~n, then all the quotients
Fp/Fq
(A~ditlvity for 'characteristic ,' fi!trationg)
Let
are exact.
F : M=' -~ M__ be an
exact functor between exact categories equipped with an admissible filtration
cF n = F
such that the quotient functors
Fp/Fp. I are exact for
1(p.<n.
0 = FoC ..
Then
n
pc1
Corollary 3.
=
(Additlvlt~r for 'characteristic' exact sequences)
0 ,
>
)
Fo
.~ Fn----+
...
is an exact sequence of exact fuuctors from M'
If
0
to M , then
n
p--0
(-,)P(~).
= o
:
~i~'--*
K K.
x=
These result from Cor. I by induction.
Applications.
We give two simple examples to illustrate the preceding results.
Let x ~ a ri~.d e y e , ~ d ~ t
vector bundles on X, (i.e. sheaves of
KIX ~ Xi~(X), where =P(X) is th. category of
_~-modules which are locally direct factors of ~X~)
equipped with the usual notion of exact sequence.
functor
KiX.
E@?
If
Given
E
in
P(X), we have an exact
: =P(X) --) =P(X) which induces a homomorphlsm of K-sToups
O --@ E' ~
Cot. I implies
E --+ E" --~ 0
(E~?).
=
(E'@?). + (E"@?), . Thus we obtain products
KxezKJ
O)
which clearly make
....>
Kix ,
i n t o a module over
KIX
(E@?).: KIX -~
is an exact sequence of v e c t o r bundles, then
(~?).x
[E]~x ~
KoX .
(Products
KiX ® 5 X --~ Ki.jX
can
also be defined, but this requires more machinery.)
Graded rin~e.
Let A
= A ° @ A! ~ .. be a graded ring and denote by
category of graded finitely generated projective
group
Ki(=Pgr(A)) is a
P = ~ Pn ' n ~ Z
Z[t,t'1]-module, where multiplication by
induced by the translation functor
Proposition.
A-modules
Pgr(A)
the
. The
t is the automorphism
P ~-~ P(-I ), P(-I )n = Pn-1 "
There is a ~ [t,t-I] -module isomorphism
~[t,t-IJe2~KiA O
Ki(Pgr(A)) , 1 @ x
b-~
(AeA ?).X
•
O
Given
P
for n~ k, and let
P
=q
Proof.
which
F_q_1P = 0
in
Pgr(A), let
and
F P = P.
A°
be the
A-submodule of
Pgr(A)
=
P generated by
consisting of those
T(P)
=
Ace A P
is considered as a graded ring concentrated in degree zero.
([Bass], p.637) that
P
is non-canonically isomorpklc to
A~AoT(P)
=
P
We have an exact functor
T • =Pgr(A) --@ =Pgr(A) ,
where
FkP
be the full subcategory of
~n~ A(-n)~AoT(P)n .
107
It is known
Pn
for
i00
It follows that
P ~-~FkP
24
is an exact functor from
~gr(A)
to itself, and that there is
a canonical isomorphism of exact functors
FnP/F_IP
=
A ( - n ) ~ A T(P)n .
o
Applying Cor. 2 to the identity functor of
P and the filtra$ion
=q
0 = F_q_IC-..~F
q
=id,
one sees that the homomorphism
tn~A
O
---~ Ki~ q
,
tn~x
~-> (A(-n)e A ?).x
o
-q~ n ~ q
is an isomorphism with inverse given by the map with components
~6T(A)
is the union of the
In this section
M
~2, (9).
Reduction by resolution
~4.
P
(Tn) . , -q~ n@ q . Since
_Pq , the proposition results from
denotes an exact category with a set of isomorphism classes, and
a full subcategory ~!osed under extensions in
object and for any exact sequence in
M
in the sense that
P
contains a zero
M
=
(I)
if
o----~M,
M'
and
M"
----~
,
~,~" -
are isomorphic to objects of
~o
P , so is
M.
Such a
category where a sequence is exact if and only if it is exact in
a subcategory of
QM= which is Sot usually a full subcategory, as
phisms and epimorphisms need not be
will always refer to
The corresponding
explicitly.
P ~
objects of
P ; it is
P'
Ki~
M-admissible monomor-
(2)
o ---.
p
P , and the symbols
M-admissible monomorphism between two
P-admissible iff the cokernel is isomorphic to an object of
when every object
M
of
M=
has a finite
.......~ ..... - . _ _ . p
n
The standard proof for
KO
o
_P in
~ ( - t ) n [ P n ~ 6 Ko=P depends only on / M ] .
(2)
P .
_M induces isomorphisms
P-resolution:
--_
~ M ----~ o .
consists in defining an inverse map
works when there exist resolutions
is
P-admissible notions will be specified
denotes an
We are interested in showing that the inclusion of
KiP= ~
QP
M-admissible monomorphisms, epimorphisms and
subobjects, respectively.
For example,
is an exact
P-admissible.
In the following, letters P, P', etc. will denote objects of
3--~, ---> , ~
P
Mr. The category
Ko~M --> Ko~P by showing
By Cor. 3 of the preceding section, this method
depending on
M
in an exact functorial fashion.
However, this situation occurs rarely, so we must proceed differently.
The following theorem handles the case where resolutions of length one exist.
example, think of
_M_ as modules of projective dimension ~< n, and
modules of projective dimension ~ n.
Theorem 3-
Let
P
The general case follows by induction (see Cor. I ).
be a full subcategor~ of an exact category
=M which is closed
under extensions and_is such that
i)
ii)
For ~
For an~
exact sequence
M"
i~
The n the inclusion
. . . .functor~~
M,
As an
=P as the subcategory of
(I), i_ff M
is in
P , then
there exists an exact sequence
M'
(I)
is in
with
P .
M
in
_P .
W2= -~Q/4= is_ a homotop,y equivalence, --s° K.PI=~-~Ki~ "
108
i01
Proof.
We factor
QP --~ QM
QP
where
g
C
and
into two inclusion functors
~ ~C
f ~
is the full subcategory of
f
25
Q~= with the same objects as
g
is a homotopy equivalence, it suffices by Theorem A to prove
contractible for any object
valent to the ordered set
P
J
in
of
C= .
The category
~P
M-admissible layers
hypothesis
i), one knows that
M1
and
MO
are in
~P
is
is easily seen to be equi-
(Mo,M I)
~th the orderin~ (~o'M1~(M'~'o"~'I) iff "~<'~o~MI~<M~
J
We will prove
are homotopy equivalences.
To show
in
QP .
in
P
such that MI/Mo6 P,
an~ ~o/M'o ' ~ / ~ ~- -P " By
=P for every
(Mo,M I)
in
J.
Hence
we have arrows
(.o,~,)
~ (o,~) ~ (o,o)
which can be viewed as natural transformations of functors from
functor
J
to
J
joining the
t o the identity and t o the constant functor with value
(Mo,M I) ~-> (O,M I)
Using Prop. 2, we see that
J, hence
~P
, is contractible, so
g
(O,0).
is a homotopy
equivalence •
To prove
QM.
=
Put
f
is a homotopy equivalence, we show
M\f
C
=
F= = M \ f ; it is the cofibred category over
u : M -->P
a map in
surjective.
Given
QM.
Let
is contractible for any
consisting of pairs
F_' be the full subcategory consisting of
X = (P,u)
in
F , write
u = i,j ! with
j : P ~M
of
F'
and
i),
i
P
is in
defines a map
arrow from an object of
F'
=
in
F'
=
P
as the notation suggests.
~ --> X.
to
Thus
X , hence
X ~-,~
dual category
which a morphiam from
triangle commutes.
Given another
extension of
F '°
P ~M
to
by
is a map
P XMP 0
which is in
Ker ( P - ~ M )
is an object
is a universal
F' is contractible.
_-
P --~ P'
P ~
M , and in
such that the obvious
ii), there is at least one such object
By hypothesis
P --~ M , the fibre product
Pc
X -->X
is the category whose objects are maps
P' - - ~ M
u
is right adjoint to the inclmsion of
F= . By Prop. 2, Cor. I, we have only to prove that
The
with
with
!
X = (P,j')
One verifies easily that
(P,u)
(P,u)
in
, i : P )--->P •
_
By hypothesis
M
is an object of
P
by hypothesis
Poem
•
P , as it is an
i).
Hence in
F 'O
we
have arrows
(P ~
(P ~ r o "-~ M) --~
M) ~
(Pc - ~ ~i)
which may be viewed as natural transformations from the functor
to the constant functor with value
we conclude that
F'
Corollary I.
(P--~M) ~-~ (P ~ P o
and to the identity functor.
Assume,. =P
is closed under extensions in
For every exact sequence
b)
Given
(I), if
- -
j : M --9@ P, there exists
M, M"
are in
j': P ' - - ~ P
(This holds, for example, if for e v e ~
M
and
M=
and further that
P , then so is
=
f : P' -->M
there exists
Let =Pn b~ the f~l su~ategory of M cons%s~i~ of ~ ~ v i ~
<~n,
.i.. e..t. such t h a t t h e r e ' ex.ists an exact sequence
Kip
-~M)
Using Prop. 2,
is contractible, finishing the proof of the theorem.
a)
jf = j'
Pc--~ M
~
ip1
Kip
109
( 2 ) , and put.
P'-~M
M'.
such that
.)
~-resolutions of length
Pco =U=Pn " Then
102
That
=nP is closed under extensions in
26
=M , and hence the groups
KiPn
are defined
results from the following standard facts (compare [Bass, p.39]).
Lemma.
For an~ exact sequence
(I)
and integer
n>~O, we have
=n
2)
]4', M" (~ =n+1
====~ ]4E-P-~+I
~)
]4 ' ~" ~ -~n+1 = ~ '
]4'~: ~n+', "
-P-n
C =Pn+1 "
Assuming this, we apply Theorem 5 to the pair
fied, for given
and by
I)
for each
]4&--Pn+1 ' there exists an
it is
n.
Pn+1-admissible.
The case of
Pco
follows by passage to the limit
Since
a simple
ii)
P--~M
The other hypotheses are clear, so
To prove the lemma, it suffices by
I):
Hypothesis
M-admissible epimorphism
is satis-
with
;
(§2, (9)).
induction to treat the case
M " ~ P I , there exists a short exact sequence
P6P
KiP_n - ~ Ki=Pn+I
n = O.
P'~--~P --~]4", so we can
form the diagram on the left with short exact rows and columns
t
and with
have
1
[
I
, F
~ P
p'
~ p ' @ P"
]4'
~M
y ]4"
]4'
~ ]4
F = ]4 xM,P .
F 6 P= .
2):
I
]4'
Since
Since
we can enlarge
P', M
F, P 6 P_-
]4"6PI,
P
Since
are in
we have from
there exists
and find
=P
P" ~
a)
P ~
M
and
=P
..... ~ M"
is closed under extensions, we
that
]4'6P= , proving
M", so applying
factoring into
,p"
b)
to
P" --~ M - + M " .
I).
pr I : P ~,,]4 - ~ P
the above diagram on the right with short exact rows and columns, and with
]4'C_PI .
3):
Applyin~
Since
I)
we see that
R"6P= , so
R~P=
and
M~P|= , proving
P', R'6 =P as
2).
ME P|, we can form the diagram with short exact rows and columns
P' -
L
impl~es
K~,
;-]4"
-~M
so
As an example o f t h e c o r o l l a r y ,
( l e f t ) A-modules.
t
P .....
M'
A~ M'E=P I , I )
P ' ~ O
t
K ~
(Better, so that
-'M"
M'E=F I , provin~
take
P = P(A)
3).
and
The l e m ~ and Cor. I are ~one.
M = ]4od(A), t h e c a t e g o r y
=
M has a set of isomorphism classes, take
of
M to be
the abelian category of a l l
A-modules of c a r d i n e l i t y <o(, where o( i s some i n f i n i t e
cardinel
P (A) be the category of
length
> card(A).)
Let
~ n, end Pco(A) = ~ =Pn(A). Then P ( A ) -
Corollary 2.
regular, then
A-modules.
,
Thus we can form
~
A-modules having P-resolutions of
as i n the corollary, so we obtain
For O ~ n ~ o ~ , we have KiA ~-~Ki(Pu(A)).
KiA ~
In p a r t i c u l a r i f
A i_~s
Ki(Modf(A)), where Modf(A) i s the category of f i n i t e l y generated
110
103
We recall that a r e ~
27
ring is a noetherian ring such that every (left) module has
finite projective dimension.
For such a ring A
we have
=Pco (A) = Modf(A).
Similarly, Cot. I implies that for a regular noetherian separated scheme the K-groups
of the category of coherent sheaves and the category of vector bundles are the same, since
every coherent sheaf has a finite resolution by vector bundles
Transfer maps.
is in
P
Let
f : A -~ B
be a ring homcmorphism such that as an
Pco(A) ° Then restriction of scalars defines an exact functor from
(A), hence
f. :
PQo(B)
B
to
KiB--~KiA
and call the transfer map with respect to
g : B -*C
A-module
by Cor. 2 it induces a homomorphiem of K-groups which we will denote
(3)
f.
Clearly given another homomorphism
with C~=Pco(B), we have
(4)
(gf). =
f.g. : KiC ~
We suppose now for simplicity that
P(A) x Pn(A)
for
~SGA 6, II, 2.2J.
A
B.
Then if
and
B
are commutative, so that we have functors
---. P=a(A) ,
O & n ( o o , which induce a product
similarly for
KiA .
(P,M) ~
PeA M .
KoA@KiA --~ KiA , [P~@z ~-h ( P ~ A ? ) . z , and
f* = (B ~A?). : KiA -+ KiB , we have the pro0ection formula
(5)
f.(f*x.y) = x . f ~
and Y ~ K i B . This results immediately from the fact that for X in
for x ~ K A
P(A)
there is an isomorphism of exact functors
Y ~
from
=P
(B)
to
(B~gAX)@BY
----- X @ J
Pea (A)"
=
Corollary 3. L e t
T = IT i , i ~ |} be an ,ex,a,c,t connected sequence of ftmctors from
an exact category M__ to an abelian category A
(i.e. fo,r, ever~ exact sequence
(I), we
have a lon~ exact sequence
- - ~ T2M"
> TIM'
Let =P be the full subcategory of
assume for each M
in M=
- -
• TIM
~ TIM" ).
T-acyclic objects
that there exists
P --~M
(Tram = 0
with
P
f o r all n~1), and
in
P , and that
--
T M = 0
n
fo_xr n ~fioient1~ ~r~e. Then Ki__P~ K ~
This results either frcm Cor. I, or better by applying Theorem 3 directly to the
inclusion
=PnC_Pn+I, where
-P-a consists of
Here is an application of this result.
and let
f : A -~ B
because
such that
TjM = O
for
K~A = Ki(Modf(A) ) for A
be a homomorphism of noetherian rings.
A-moduie, then we obtain a homomorphism of
(6)
M
Put
If
B
j>n.
noetherian,
is flat as a right
K-groups
(B~A?) . : K~A --9" K~B
B~A?
is exact.
A-module, then applying
But mere generally if
Cot. 3
to M = Modf(A)
111
B
is of finite Tot-dimension as a right
104
we find that
M
K P "~ K:A , where
I-l
TnM = 0 for n > O .
such that
(6)
P
28
is the full subcategory of
Since
B~A?
is exact on
Modf(A)
consisting of
=P , we obtain a homomorphism
in this more general situation.
~5°
Devissa~e and localization in abelian categories
In this section
A
will denote an abelian category having a set of isomorphism
classes of objects, and
B
will be a non-empty full subcategory closed under taking
subobjects, quotient objects, and finite products in
gory and the inclusion functor
B --~ A
A.
Clearly B----. is an abelian care=
We regard A and B as exact cate-
is exact.
gories in the obvious way, so that all monomorphisms and epimorphisms are admissible.
Then
~
is the full subcategory of
objects of
Theorem 4.
(Devissage)
0 = MoCMIC..CM
functor
n = M
QB_ --p ~
Proof.
that
f/M
Suppose that every object
such that
Mj/Mj_ I
is a homotopy equivalence, so
is contractible for any object
QB= consisting of pairs
By associating to
such that
u
equivalence of
such that
u
M
with the ordered set
MI/Mo£B,
with the ordering
A
of
A.
=
has a finite filtration
J.
Then the inclusion
KiA= .
f, it suffices by
Theorem A
The category
N~ Q~= and
f/M
is such that
to prove
is the fibred
u : N --~M
is a map in
(Mo,M I)
consisting of layers
(Mo,M1)<~ (Mo,M ~)
iff
(Mo,M I)
MoCMoCMICM~
in
M
•
has a finite filtration with quotients in
i : J(M') -eJ(M)
M/M' 6B.
=
of
N -~ MI/M ° , it is clear that we obtain an
J(M)
M
it will suffice to show the inclusion
M'C M
of
what might be called its image, that is,the layer
is given by an isomorphism
f/M
M
for each
Ki_B
- ~
(N,u), where
By virtue of the hypothesis that
whenever
is in. B
Denoting the inclusion functor by
category over
M
QA= consisting of those objects which are also
B.
B ,
is a homotopy equivalence
We define functors
r : J(M) --~ J(M')
,
(Mo,M1) ~-~ (Mo~M', MI~M')
s-~(M)-~(z~)
, (Mo,M, ) ~
(MonM',M1).
These are well-defined because
MI~M'/Mon~' C MI /MonM'
and because
idj(M, )
B
C MI/MoxM/M'
is closed under subobjects and products by assumption.
and that there are natural transformations
ir--~ s 4-- idj(M)
Note that
ri =
represented by
(MonM', MlnM') ~ (MonM', M1) ~ (Mo, M I) •
Hence by
Prop. 2, r
Corollary I.
is a homotopy inverse for
Let
A
be an abelian cate$or~ (with a set of isomorphism classes) such
that every object has finite len6th.
Then
~A
~--X=
where
fxj, j 6 J~
objects of
A , and
i, so the proof is complete.
II
KiDj
j~j
is a set of representatives for the isomorphism classes of simple
Dj
is the sfield
End(Xj) °p.
112
105
Proof.
F r ~ the theorem we have
29
K B = K A , where
B
is the subcategory of semi-
simple objects, so we reduce to the case where every object of =A
the fact that
we reduce to the case where =A
then
is semi-simple.
K-groups commute with products and filtered inductive limits
M~-~HOm(X,M)
has a single simple object
is an equivalence of ~
X
Using
(§2, (8),(9))
up to isomorphism.
But
with ~(D), D = End(X)°P, so the corollary
follows.
Corollar~ 2. If
I
is a nilpotent two-sided ideal in a noetherian rin~
Kx(A/I) -~ K~A , (notation as in
~4,(6)).
This results by applying the theorem to the inclusion
Theorem 5.
(Localization)
A, then
Let
B
Modf(A/I) C Modf(A).
be a Serre subcategory of
A , let =A/B be the
associated quotient abelian categor~ (see for example [Gabriel], [SwanJ), and let
• : B
--, A= , s : =A -* ~ B
=
denote the canonical functors.
Then there is a long exact
_
sequence
s.
.....
e.
K I (~_B)=
_
.
~ ~
s.
;
KA
,
Ko(~B)= ----. 0 .
(It will be clear from the proof that this exact sequence is functorial for exact
functors
(A,B) --~ (A',=B'). Unfortu~uately the proof does not shed much light on the
nature of the boundary map ~ : Ki÷I (_~__B)~
Ki(B ) , and further work remains to be done
in this direction.)
Before taking up the proof of the theorem, we give an example.
Corollary.
If
A
F, there is a loW
is a Dedekind domain with quotient field
exact
sequence
> Ki+IF
> ~
Ki(~m)
• KiA
~ KiF
~ . .
m
where
m
runs over the maximal ideals of A.
This follows by applying the theorem to A
= Modf(A), with
=
torsion modttles, whence
We have
Ki~ = KiA
~ote that the map
phism A --~F
by Cor. 2 of Theorem 3, and
__~. One knows that
aM ---~O.
the subcategory of
A--~A/m
Ki(~m) - ~ K i A
0
in A= , and let
is the full subcategory of ~
Hence the composite of
the constant functor with value
is the transfer map associ-
in the sense of the preceding section.
Fix a zero object
B
Ki~ = ~ K i ( A / m ) by Theorem 4, Cor. I.
in the exact sequence is the one induced by the homomor-
as in ~2, (10), and the ~ p
Proof of Theorem 5.
B
=
is equivalent to Modf(F)= ~(F), (compare[Swan, p. 115]).
KiA --*KiF
ated to the homomorphism
in
~
Qe : ~B
--~QA
with
=
=
O,
so
>
Qe
O\Qe
O
also denote its image
consisting of M
Qs : QA
--~ Q(~B)
=
_
=
such that
is isomorphic to
factors
~ QA=
In view of Theorem B, ~I, it suffices to establish the followi~ assertions.
a)
For ever~
b)
The functor
u : V'--* V
in
Q~_- ~2 0 \Qs
Q(_A/B), u*: V \ Q s -@V'\Qs
is a homotopy equivalence.
113
is a homotopy ea~ivalence.
106
Factoring
when
u
u
into injective and surjective maps, one sees that it suffices to prove
is either injective or surjective.
its dual does not change the
iajective in
u = i! ,
Q-category
(§2,(7))°
the i n j e c t i v s map ~ :
FV
Finally we have
for ~ y
As surjective maps in
is an isomorphism.
when
u
become
is injective, say
a)
for
__A/B .
be the full subcategory of
u : V - ~ sM
a)
Q(_~B)
i!~,: = iV! , so it suffices to prove
V in
a)
On the other hand, replacing a category by
Q((_~B) O) = Q(A°/=B°), it is enough to prove
i : V'~--~V.
Let
30
V\Qs
Clearly
~
consisting of pairs
is isomorphic to
~
(M,u)
such that
, so assertion
b)
results from the following.
Lemma I.
The inclusion functor
Denoting this functor by
F V -@ V \ i s
f, it suffices by
is contractible for any object
(M,u)
of
Theorem A
V\~s.
represented by an isomorphism
V -~ VI/V ° , where
easily seen that the category
f/(M,u)
(Mo,M I)
in
M
such that
is a homotopy equivalence.
to show the category
Let the map
(Vo,V I)
u : V -~ sM
is a layer in
in
eM.
f/(M,u)
Q(=A/B) be
It is
is equivalent to the ordered set of layers
(aMo,SM1) = (Vo,VI), with the ordering
(Mo,MI)~< (M~,M~)
iff
MoCMoC M|CM; . This ordered set is directed because
(~o.M,) <. (~%~M° , MI+ . ; ) ~ (M~,M~)
It is non-empty because any subobjsct
In effect,
V I = aN
s(g)s(i) -I
where
one can take
then
for some
N
i : N')--~N
MI
VI
in
of
sM
is of the form
A , and the map
has its cokernel in =B
to be the ~ms~e of
g.
Thus
V I --~ sM
and
sM I
MIC'M.
can be represented as
g : N' - ~ M
f/(M,u)
for some
is a map in
A ;
_-
is a filtering category, so
it is contractible by Prop. 3, Cot. 2, proving the lemma.
The next four lemmae will be devoted to proving that the category
equivalent to
~_.
h : M ~N
is homotopy
To this end we introduce the following auxiliary categories.
be a given object of
where
=FV
=A , and let
~
be the category having as objects pairs
is a mod-B_-- isomorphism, i.e. a map in
A
Let
whose kernel and cokernel are
=
in B ' or equlvalently one which becomes an isomorphi= in _~. A mOrphiem f = ~
to (M',h') in ~
is by definition a map u : M --~M' in Q~= such that
(*)
1~ i
j
h
M
"i'
N
(M,h),
(M,h)
h'
~N
!
commutes i f
of
B
=
by
(*)
u = i!j'.
TO each
(M,h)
in
_~
we associate
determined up to canonical isomorphism.
we associate the map in
~_
Ker(h)~<
induced by
j
and
i
respectively.
To the map
Ker(h), which is an object
(M,h) -->(M',h')
represented
represented by the maps
_
Ker(hj) 3
~ Ker(h')
It is easily checked that in this way we obtain a
functor
,
determined up to canonical isomorphism.
(M,h)
~
We prove
114
Ker(h)
~
is a homotopy equivalence in two
31
107
steps.
Lemma 2.
Let E~
that h : M --~ s
be the full subcate~or~ of
ia an e p ~ o r p ~ .
_~
consistiu~ of pairs
~hen the restriction
~
(M,h) such
: ~ --* ~
o__r ~
18 a
homotopy e~uivalence.
I t s u f f i c e s to prove ~ / ~
i s contractible for any ~
is the fibred category over ~
where
u : Ker(h) - ~ T
((M,h),u)
with u
consisting of pairs
is a map in
surjective.
QB
=
and define
Her(h) > .... ~ M
ITo ~....
~t
((M,h),u), with
~ = ~/~
; it
(M,h) in ~
, and
!
in C ,
write
u=
j'i! with
(i.M,h) by 'pushout':
h ~> N
I
lJ
> i.M
,1~ ~ l I
!
Let X = ((i.M,h),j'); it belongs to C'
and there is an evident map X -@ ~.
verifies as in the proof of Theorem 3 that X - ~ X
object of C'.
~.
be the full subcategory consisting of
Given X = ((M,h),u)
i : Ker(h) > - ~ T ° , j : T - - ~ T °
--
Let C'
=
in
Hence the inclusion
reduced to proving that
C'
C'--~ C
has the left adjoint
is contractible.
One
is a universal arrow from X
But
C'
to an
X ~-~, so we have
has the initial object
((N,i%), jT ) , so this is clear, whence the le~aa.
Lemma 3.
The functor ~
:~
-@ ~
is a homotop~ e~uivalence.
Thanks to the preceding lemma, it suffices to show the inclusion
homotopy equivalence.
is in
I
be the ordered set of subobjects
=B , and consider the functor
easily that
=E~ to
and
Let
_~
f
being
in
I.
I
being
J xi? : (M--*,I) ~-~ (J l ~
kj , it follows from
JCI
f : _~ --+ =I sending
is fibred, the fibre over
Lemma 2 that
f
over
I.
Since
I
Im(h).
One verifies
Since
commutes with
J Xl?
kI
_E_i is homotopy equivalent to the
is contractible (it has
one knows from homotopy theory that the inclusion
for each
(M,h) to
is a
such that N/I
is a homotopy equivalence for every arrow
From Theorem B, Cot., we conclude
homotopy-fibre of
_~ -~ _~
of N
E~ , and the base change functor from
-*,J).
J Xl?
I
E~ -* _~
N
for final object),
is a homotopy equivalence
I, proving the lemma.
We now want to show FV
is homotopy equivalent to _~
when
aN ~
V.
First we note
a simple consequence of the preceding.
Lemma 4.
functor
Let
g : N --~ N*
g. : _~ --* ~ ,
be a map in A
, (M,h) ~-> (M,gh)
which is a
mod-=B isomorphism.
Then the
is a homotopy equivalence.
One verifies easily that by associating to (M,h)6 ~
the obvious injective map
Ker(h)---> Ker(gh), one obtains a natural trar~sformation from ~
to ~ , g . °
(Observe:
In 'lower* K-theory one calculates with matrices- in 'higher* K-theory with functors.)
~h~
%
~
%g.
~ s homotopic, and since %
and % ,
are h~otopy equivalenoes.
so is g. , whence the lemma.
Now given V
where
~
is in A
in ~B_ , let
and
_~
~ : aN ~ V
be the category having as objects pairs
is an isomorphism in
115
(~,~),
_~_B , in which a morphism
108
32
(~.#) -, (~'.#') ie a =~p g : N -, ~' such that d,s(g) = ¢ .
construction of
=A/B that
g2 : (N,~) -~ (N',d')
__~ is a filtering category.
we have
s(gl-g2) = O, so
(N',~') --~ (N",#") equalizing
We have a functor from
(N',~')
to
gl 'g2
g. : _~ --) _~, . Further, for each
P(N',#') g~
=
P(N,#)
_-;v '
g1'
/m(gl-g2)~=B, hence we obtain a map
N" = N'/Im(gl-g2).
_Iv to categories sending
P(~,~) ' -~ - +
Since
with
It is clear from the
For example, given two maps
(N,~)
(~,h) ~
for any map
(N,~
to
_~
and
g : (N,~) --~
we have a functor
(~, s(h)-1~-1: V ~
g : (~,~) -~ (N',~')
in
~
-~ .~)
.
__~ , we obtain a
func tor
=Iv
which we claim is an isomorphism of caBegories.
(~,
for any
(M,@)
in
e : v = ~)
we otain a map
(**)
is surjective on objects.
M = M ~ and
g : (N,#) -.(u',d')
is injective on objects.
Applying
Prop. 3, Cor. I, we obtain from
Lemma 5.
For any
~ : ~i -~V, the functor
: V \ O ~ -@ O \ Q s
s(hi, = e(h') .
Also given
Letting
N' = N/Im(h-h')
such tha~ g.(~,h) = g.(M',h'), ~cwing t~at (**)
The verification that
The end is now near.
(~!)*
= p(M,O-I)(M,i%)
FV , s h o w i ~ that
p(N,~)(M,h) = p(N,~)(g',h'), then
In effect
(**)
is bijective on arrows is similar.
Lemma 4 and
P(N,~)
(**)
the following.
is a homotopy equivalence.
To finish the proof of the theorem, we have only to show
i S a homotopy equivalence.
Choose
(N,~)
a s i n Lemma 5 and form
t h e diagram
_~
P(N,~) ~ Fv = v \ ~
~£
I
(~v,)*
The diagram is not commutative, for the lower-left and upper-right paths are respectively
the functors
(M,h)
,L _~
(Ker(h), 0 -~ s(Ker(h)) )
(M,h)
~
(M, (JAM) ,. : O --~ aM) .
However it is easily checked that by associating to
(M,h)
the obvious injective map
Ker(h) --, M, one obtains a natural transformation between these two functors.
Thus the
diagram is homotopy commutative, and since all the arrows in the diagram are homotopy
equivalences except possibly
one a l s o .
(~!)*
by Lenmaas I, 3, and 5, it f o l l o w s that
The p r o o f o f t h e l o c a l i z a t i o n
theorem i s now c o m p l e t e .
116
(~!)"
is
33
109
§6.
Filtered rings @nd the homotopy property for regular rings
This section contains some important applications of the preceding results to the
groups K~A = Ki(Modf(A))
for
A
noetherian.
If A
is regular, we have
the resolution theorem (Th. 3, Cot. 2), so we also obtain results about
regular.
In particular, we prove the homotopy theorem:
K-groups of Kareubi and Villamayor for
KiA
by
K A for A
i
for A regular.
KiA = Ki(A[t])
According to [Gersten I], this signifies that the groups
KiA = K~A
are the same as the
A regular (assuming Theorem I of the announcement
[Quillen 11 which asserts that the groups
KiA
are the same as the Quillen K-groups of
[ reten I]).
Graded rings.
Let
B = ~
B
, n~O
we consider only graded B-modules
Ti(N)
where
k
is regarded as a right
Ti(N)
is a graded
Denote by FpN
be a graded ring and put
N = ~
=
Nn
with
Tor~(k,S)
B-module by means of the augmentation
k-module in a natural way, e.g.
the submodule of N
o-F_,NC;oN
To4F ) n
To(N) n =
generated by Nn for
It
fl)
k = B° . From now on
n>. O, unless specified otherwise. Put
Then
Nn/(AINn_I + .. + AnNe).
n.~ p, so that we have
that
is clear
_-
B --~ k.
o
( To(~) n
n>p
n<, p
and that there are canonical epimorphisms
(2)
B(-p) ~k To(N)p
where
B(-p) n = Bn_p .
Lemma I.
I_~f TI4N) = 0 an__d Torlk(B,To(N)) = 0
isomorphism for all
Proof.
(5)
':" F~/F~-IN
k-module
resolution of
X
is a
~
is an
Ti(B~kX)= 0 for i>O.
k-projective resolution of X, then
B ®kX , and
B ®k P.
Ti(B~k x) = Hi(k ~B B ®kP.) = Hi(P. ) = 0
particular by the hypothesis on
44)
42)
we have
for i>O
Tor~(B,X) = 0
P.
i>O, then
p.
For any
In effect, if
for all
is a
for
B-projective
i>O.
In
To(N), we have
Ti(~®k~o(~)) = 0 for i > O .
Let R p
be the kernel of
42). Since
(2)
clearly induces an isomorphism on
To, we
obgain from the Tor long exact sequence an exact sequence
T1 (B(-p) ~kTo(N)p)n --~ T1 (FpN/Fp.lN)n ~-~ To(RP)n----> OThe first group is zero by
Fix an integer
s.
also that T I(FpN)n = 0
true, because
(4), so
for n~<s
TI(FpN)n= TI(N) n
Assuming TI(FN) n = O
~
is an isomorphism.
We will show that
(2) is an isomorphism in degrees ~<s and
by decreasing induction on
for
p~.n, and because
for n.<s, we find from
117
(I)
p.
TI(N) = 0
For large
p, this is
by hypothesis.
and the exact sequence
ii0
34
TI(FpN) n --'~TI(FpN/Fp_I~) n --~To(Fp_IN) n "-~To(FpN) n
that
T I(FpN/Fp_IN) n = To(RP) n = 0
~s, showing that
(2)
for
n~ s.
It follows that R p
is an isomorphis~ in degrees
find o = T2(B(-p) ~kTo(N )p)n
~---T2(FpN/Fp_IN)
n -
is zero in degrees
..~s as claimed.
for
In addition we
n ~ s, whence from the exact sequence
T2(rpN/Fp_1~) n --> T I (Fp.IN)n --* T I (FpN)n
we have
TI(Fp_IN) n = O
for n~s, completing the induction.
Since
s
is arbitrary,
the lemma is proved.
Suppose now that B
is (left) noetherian, and let Modfgr(B)
category of finitely generated graded
over ~[t], where the action of
The ring k
t
induced by the exact functor
F
be the abelian
K-groups are naturally modules
B
N ~N(-I).
has finite Tor dimension as a right
k-module,
(~4,(6))
(B ~k?). : K~k
(~)
Its
is induced by the translation functor
is also noetherian, so if
we have a homomorphism
k-modules
B-modules.
such that
,~ Ki(Mo~gr(~))
B ~k ?
on the subcategory
Tork(B,F) = O
for
~
of Modf(k)
consisting of
i>O.
i
Theorem 6.
Suppose
B
is a graded noetherian ring such that
dimension as a right
k-module, and such that k
B-module.
extends to a ~[t]-module isomorphism
Then
(5)
~[t] ~i~ K~k
(The hypothesis that k
For example, if k
ring over k.
over k.
"~
has finite Tor
Ki(Modfgr(B) ) .
be of finite Tor dimension over
is a field and
B
has finite Tor dimension as a right
B
is commutative, then
B
B
is very restrictive.
has to be a polynomial
In all situations where this theorem is used, it happens that
Does this follow from the assumpltion that
B
and
k
B
is flat
are of finite Tor dimen-
sion over each other?)
Proof.
Ti(N) = 0
that
Let
for
N'
=
be the full subcategory of Modfgr(B)
i>O, and let ~"
consisting of
N
such that
be the full subcategory of N' consisting of
N
such
To(N) E ~ . By the finite Tot dimension hypotheses and the resolution theorem
one has isomorphisms
subcategory of N"
K.Fx== K~k , K.N"x== K N'x= = Ki(M°dfgr(B))"
consisting of N
such that
=
F N = N.
(~4)
Let N"=n be the full
We have homomorphisms
n
(KiF)n= =
Ki(Fn
1=
induced by the exact functors
and ~ ~-* (To(N)j)
b ) Ki(~_~
)_
(Fj , O ~ j ~ n) ~
respectively.
Clearly
c , (Ki{)n=
~
cb = id.
in =nN" has an exact characteristic filtration
B(-j) ®~Fj__ (this is in ~"
OCFoNCJ..C-FnN = N
B(-p) ®kTo(~)p , so applying Th. 2, Cor. 2, one finds that
isomorphism, so by passing to the limit over
n
bc = id.
with
F~/Fp_1~p_ =
Thus
b
is an
we have ~It]® Ki~ ~*Ki~", which proves
the theorem.
The following will be used in the proof of Theorem 7.
118
by (3))
On the other hand, by Lemma I any N
iii
Lemma 2.
Suppose
dimension as a risht
B
is noetherian, k
B-module.
Then any
finitely ~enerated pro~ective ~raded
Proof.
Pr-1
)
N
~
r
is projective.
Ti(N r) = Ti+1(Nr_1)
d
for
O
for
k
in
Modfgr(B)
)
k
To(Nd)
is projective for
over
B.
Then for
r~d
~To(Pr_ I) ~ T o ( N r _
Filtered rin~s.
Let
B = gr(A) = ~
A
M
-~,=- F_IM
O
~FoMC..
~4FpM/Fp_IM
Le,ma 3.
ii) If
s, so
To(Nr)
is projective
is projective, whence the lemma.
p+qA , and ~ F p A = A.
k = FoA = B °
Let
By a
A-module equipped with an increasing filtration
FpA.FqMCFp+qM
and U F p M = M . Then
gr(M) =
is a finitely ffenerated B-module, then M
In particular, if ever~ sTaded left ideal in B
is a projective
gr(M)
is a finitely
is finitely
B-module, then M
is a pro~ective
A-module.
has a resolution by finitely 6enerated pro~ective 6raded
of length ~ n, then M
Proof.
r~ d, where
is noetherian.
gr(M)
iii) If
and
B-module in anatural way.
gr(M)
A-module.
~enerated, then A
i>O
Since
~ 0 .
be the associated graded ring and put
such that
i) If
I)
I ~ F Ao ' F AA-F
C qF p
we will mean an
is a graded
large.
be a ring equipped with an increasing filtration by subgroups
such that
FpA/Fp_IA
A-module
for
r
we have exact sequences
has finite projective dimension
O = F_IAC-FACFIAC...
~enerated
has finite Tor
) 0
Nr-1
i~O, it follows that Ti(N r) = O
~To(N r)
is regular,
,
Pr-1
r ~ d+s . It follows from Lemma I that Nd+ s
filtered
k
has a finite resolution by
B-modules.
We have to show N r
is the Tor dimension of
As
is re$,ular, and that
N
Starting with N O = N, we recursively construct exact sequences in Modfgr(B)
0
where
35
has a
B-modules
~(A)-resolution of length ~ n.
We use the following construction.
Suppose given k-modules
L
and maps
J
of k-modules L - - ~ F M
J
J
for each
Lj --~FjM
is onto. Let
P
j~O
suchthat the composition
>grj(M)
be the filtered
~ To(gr(M)) j
A-module with
be such that
~
Lj
Then To(gr(P)) j = Lj , and
to FjM.
restricted to A @kLj
~
gr(~)
FnP = I ~
Fn_jA ~kLj
and let
~ :P--.M
A-linear extension of the given map from
is a map of filtered
is onto, hence
Fn(~)
A-modules such that
To(gr(~))
is onto.
and so
is onto. Thus if K = Ker(~) , F K = KnFnM , we have an exact sequence of
~
It follows that
is the
is onto for all
n,
A-modules
O ~
>K
)P
~M
~O
such that
,o
o
(6)
o
are exact for all
i):
If
gr(M)
~K
)~nP~g~M
~0
n.
is a finitely generated
119
B-module, then
To(~(M))
is a finitely
112
generated
k-module, hence we can take
which is zero for large
j.
Then
P
56
L
to be a free finitely generated k-module
J
is a free finitely generated A-module, so M is
finitely generated, proving the first part of
to be a left ideal of
ii):
If
we can take
gr(M)
A
i).
The second part follows by taking
M
and endowing it with the induced filtration
is projective over
Lj = To(gr(M))j ,
Then
B, then
To(gr(~))
F M = MnF A .
n
n
is projective over k, and
To(g~r(M))
is an isomorphism, so from the exact
sequence
we conclude that
To(gr(K)) = O.
projective over
iii):
A, proving
We use induction on
Assuming
gr(M)
B-module.
P
as in the proof of
From the exact sequence
we know that
graded projective
a
n, the case
n = 0
has a resolution of length
B-modules, choose
le~aua),
Then gr(K) = O, so
K = 0 , M = P , and
M
is
ii).
gr(K)
~ n
being clear from
and
ii).
i), so that
gr(P)
is a free finitely generated
(6), and the lemma after Th. 3, Cot. I, (or Schanuel's
has a resolution of length ~<n-1
B-modules.
i)
by finitely generated graded projective
by finitely generated
Applying the induction hypothesis, it follows that
P(A)-resolution of length $ n-l, so
=
M
has a
K
has
P(A)-resolution of length ~ n, as was
=
to be shown.
Lamina 4.
as a right
If
B
is noetherian,
B-module, then
A
k
is regular, and if
k
has finite Tot dimension
is re$ular.
This is an immediate consequence of Lemma 2 and Lamina 3 iii).
We can now prove the main result of this section.
Theorem 7.
Let
0 = F_mA<FACFIAC
B = gr(A)
such that
and
A
~/so if
B
16FoA , FA.FACFp+A
k = FoA.
FoA
FoACA
has Tor dimension
d
n, so the same is true for
B
A
Suppose
is of finite Tot dimension as a
induces isomor~hisms
K~(FoA ) ~" KIA .
is noetherian, we h o w
over
k, then
If further
Ki(FoA ) - ~ KiA .
FnA/F_IA
A
is also by Lemma 3 i).
has Tot dimension
FnA , and hence also for
A.
Lemma 4 and the fact that
~d
Thus the map
is defined, and we have only to prove that it is an isomorphism.
tion of the theorem results f r ~
= A.
is of finite Tor dimension as a right
A, and we have isomorphisms
Since
,and ~ F p A
is of finite Tot dimension as a risht module over
are noetherian and
Then the inclusion
i~t
B
Suppose also that
is re~alar, then so is
Proof.
each
FoA
FoA-modtule ) .
B-module.
FoA
be a ring equipped with an increasin~ filtration
...
is noetherian and that
B o = FoA, (hence
right
A
for
K~k -~ KIA
Indeed, the last asser-
KiA = K~A
for regular
A
by
the resolution theorem (Th. 3, Cor. 2).
Let
z
be an indeterminate and let
show the graded ring
A'
central non-zero-divisor in
As
k
be the subring
~(FA)z n
satisfies the hypotheses of Theorem 6.
finite Tor dimension over
A'.
A'
k
is clear from the preceding paragraph.
A', we have that
has finite Tor dimension over
B = A'/zA'
A[z]° We
Since
z
A'
has
is a
is of Tor dimension one over
B, it follows that
120
of
The fact that
k
has finite Tot dimension
37
113
over A'.
Finally to show A'
is noetherian, we filter A'
those polynomials whose coefficients are in F A.
P
gr(A') =
II (grpA)zn
p~<n
is isomorphic to gr(A)[zJ, which is noetherian, h@nce A'
= O
Let
F__ be the full subcategory of Modf(k)
for
i>O, whence
Theorem 6 to B
and
KiF = K;k
by letting F A'
P
The ring
consist of
is noetherian by Lemma 3 i).
consisting of F
such that
By the resolution theorem (Th.3, Cor. 3).
Tork(B,F)
Applying
A', we obtain ~[t]-modtLle isomorphiems
(7)
Z~[t]® Ki~
Let
z
B
Ki(Modfgr(A')) ,
~
be t h e S e r r e s u b c a t e g o r y o f
is nilpotent.
(A'®k?),x .
, ® x ~
A = Modfgr(A' )
consisting
o f modules on which
The functor
j = Modfgr(A')
--~ Modf(A)
, M~
M/(z-I)M
is exact and induces an equivalence of the quotient category _~B
[Swan, p. 114, 130]; note that if
S =
Iz n} , then
with Modf(A). (Compare
S'IA ' is the Laurent polynomial ring
Air,z-13, and a graded module over A[z,z-lJ is the same as a module over
Since
A = A'/(z-1 )A'.)
A'/zA' = B, we have an embedding
i • Mo~r(B)
---~ Mo~gr(A')
identifying the former with the full subcategory of the latter consisting of modules
killed by
z.
The devissage theorem implies that Ki(Modfgr(B)) = Ki=B . Thus the exact
sequence of the localization thorem for the pair
(8)
i .:.~
~.~ Ki(Modfgr(B) )
We next compute
i.
(A,B) takes the form
j.
Ki(Modfgr(A'))
~ K:A
with respect to the isomorphisms
~.
(7). Associating to F
in __F
the exact sequence
O---~A(-~)~#
"-~- A=kr
>,~#---.-,.o
we obtain an exact sequence of exact functors from
F
=
to Modfgr(A').
Applying Th. 2.
Cor. I, we conclude that the square of Z~[t]-module homomorphiems
~-t I
~i.
=It] o Ki~ ~ ,
is commutative.
sequence
(8)
Since
1-t
Kimodf~r(A))
is injective with cokernel
K.F , we conclude from the exact
that the composition
Ki~
~
Ki(MoGfgr(A')) - - ~
induced by F~-~ A' ~k F W-? A ®kF
K~A
is an isomorphic.
the theorem.
121
Since
K.Fz== K:k , this proves
114
38
The preceding theorem enables one to compute the K-groups of some interesting
non-commutative rings.
Examples.
U(~)
Let
c~
be a finite dimensional Lie algebra over a field
be its universal enveloping algebra.
that U(~)
is a filtered algrebra such that
Thus Theorem 7 implies that Kik = KiU(~).
algebra over k
with generators
Pi ' ~
i)
ii)
If
K i(A[t])
A
i)
is a polynomial ring over
~
k.
is the Heisenberg-Weyl
~i,Pj] =
Kik -- KiHn .
is noetherian, then there are canonical isomorphisms
=
KIA
Ki(A[t,t'1])~
Proof.
gr(U(~))
Similarly if
' 1~i~<n, subject to the relations
[~,qj] = O , /pi,qj ] = ~ij ' then we have
Theorem 8.
k, and let
The Poincare-Birkhoff-Witt theorem asserts
KIA (~ KI_mA
follows immediately from the preceding theorem.
ii): Applying the localization theorem to the Serre subcategory
consisting of modules on which
~
Ki=B
t
~
is nilpotent, we get a long exact sequence
K~(A[t])
K'A
1
_B of Modf(A~t])
~
Kl(A[t,t-1])
>
K'A
1
where the first vertical isomorphism results from applying the devissage theorem to the
embedding Modf(A)=Modf(A[t]/tA[t])C_B
to
I makes
A
Ki(A[t,t-1]) ~
. The homomorphism A [ t , t - 1 ] ~ A
left inverse to the oblique arrow.
Thus the exact sequence breaks
ii).
(#~undamental theorem for regular rin6s) If
are canonical isomorphisms
This is clear from
Exercise.
t
a right module of Tor dimension one over A[t,t-1], so it induces a map
K'AI
up into split short exact sequences provir~
Corollary.
sending
Let
~
Ki(A[t]) = KiA
and
Th. 3, Cor. 2, since
A ~
,
then there
Ki(A[t,t-1~ ) = KiA ~ Ki_IA .
A[t] and A[t,t -1] are regular if A
be an automorphism of a noetherian ring
A, and let
is.
A~[t],
A~it,t -I] be the associated twisted polynomial and Laurent polynomial rings in which
t.a = ~(a).t , ([Farrsll-Hsiang]).
Show that KIA = Ki(A~[t])
and that there is a long
exact sequence
(9)
~ Ki A .I-~,
.............K:AI
~ KI(A~ [t't-1]) ~
Ki-IA'
)
We finish this section by showing how the preceding results can be used to compute
the
K-groups of certain skew-fields.
already in the case of
is not known.
Keith Dennis points out that this has some interest
K2 , since a non-c~mmutative generalization of Matsumoto's theorem
(Here and in the computation to follo~, we will be assuming Theorem I
the announcement [Quillen I ] , which implies that the ~ A
and that the groups
K.R
l q~
Example I.
k
Let
of
here is the same as Milnor's,
are the same as the ones computed in [Quillen 23.)
be the algebraic closure of the finite field R~p, and let
be the twisted polyno~al ring
k~[F~
with
122
Fx = xqF
for
x
in k, where
q = pd
A
be
115
Then
A
is a non-commutative domain in which eve/y left ideal is principal.
the quotient skew-field of
category consisting of
dimensional over
(1o)
(A
39
k.
D
MOdf(D) = Modf(A)/B, where
additive map
i. ~ KiA
bijective on
B
~ KiD
*
D
be
KiA = Kik
Ki-IB
by
Theorem 7.
is a finite dimensional vector space
F : V --, V
well-known that
Let
is the Serre sub-
A-modules which are torsion, or equivalently, which are finite
are regular), and we have
An object of
B
The localization theorem gives an exact sequence
~ K x=
B
and
A, whence
V
such that
F(xv) = xqF(v)
splits canonically:
for
¥
x
over
k
k
and
in
V = V ° @9 V I , where
F
equipped with an
v
in V.
is nilpotent on
It is
V°
and
V I , and moreover that
k
v'
vI
q
where
~
of
with
k
= i v ~ V [ Fv = ~
q
elements.
B
=
is a finite dimensional vector space over the subfield
q
Thus we have an equivalence of categories
~
Modf
x
Modf
.
n
Applying the devissage theorem to the first factor, we obtain
Let
d : k --~ k
base extension of the
k
k-vector space
is regarded as a right
,
On t h e o t h e r h a n d , i f
A®kd(V)
~, i.e.
~(V) = k ~k V, where
V
is regarded as an
,
V
~ A®kV
A-module killed by
: 0
~-~ a x P ® v
W is a finite
~ A~W
dimensional vector
.~ A ~ W
q
a ® w
F
If
space over
I r , we h a v e an
q
A-modules
0
where
with respect to
~.
A-modules
a®(x®v)
exact sequence of
V
k-module via
F, we have an exact sequence of
0
Ki__B= K.k ~ K.K~ .
1
I q
~(x) : x q, and let ~(V) denote the
be the Frobenius automorphism:
}-~
acts on the cokernel by
,~ k ~ W
q
a(F-1)Ow
~ > 0
q
F(x ~ w) = x q ~ w .
Applying
Th. 2, Cor. I, to these
"characteristic" sequences, one easily deduces that the composite
i,
Kik d) KiF q = KiB=
~ KiA = K.kx
is zero on the factor
K.F
xq
and the map
I - ~.
on
Kik . From [Quillen 2] one has
exact sequences
1-d.
0
for
i > O.
~
K.F
i q
•
Combining this with
KoD
(11)
K2iD
K.k
l
(10)
=
.~ K . k
l
K2i+ID =
0
we obtain the formulas
2~ , KID
=
•
(K2i_,~12
=
7Z.~2Z
=
( ~ ( q i _ 1 1~.)2
(K2i )2 = o
123
i>o
i>o .
116
Example 2.
pq - qp = I
of
H.
Let
H
40
be the Heisenberg-Weyl algebra with generators
over an algebraically closed field
k, and let
D
p,q
In this case, one can prove that the localization exact sequence associated to
Modf(R)
and the Serre subcategory of torsion modules breaks up into short exact sequences
O
~" K.kl
~ K DI
~
~I Ki_Ik
~
0
where the direct sum is taken over the set of isomorphism classes of simple
The proof is similar to the preceding, the essential points being
generated
over
such that
be the quotient skew-field
R-modules are of finite length, because
k, and
b)
k
J.
If
X
where
M(X)
KqX = KqP(X),= where
(= locally free sheaves of
usual notion of exact sequence.
([Quillen 3]).
If
X
P(X)= is the category of vector
_Ox-modules of finite rank) equipped with the
is a noetherian scheme, we put
is the abelian category of coherent sheaves on
concerns primarily the groups
H-module
K~-theor~ for schemes
is a scheme, we put
X
has no modules finite dimensional
is the ring of endomorphisms of any simple
§7.
bundles over
H
H-modules.
a) torsion finitely
X.
K'Xq = K¢~(X),
The following theory
KqX, so for the rest of this section we will assume all
schemes to be noetherian and separated, unless stated otherwise.
As the inclusion functor from
(1.1)
KX ~
X
M(X)
is exact, it induces a homomorphism
~'X
q
When
F
=P(X) to
q
is regular this is an isomorphism.
In effect, one knows that any coherent sheaf
is a quotient of a vector bundle [SGA 6 II 2.2.3 - 2.2.7.1] , hence it has a resolution
by vector bundles, in fact a finite resolution as
[SGA 2 VIII 2.4]). Thus
If
E
1.1
is a vector bundle on
itself, hence as in
(I .2)
§3 ,(I),
KX
o
X
is regular and quasi-compact
(see
is an isomorphism by the resolution theorem (Th. 3, Cor. I).
X, then
F ~-~ E ® F
is an exact functor from
M__(X) to
we obtain pairings
® K'X
q
--*
K'X
q
making
K'X
a module over the ring KoX. (In a later paper I plan to extend this idea to
q
define a graded anti-commutative ring structure on K,X such that K~X is a graded
module over
2.
K,X.)
Functorial behavior.
If
f : X --, Y
morphism), then the inverse image functor
is exact, hence it induces a homomorphism of
42.1)
f~ : K Y - ~ K J
q
K'q
f* : M(Y) -~ M(X) )
K-groups which will be denoted
( resp. f* : K ' Y - , K'X ) .
q
q
It is clear that in this way
groups, and that
is a morphism of schemes (resp. a flat
f* : P(Y) -@ =P(X) (resp.
K
becomes a contravarlant functor from schemes to abelian
q
is a contravariant functor on the subcategory of schemes and flat
morphisms.
124
ll7
Propositien 2.2. Let
i ~-~ X i
the
. . . .transition.
.
.
morphiams
.
.
.Xi-#X j
(2.5)
KX
be a filtered pr0jeqtive s~stem of schemes such that
are affine, and let X = ~limX''x Then
=
q
41
limKX
--~
q~
.
If in addition the transitlon mo~hisms are flat~ then
(2.4)
K'X
q
Proof.
=
We wish to apply
inductive limit of the
lira
--~ K'X..
q~
~2 (9), using the fact that
~(Xi)
by
Let
I
is essentially the
[EGA IV 8.51 . In order to obtain an honest inductive
system of categories, we replace
as follows.
~(X)
=P(Xi)
by an equivalent category using Giraud's method
be the index category of the system
category obtained by adjoining an initial object
~
to
X i , and let
I.
I'
be the
We extend the system
X.
to
I
I'
by putting X% = X, and let ~
=P(Xi)
over
object of
i.
Pi
Let
i ~
I'
having the fibre
=IP' be the category of cartesian sections of
is a family of pairs
(j -~i)*Ei ---~E
and
be the fibred category over
for each object
=~i is a functor from
(Ej, @j)
j .i
I°
of
with
Ej6=P(Xj)
I'/i.)
Clearly
to categories.
=P over
and
P:
ej
I'/i. (An
an isomorphism
is equivalent to
P(X~)
Using [EGA IV 8.5] it is not hard
to see that we have an equivalence of categories
i _ ~ ( i ~ ~i ) --.-, ~(X)
I
such t~at a sequence is exact in
some
=xP'" Thus from §2 (9)
P(X)
we have
if and only if it comes from an exact sequence in
Kq=P(X) = lim._~KqP
i= , proving 2.3. The proof of
2.4
is similar.
2.5. Suppose that
f : X -, Y
is a morphism of finite Tor dimension (i.e. OX
of finite Tot dimension as a module over
category of M=(Y) consisting of sheaves
f-1(O ) ), and let
Y
such that
P(Y,f)
=
is
be the full sub-
F
%
Tori= ( ~ , F ) = O
Assuming that every
theorem
implies that the inclusion
induces isomorphisms
Combining this isomorphism with the homomorphism induced by the exact
f* : K'Y
q
--,
The assumption holds if either f
sheaf on Y
K'X .
q
is flat (whence P(Y,f) = M(Y) ), or if every coherent
is the quotient of a vector bundle (e.g. if Y
In both of these cases the formula
247. Let
f : X -* Y
for
=F(X'f)°'#
~(X)
°=
i
large
(fg)* = g'f*
has an ample line bundle).
is easily verified.
be a proper morphism, so that the higher direct image functors
carry coherent sheaves on X
full subcategory of M(X)
Rif. = 0
__P(Y,f),the resolution
=P(Y,f)--~ M(Y)
f* : =P(Y,f) -, M(X), we obtain a homomorphism which will be denoted
(2.6)
Rlf.
i>O .
in M=(Y) is a quotient of a member of
(Th. 3, Cot. 3)
on K-groups.
functor
F
for
to coherent sheaves on
consisting of F
such that
Y.
Let
Rif.(F) = O
F(X,f)
for
[EGA iII 1.4.12], we can apply Th. 3, Cot. 3
to get an isomorphism
denote the
i>O.
Since
to the inclusion
Kq=F(X,f)-~KqX, provided we assume that every
125
42
118
coherent sheaf on
X
can be embedded in a member of
with the homomorphism of
F(X,f).
Composing this isomorphism
K-groups induced by the exact functor
f. : =F(X,f) -~ M(Y),
we
_-
a homomorphism which will be denoted
obtain
(2.8)
f. : K'X ---* K'Y .
q
q
The assumption is satisfied in the following cases:
~nen
i)
Rif. = O
for
ii) When
on
f
is finite, in particular, when
i>O
X
[EGA III 1.3.2] , so
f
is a closed immersion.
In this case
F(X,f)
= M(X).
=
=
has an ample line bundle
[EGA II 4.5.3].
In effect if
L
is ample
X , then it is ample when restricted to any open subset, and in particular, it is
ample relative to
f.
ample relative to
f , and further that
any
n
Replacing
we have an epimorphism
L
by a high tensor power, we can suppose
L
L
is very
is generated by its global sections.
Then for
(O_l)rn--. L~n, hence dualizing and tensoring with
L f~n, we
obtain an exact sequence of vector bundles
o --.
~-~
Hence for any coherent sheaf
(2.9)
0
where
that
--~
F(n) = r ® L~n.
Rif.(F(n)) = 0
F
(T~)=-~
F
on
--*
X
~.
we have an exact sequence
F(n)rn---~
F ® E
But by Serre's theorem
for
embedded in a member of
o.
-~
i)bO, n ~ n O,
so
--,
0
6EGA IIl 2.2.1], there is an
F(n) 6 =F(X'f) for
n ~ n O.
no
Thus
such
F
can be
=F(X,f) as asserted.
The verification of the formula
(fg). = f.g.
in cases
i)
and
ii)
is straight-
forward and will be omitted.
Proposition 2.10. (Projection formula)
Tcr dimension, and assume
defined.
f.(x)
Then for
and
x e KoX
is the ~ma~e of
Proof.
X
•
Y
and
y~
K~
Rf.(E) ~
We recall that if
f.E , and
such that
Y.
Let
L
F ~ =L,
exact functor
we have
2.6
f.(x.f*y) = f.(x).y
f. : KoX - @ K o Y
Rf.(E).
[E]
Arguing as in case
such that
and
in
__
o_~f [SGA 6
ii)
Rif.(E) = 0
denote the full subcategory of
=
By the resolution theorem we have
--*
proper and of finite
2.8
K 'q Y ,
are
where
2.12.33 .
E, then
f.(x)
above, one sees
for
i>O.
Then
f.(x) = E(-1)i[Pi] ~ KoX , where {P.} is a finite resolution of
Oy
Tori= (f.E,F)
O
f : X -~ Y
• = [E] is the class of a vector bundle
is generated by the elements
by vector bundles on
for
K'J
b~ the homomorphism
is the class of the perfect complex
that
Suppose
have ample line bundles so that
Pn ~ F
---, ..
one sees t ~ t
F~-~f.E ~ F
O
=
K L = K'Y.
q=
q
--*
Po ® F
y ~-~ f.(x)-y
from
Oy
Tori = (_O_x,F)
L
to
F
in
.
--~
is the endomorphism of
to
O
K'Y
q
induced by the
M(Y).
From the projection formula in the derived category:
(see [SGA 6 I i I 2 . 7 J ) , we f i n d f o r
i>O
Moreover, applying Th. 2, Cor. 3
---* f.E ~ F
f.E
M__(Y) consisting of
L
126
that
L
Rf.(E ® ~ )
=
L
Rf.(E) ~
F
F
119
43
C
q
O
[ f.E M F
Thus
E ® f*F
is in
y~f.(x.f*y)
from
L
=
F(X,f), so by the definition of
O
q = 0 •
2.6
and
2.8, we have that
is the endomorphism of
to
K'Y induced by the exact functor F~-~f.(E ®f'F)
q
Since we have an isomorphism f.(E ® f'F) ~ f.E ® F , the projection
M(Y).
=
formula follows.
P r o p o s i t i o n 2.11.
Let
X'
g' ~
y,
g
X
L
~ Y
be a cartesian s,quare of schemes havin~ ample line bundles.
of finite Tor d~ension,
and that
Y'
an__~d X
x E X , y'~Y', y ~ Y
such that
g* f.
Sketch of proof.
Set
in the derived category
=
=
0
:
K'X
q
L = P(X,g')O=F(X,f).
--~
the inclusion
K L-~
L--~ P
From the formula
KqX .
,g')
3.
Rif.
on the category
Closed subschsmes.
Since
we have that
g*f.(F) = f'.g'*(F).
KqP(X,g')~K~X
induces isomorphisms on
functor
X
killed by
Proposition 3.1.
-If
-
K (Xre
Z
I
i. : M(Z) -# M(X)
sheaves on
__P(X,g') are effaceable for
Let
canonical immersion, and let
Y, (i.e.
Lg*Rf. = Rf'.Lg'*
F6L
this follows from the resolutien theorem, because the exact sequence
functors
i_~s
K'Y' .
q
f.F 6 P(Y,g), g'*F ~ F(X',f'), and that there is an isomorphism
only to check t ~ t
g
Then
[SGA 6 IV 3.1.0]. one deduces that for
Thus everythir~g comes to showing that
is proper,
fo_!r i > O
f(x) = y = g(y').)
f'. g'*
f
are Tot independent over
ToriO-y'Y(Oy,,y,, OX, x)
for ar~
Assume
, we have
K-groups.
2.9
But
shows that the
i>O.
be a closed subscheme of
X, let
be the coherent sheaf of ideals in
i : Z --~ X
_~
allows us to identify coherent sheaves on
defining
Z
be the
Z.
The
with coherent
I.
I
is nilpotent, then
i. : K'Z
q --~ K'X
q
is an isomorphism.
K'X.
This is an immediate consequence of Theorem 4.
PropOsition 3.2.
Let
canonical open ~ e r s i o n .
(3.3)
.~
Proof.
of
~(U)
K~+IU ~
U
be the complement of
Z
i__nn X, and
j : U -~ X
th__~e
Then there is a lon~ exact sequence
K'Z
q
i.
One knows [Gabriel, Ch. V]
with the quoZient category
~ K'X ~
q
that
*
Z.
127
,
j* : M(X) - - ~ _M(U) induces an equivalence
M=(X)/~, where
t i n g of coherent sheaves with support in
K'U
q
B i s the Serre subcategory c o n s i s -
Theorem 4
implies that
i. : M(Z)--@ B
In
120
induces isomorphisms on
Remark 3.4.
44
K-groups, so the desired exact sequence results from
Theorem 5.
The exact sequence 3.3 has some evident naturality properties which
follow from the fact that it is the homotopy exact sequence of the "fibration"
For example, if
Z'
is a closed subscheme of
the exact sequence of
(X,Z)
to the one for
From
3.3
containing
(X,Z').
(X,Z)
induces a map from the exact sequence for
Remark 3.5.
X
q
U
Also a flat map
to the one for
f : X'--~ X
(X',f-IZ).
one deduces in a well-known fashion a Ma~er-Vietoris sequence
q+1
for any two open sets
Z, then there is a map from
and
V
of
X.
q
Starting essentially from this point, Brown and
Gersten (see their paper in this precedings) construct a spectral sequence
E~q=
which reflects the fact that
a certain sense.
higher
aPCX, K' )
=-q
= - - - ~ K' X
-n
K'-theory is a sheaf of generalized cohomology theories in
In connection with this, we mention that Gersten has proposed defining
K-groups for regular schemes by piecing together the Ymroubi-Villamayor theories
belonging to the open affine subschemes (see [Gersten 2]).
sequence and the fact that Karoubi-Villamayor
Using the above Mayer-Vietoris
K-theory coincides with ours for regular
rings, Gersten has shown that his method leads to the groups
4.
K X = K'X
q
q
studied here.
Affine an d projective space bundles.
Proposition 4.1.
(Homotopy property)
Let
f : P ---~ X
be a flat map whose fibres
are affine spaces (for example, a vector bundle or a terser under a vector bundle).
f* : K'X ---@ KqP
q
Proof.
If
Then
is an isomorphism.
Z
is a closed subset of
X
with complement
U, then because
f
is flat
we have a map of exact sequences
.... ~
K'Z
q
.......
- K'X
q
$
- -=
K'U
q
$
KqPZ
~ KiP
$
~
KqPU ---1
By the five lemma, the proposition is true for one of
other two.
subsets
X, Z, and U
Z ~ X.
We can suppose
then the proposition holds for
can also suppose
X
X
ZI
reduced by
is irreducible, for if
and
Then by
2.4,
residue field at
the case where
lira_
> K'Uq = Kq(k(x))
x, and where
X : Spec(k), k
But this follows from
X = ZIUZ 2
with
Z I,Z 2 ~ X,
X - Z 1 = Z 2 - (ZI~Z2) , hence also for
X.
We
3.1.
Now take the inductive limit in the above diagram as
CX.
if it is true for the
Using noetherian induction we can assume the proposition holds for all closed
x
and
Z
I ~ K q P U = Kq(k(x) XxP) , where
is the generic point of
X.
a field, and we want to prove
§6 Th. 8 , so the proof is complete.
128
runs over all closed subsets
k(x)
is the
Thus we have reduced to
K'kq -T~Kq(k[t I .... tn]).
45
121
4.2.
scheme
that
Jouanolou's device.
X
P
Jouanolou has shown chat at least for a quasi-projective
over a field, there is a torsor
is an affine scheme.
Karoubi-Villamayor
He defines higher
X
with group a vector bundle such
K-6Toups for smooth
P.
From
4.1
P
X
by taking the
and showing that these do not
it is clear that his method yields the groups
considered here.
Proposition 4.3.
Let
E
be a vector bundle of rank
be the associated projective bundle, where
f : PE -~ X
be the structural ma~.
SE
y ~ x ~-~ y-f*x .
line bundle
r
over
X, le__~t PE = Proj(SE)
is, the s~ametric algebra of
Then we have a
K°(PE) ~oX K'Xq ~
(4.4)
given by
over
K-groups of the coordinate ring of
depend on the choice of
K X = K'X
q
q
P
E, and let
Ko(PE)-module isomor2hism
K'(PE)q
Equivalentl~,
if
z~ Ko(PE)
is the class of the canonical
0(-I ), then we have an isomorphism
r-1
(4.5)
(K~X)r ~
Sketch of proof. The equivalence of
Ko(PE)
is a free
sequence
field.
3.3
(Xi)o~'i<r ~-* >~i=o
z~f*x
K'(~)q '
4.4
KoX-module with basis
as in the proof of
and
4.5
results from the fact that
I .... r-l, [SGA 6 Vl 1.1] . Using the exact
4.1, one reduces to the case where
By the standard correspondence between coherent sheaves on
generated graded
Modfgr(SE)
SE-modules, one knows that
by the aubcategory of
has the same
M
PE
k
a
and finitely
M=(PE) is equivalent to the quotient of
such that
K-groups as the category
X = Spec(k),
M
Modfgr(k)
SE-modules killed by the augmentation ideal.
n
= O
for
n
large. This subcategory
by Theorem 4, where we view k-modulesas
Thus from the localization theorem we have
an exact sequence
where
PE.
i. , K q(Modfgr(SE) )
~ Kq (Modfgr(k))
(4.6)
i
From
is the inclusion and
Theorem 6
j
J* ~ K'(PE)q
associates to a module
M
.
the assoclated sheaf
M
on
we have the vertical isomorphiams in the square
K q (Modfgr(k))
~
2~/t] ® K'qk
i. ~ K q (Modfgr(SE))
~I
= h
~ ~Z[t] ~ Kqk
Using the Koszul resolution
O ~
and
SE(-r) ® A r E ® M
~
.....
Th. 2, Cor. 3, one shows that the map
multiplication by ~_t(E) = Z
h
~ SE~ M
~ M
~
O
rendering the above square commutative is
(-t)i[AiE]-- . Thus
i.
is injective, so from
4.6
we
get an isomorphism
I I ti • K'k
q
K;(PE)
O~i<r
induced by the functors
M ~-~ O(-1)®iOkM , O~<i<r
gives the desired isomorphism
4.5.
129
from
Modf(k)
to
M(PE).
This
122
The following generalizes
Proposition 4.7.
x
in
the residue field extension
plicative system in
f. : K~(X') -,
Proof.
3.1.
Lent f : X' --~X
surjective (i.e. for each
~
46
X
be a finite morphism which is radicial and
the fibre
k(x')/k(x)
f-1(x)
generated by the degrees
K'Xq induces an isomorphism
If
Z
Z
[k(x'):k(x~
S-IK'(X
')q
is a closed subscheme of
are the respective inverse ~m~#es~ of
has exactly one point
is purely inseparable).
and
Let
S
x'
an__~d
be the multi-
for all
x
in
X.
Then
--~S-IK'Xq .
X
with complement
U
in
U, and if
Z'
and
U'
X', then we have a map of exact
sequences
> K,(z,)
.-- K,(x,)
~
(fz).
Localizing with respect to
holds for two of
K'X
q
S
fz' f' fu
k'
~ K'U
q
,
and using the five lemma, we see that if the proposition
it holds for the third.
we can reduce to the case where
X' = Spec(k'), where
_
(fu).
f.
K'Z ~
q
.)
) K'(U')
X = Spec(k),
k
Thus arguing as in the proof of
a field.
By
3.1
4.1
we can suppose
is a purely inseparable finite extension of
k.
Thus we have
reduced to the following.
Proposition 4.8. Let f : k--~k'
be a purely inseparable finite extension @f
d
d
degree p . Then f.f* = multiplication by p
o~n K k and f*f. = multiplication by
d
q
p
o===nn K (k').
q
Proof.
The fact that
of the projection formula
The homomorphism
f*f.
~(k') to itself.
k' ®k k'
~4 (5)
[k':k] is an immediate consequence
and does not use the purely inseparable hypothesis.
is induced by the exact functor
v ~
from
f)f* = multiplication by
k,%v
Since
is nilpotent.
(k,%k')%,v
=
k'/k
is purely inseparable, the augmentation ideal
Filtering by powers of
I
of
I, one obtains a filtration of the above
functor with
gr((k' ®k k') ®k' V)
=
~
(In/I n+m ) Ik, V .
n
But because the two
k'-module structures on
isomorphic to the functor
Cor. 2
V ~-~V r, where
Filtration b2 support~ Gersten's conjecture! and the Chow rln~.
denote the Serre subcategory of
is of codimension
~ p.
M(X)
(5.1)
~2 (9)
and
-~,z
where
3.1, it is clear that we have
Kq(Mp(X))=_
=
Let
Mp(X)
consisting of those coherent sheaves whose support
(The codimension of a closed subset
the dimensions of the local rings
From
®k k')) = p d
Applying Th. 2,
d
f*f, = multiplication by p , completing the proof.
to this filtration, we find
5.
In/I n+1 coincide, this graded functor is
r = d~k,(gr(k'
lim, qK'Z
}.30
z
Z
of
X
is the infimum of
runs over the generic points of
Z.)
47
123
Z
where
runs over the closed subsets of codimension
(5.2)
f*(~p(X)) c- Mp(X')
In effect, one has to show that if
codimension
~ p
in
X'.
But if
Z
if
~ p.
f : X'-~X
has codimension
z'
We also have
is flat.
~ p
in
is a generic point of
f-Iz
X, then
f-IZ, and
has
z = f(z'), then
the homomorphism -~,z " ~ - ~ ' , z ' is a f l a t local homomorphism such that rad(_~,z)"_~,,Z,
is primary for r a d ( ~ , , z , ) ; hence dim(_Ox,z) = dim(Ox,,z,) by [EGA IV 6.1.3], proving
the assertion.
If
X = lim X i
where
i ~-~ X.
is a filtered projective system with affine flat
transition morphisms, then we have isomorphisms
(5.3)
Kq(M__p(X)) =
In view of
form
5.1
this reduces to showing that any
f-l~(Zi) for some
fi : X -~ X i
with
i, where
fi(Z).
Theorem 5.4. -Let
-
Xp
Z
of codimension
is of codimension
But for
i
p
in
Thus
z'
Z.
x
p
in
X
and
z
z'
is of the
X i , and where
large enough, one has
Hence any generic point
z of Z, so the local rings at
result about dimension used above.
spectral
Zi
denotes the canonical map.
Z i = the closure of
generic point
i ~ Kq(_M_p(Xi))
of
Z = f.~1(Zi)
Z.l is the image of a
have the same dimension by the
also has codimension
be the set of points of codimension
p
p, proving
in
X.
5.3.
There is a
sequence
ElPqCx)
(5.5)
I I K-p-q k(x)
:
x~X
which is convergent when
X
>K
-n x
P
has finite (Krull) dimension.
contravariant for flat morphiams.
Furthermore, -if
-
This spectral sequence is
X = ~lim X I , where
i ~-~X.1
is a-
filtered pro~ective s2stem with affine flat transition morphisme, then the spectral
sequence for
X
is the inductive limit of the spectral sequences for the
In this spectral sequence we interpret
sequence is concentrated in the range
Proof.
of
M(X)
=
Kn
p>~O ,
as zero for
n~O.
Xi .
Thus the spectral
p+q ~ O.
We consider the filtration
by Serre subcategories.
There is an equivalence
,(x)
I1 U Modf( ,/rad( ,x )n)
x~X
so from
n
P
Th. 4, Cor. I, one has an isomorphism
Ki(~p(x)/~p+,ix)) ~
I | ~ik(x)
x~X
where
k(x)
--*
is the residue field at
Ki(Mp+I(X))
--~
Ki(Mp(X))
x.
P
From
--*
131
Th. 5
we get exact sequences
I I Kik(x)
xgX
P
~
Ki_I(_M_p+I(X))
124
48
which give rise to the desired spectral sequence in a standard way. The functorality
assertions of the theorem follow immediately from
5.2
and
5.3.
We will now take up a line of investigation initiated by Gersten in his talk at this
conference [Gersten 3].
Proposition 5.6.
i)
For every
ii)
For all
The following conditions are e~uivalent:
p>/O, the inclusion
q , EPq(X) =O
M__p+I (X)--> M=p(X) induces zero on
if p ~ O
--
and the edge homomorphism
K-~roups.
K'- - q X ~
EOq(x)
is an isomorphism.
iii) For every
(5.7>
o
~
n
the sequence
~'# ~
I I ~k(x>
d,
I1 ~_,k(x> d,,
x ~ Xo
is exact.
Rere
dI
is the differential on
E I (X)
and
pulling-back with respect to the canonical morphisms
e
Proposition 5.8.
for all
(Gersten)
in X.
E2Pq(x)
is the map obtained b.y
5.5
and its construction.
K' denote the sheaf on X associated to the
=n
Spec(~,x) satisfies the equivalent conditions of
5.6
Then there is a canonical isomorphism
EPq(X)
with
.
Let
U ~-~ K'nU . Assume that
x
.
Spec k(x) -=@ X.
This follows immediately from the spectral sequence
presheaf
.
xE X l
as in
=
HP(x,K=~'q)
5.5.
Proof. We view the sequences 5.7 for the different open subsets of
X
as a sequence
of presheaves, and we sheafify to get a sequence of sheaves
o -+
(5.9)
~, - +
II (ix).(~k(~)) ~
I] (ix)~(~n ,k(xl)
x~ X °
where
ix : Spec k(x) --)X
sequence
5.7
for
denotes the canonical map.
Spec(_Ox,x),__ because
affine open neighborhoods of
such projective limits.
K'
"
x~X I
The stalk of
Spec(_~,x)=~U,
where
x, and because the spectral sequence
By hypothesis, 5.9
5.9
over
x
is the
U
runs over the
5.5
commutes with
is exact, hence it is a flask resolution of
, SO
=n
xEX
:
{s
S
_-
as asserted.
The following conjecture has been verified by Gersten in certain cases [Gersten 3].
Conjecture 5.10.
of a r ~ l a r
(Gersten)
The conditions of
5.6
are satisfied for the spectrum
local ring.
Actually, it seems reasonable to conjecture that the conditions of
5°6
hold more
generally for semi-local regular rings, for in the cases where the conjecture has been
132
125
49
proved, the arguments also apply to the corresponding semi-local situation.
On the other
hand there are examples suggesting that it is unreasonable to expect the conditions of
5.6
to hold for any general class of local rings besides the regular local rings.
We will now prove Gersten's conjecture in some important equi-characteristic cases.
Theorem 5.11.
Let
set of primes in
R
R
be a finite type alsebra over a field
such that
R
is regular for each
p
regular semi-local rin~ obtained by localizin~
satisfies the conditions of
Proof.
subfield
of
k
S
S'
R
with respect to
R
is smooth over
of
Spec R'
such that
are the base extensions of the primes in
with respect to
S', then
subfields of
containing
k
A = lim ki®k,A'
and
~
k, let
S
S, and let
S.
be a finite
A
be the
Then
Spec A
k.
There exists a
finitely generated over the prime field, a finite type
R', and a finite subset
in
in
5.6.
We first reduce to the case where
k'
p
A = k ~,A'
k'
and
R = k ~k,R'
S'.
A'
If
A'
k'-algebra
and such that the primes
is the localization of
is regular.
k.1 run
Letting
R'
over the
and finitely generated over the prime field, we have
K.(M (A)) = lim K.(M (k M_,A'))
=p
•
=p
by
1 K
5.3, where here and in the
following we write
when
M=p (A) instead of M=p (Spec A). Thus it suffices to prove the theorem
is finitely generated over the prime field. In this case A is a localization
k
of a finite type algebra over the prime field, so by changing
the prime field.
the points of
some
f
R, we can suppose
As prime fields are perfect, it follows that
S, hence also in an open neighborhood of
not vanishing at the points in
S.
S, we can suppose
R
Replacing
R
k
is smooth over
R
by
is smooth over
Rf
k
is
k
at
for
as
asserted.
We wish to prove that for any
K-groups.
By
5.3
K . ( M I(A))
where
f
p>.O
runs over elements not vanishing at the points of
K.(Mp+I(R))=
t
induces zero on
~p+1(A) -@ ~p(A)
= 1~. K.(~p+l(Rf))
we reduce to showing that the functor
where
the inclusion
we have
=
M=p+1(R) --> Mp(A)
element
t, there exists an
from
Rf,
As
R, it suffices to show that given a regular
f, not vanishing at the points of
Mp(R/tR) to Mp(R)
by
K-groups.
~lim K.(Mp(R/tR))=
runs over the regular elements of
M ~-, Mf
S, hence replacing R
induces zero on
induces zero on
S, such that the functor
K-groups.
We will need the following variant of the normalization lemma.
Lemma 5.12.
k, let
t
Let
R
be a smooth finite type algebra of dimension
be a regular element of
there exist elements
B = k[x I .... Xr_1]c-R,
at the points of
R, and let
x I ,..,Xr_ I o f
then
i)
R/tR
R
S
r
be a finite subset of
algebraicall 2 independent over
is finite over
B, and ii)
R
over a field
Spec R .
k
such that if
is smooth over
S.
Granting this for the moment, put
B' = k/tR
arrows
133
and
R' = R ®B B'
Then
so that we have
B
126
R'
~
R
B'
~,
B
where the horizontal arrows are finite.
lying ever the points in
S, u'
S.
As
u
50
Let
S'
be the finite set of points of
is smooth of relative dimension one at the points of
[SGA I I I
4.15]
that the ideal
I = Ker (R'--~B')
hence principal in a neighborhood of
S'.
not vanishing at the points of
module.
We can also suppose
R'/R
Then for any
(*1
B'-module
Since
R'f
is flat over
phism
If
we have an exact sequence of
B', if
M
is in
S
smooth over
R
over
M__p(R'f), so
is an exact sequence
Mp(Rf)= induces the
a maximal ideal containing it, we
R.
Let ~
be the module of Kahler
It is a projective
R-module of rank
B = k[Xl..,Xr_1]
at the points of S
means that the differentials
R/m n , m £ S,
V
of
R
suppose also that
S.
Let
J
such that for each
V
generates
R
m
at
S, there exists
by letting
r-1.
R/tR
has dimension
is of dimension
Let
be a system of parameters for
z 1,..,zr_ I
>I 2.
are lifted to elements
By the choice of
Then
x l' of
•
r-2, so
gr(R/tR)
gr(R/tR)
R, then
V, we can choose
gr(R/tR)
is finite over
Vl,..,Vr_ I
On the other hand, the
is finite over
k[Xl,..,Xr_1].
xi
in
134
V
S.
whose
We can
in
Proj(~Fn(R/tR))
of the affine space
Proj
at infinity, namely
r-1
as asserted.
V
zi
is
kin I .... Zr_1], so if the
zi
k x ,.., r-1
such that
x i = x! + v i
l
S, whence condition ii) of the
have the leading terms
The proof of the lemma and
now complete.
As
be the subspace
such that each
is finite over
R/tR
S.
Then the associated
has dimension
I ~ i ~ r, have independent differentials at the points of
lemma is satisfied.
V.
Spec (R/tR)
r-l, the part of this
Proj(gr(R/tR)),
homogeneous of degree
Fn(R/tR )
To see this, note that
is the closure in projective space of the subscheme
Since
to be
dxi6 j ~ 1
k.
in the elements of
is of dimension
v I ,..,v r
vanishin~ at the other points of
R/tR
~ n
R
k, we can find a finite dimensional
as an algebra over
Define an increasing filtration of
gr(R/tR)
in
m
r, and for
be the intersection of the ideals in
is finite dimensional over
spanned by the monomials of degree
graded ring
S
k.
differentials form a basis for ~ I
R/tR
is in
(*)
M__p(B') to
Choosing for each prime in
are independent at the points of
so
R'~B,M
Th. 2, Cor. I, and using the isomor-
is a finite set of maximal ideals of
differentials of
Spec S(V).
B'.
K-groups, as was to be shown.
can suppose
k-subspace
R'f-
Rf-modules
M=p(Rf). Thus
M__p(B') to M=p(Rf). Applying
Proof of the lemma.
R/J n = ~
as an
in
O.
M=p(B'), then
is in
R'f
f
is smooth, hence flat, ever
---,. Mf
R'~B,M
S',
Thus we can find
is isomorphic to
R'f
If ~---R'f , we conclude that the functor from
zero map on
Spec R.
M
Rf-module, we have
of exact functors from
in
chosen so that
R,#B,M
One knows then
is finite, this neighborhood
f
o
viewed as an
S
S such that
S'.
is principal at the points of
Since
contains the inverse ~m~ge of a neighborhood of
R
Spec R'
is smooth of relative dimension one at the points of
zi
in
gr(R/tR),
Theorem 5.11
is
51
127
Theorem 5.13.
The conditions of
formal power series
power series in
k ~ X I .... X n ~
XI,..,X n
5.6
hold for
over a field
Spec A
when
k, and when
A
is the rin~ of
l e t h e ring of convergent
A
with coefficients in a field complete with respect to a
non-trivial valuation.
The proof is analogous to the preceding.
then after a change of coordinates,
Weierstrass preparation theorem.
A/tA
Indeed, given
Further, if we put
K-groups.
B = kI[X I ....Xr_1] ] by the
A' = A ~ / t A ,
is principal, so arguing as before, we can conclude that
on
O ~ t e A = k[[X1,,,Xn~,
becomes finite over
then Ker(A' --~A/tA)
Mp(A/tA) -* Mp(A)
induces zero
The argument also works for convergent power series, since the preparation
theorem is still available.
We now want to give an application of
the fact tha~ the
particular that
KIA
KIA
5.11
to the Chow ring.
We will assume known
defined here is canonically isomorphic to the Bass
is canonically isomorphic to the group of units
K I , and in
A', when
A
is a
local ring or a ~.~aclidean domain.
Proposition 5.14.
Le.t X
be a regular sc.h.eme...~.o.f....finite
t~pe over a field.
Then the
ima~e~
dl : I I Klk(X)
~ ~
x{Xp_ I
in the spectral sequence
AP(x)
P
is the subgroup of codimens%on
5.5
linearly e~uivalent to zero.
~rou~
Consequently
of cycles of codimension
Proof.
Let
p1
form
W ° - W o, where
known formula for
Let
W
W
or
k(y).
and let
=
o
- W
dim(W) - I .
p1.
Let
cP(x)
Let
x
W ~ ( X x O)
t
denote the
denote the group of codimension
Y
t'
and
W
w
Y
the generic point of
: k(y)" -9-~
ordyx(f)
for
f ¢ o.
5.15
k(y).
p
x
in
of codimension
are proper.
p
such
We need a
p1
X x
-~ X, so that
W = Y x p1
and
p - I
W, so that
k(w)
Y,
X x p1
Wr~(X x co)
has codimension
be the non-zero element of
(multiplicity of
yx
=
In the latter case we have
be a point of codimension one in
(5.15)
From
W
under the projection
dimension one with quotient field
ord
and
Go
which we now recall.
dim(W) = dim(Y), whence
generic point of
where
iscanon!9.ally isomorphic to the
modulo linear equivalence.
is an irreducible subvariety of
W
o
co
be the image of
Y
we may assume
of
E~'-P(x)
P
cTcles which are
p
The subgroup of cycles linearly equivalent to zero is generated by cycles of the
that the intersections
dim(W)
p
x~X
be the projective line over the ground field, and let
canonical rational function on
cycles.
Kok(X) =-l--t
x~X
W
o
X.
in
k(w)
Oy, x
Go
Let
=
O, so
y
be the
is a finite extension
obtained by pulling
whence
dim(Y) =
- W
t
back to
W,
is a local domain of
Then the formula we want is
WO -Wco)
=
ordyx(ROrm~(w)/k(y ) t')
is the unique homomorphism such that
=
length(Oy,/fOg, x)
For a proof of
5.15
see
it is clear that the subgroup
135
[Chevalley, p. 2-12].
of cycles linearly equivalent to zero is
128
52
the image ef the homemorphism
ii
¢
k(y)"
tl
where i f
f~k(y)',
Klk(y)
=
¢(f)=
then
k(y)', we see
Z ord. (f)-~
~
is a map from
~
~ d w e put
E F I '-P(x)
remains to prove the preposition is to show that
Let
fer
y
Xp_ I
dI
in
Xp_ I
Y
(dl)y x
: k(y)" = K1k(Y)
and
in
X
x
Xp .
be its closure.
for all
the ene fer
ord
--0
if
x¢E}-
Since
YXE IP'-P(x),so all that
to
~ = dI .
have the components
and let
Mj+p_I(X)
=~ = cp(x)
xeX P
Y ~ Xp_ I
~
Kok(X)=~,
We want te show that
The closed immersien
(dl)yx = °rdyx "
Y --~X
carries
j, hence it induces a map from the spectral sequence
augmenting the filtration by
dI
p-1.
which shows that
_o,-l(y)
~:1
(dl)yx = O
codimensien one in
d~ ~ ~I
J ,-1 (y)
unless
x
is in
Y, then the flat map
5.5
in
te
fer
Y
te
cP(x)
]-
=
y
Thus we get a commutative diagram
E~ 1']p(x) ---. :'-P(x)
KIk(Y)
Fix
Mj(Y)
Y.
:
C1 (y)
On the other hand, if
Spec(Oy, x) ---~Y
x
is of
induces a map of spectral
sequences, so we get a commutative diagram
KIk(Y)
_0,-I (y)
~I
=
II
KIk(Y)
which shows that
the equality
Lemma 5.16.
=
~I
is the map
dI
~'I
and residue field
is isomorphic to
ord(x)= length(A/xA)
We have isomorphiams
o
k.
x
If
since
x
is not a unit.
x
--~
in
Speck
KIP = F"
and
for
x
KIA = A"
in
By hypothesis
By naturality of the exact sequence
Spec A.
Then
is the hom0morphism such
k
e
A
since
A, x { O.
is in the image of the map
were algebraic over
ord
of
A, x ~ O .
algebraic, se we have a flat homomorphism
x.
KoA
ord : F" -* ZZ , where
x
We wish te show ~(x) = ord(x)
can suppose
of
for
Kk
" KeF --~ 0
associated to the closed set
k
~-~
k, and let
3.3
clear, as ~(x) = O
Oy, x . Therefore
be an equi-characteristic local noetherian domain of dimensien
F
be the exact sequence
Proof.
2Z
in the spectral sequence for
K I,
rings.
[multiplicity
oe x
is a consequence of the fellewing.
--~
: KIF ~ K k
C I (y)
(Oy,x) =
K;A
that
=
1
_~,_~
=
erdyx
.Let A
ene with quotient field
E I ,-1 (y)
l
~o,_~(%,x) ~
=
(dl)yx
(dl)yx
dl >
If
is in
F
are local
A', this is
.
Thus we
is an algebra over the prime subfield
ko[t ] --~ A
136
and
x
KIA -~ K~A - + K I F
, it would be a unit in
3.3
A
A.
Thus
x
sending the indeterminate
is net
t
to
for flat maps, we get a commutative diagram
129
'~ Klko[t]
53
~ Klko[t,t-1 ]
1
8 , Kok°
1u
K~A
-
~
tv
KI F
9
~
K k
O
such that
u(t) = x.
The homomorphism
v
is induced by sending a
k -vector space
V
to
O
the
A-module
A ®ko[t]V
=
A/xA ®k V
O
and using devissage to identify the
length with those of
P(k).
K-groups of the category of
A-modules of finite
Thus with respect to the isomorphisms
K k
=
is multiplication by
top row of the above diagram, one has
~ (t) = ~ 1 . But this is easily verified by
KoR = Z
and
Remark 5.17.
In another paper, along with the proof of Theorem I of
local noetherian domain
for
A
of dimension one with
By the universal property of the
(5.18)
B(Aut(V))
compatible with direct sums.
--~
A-submodules
of layers
such that
(Lo,LI)
Then
G
X(V).
acts on
there is a functor
desired map
X(V) G --~
F
for a
and residue field
V
over
F
a homotopy class of maps
BQ(~(k))
L
such that
LI/L °
F ~A L = V.
A-lattices in
Let
X(V)
is killed by the maximal ideal of
X(V), so we can form a cofibred category
One can show that
'bttilding'), hence the functor
[Quillen I],
~ : KnF - + K _ i k
quotient field
To do this consider the set of
finitely generated
with fibre
for a
K-theory of a ring, such a map is defined by giving
every finite dimensional vector space
G = Aut(V).
KIR = R"
q.e.d.
I plan to justify the following description of the boundary map
k.
, v
O
Therefore it suffices to show that in the
explicitly computing the top row, usiz~ the fact that
Euclidean domain,
= K k =~
O0
length(A/xA) = ord(x) .
X(V)
be the ordered set
A, and put
X(V) G
over
G
is contractible (it is essentially a
X(V) G -~ G
~(P(k))
V, i.e.
is a homotopy equivalence.
sending
(Lo,LI)
to
On the other hand
LI/L O , hence we obtain the
5.18.
It can be deduced from this description that the
Lemma 5.16
is valid without the
equi-characteristic hypothesis.
Combining
5.8, 5.11, and
Theorem 5.19.
5.14
we obtain the following.
For a re~alar scheme
X
of finite type over a field, there is a
canonical isomorphism
HP(X,Kp)
For
HI(x,_~) =
p = O
and
Pic(X).
I
For
=
AP(x) .
this amounts to the trivial formulas
p = 2
H°(X,~) = C°(X)
and
this formula has been established by Spencer Bloch in
certain cases (see his paper in this proced~gs).
One noteworthy feature about the formula
contravariant in
X~
5.19
which suggests that higher
is that the left side is manifestly
K-theory will eventually provide hhe tool
for a theory of the Chow ring for non-projective nonsingular varieties.
137
130
~8.
54
Pro~ective fibre bundles
The main result of this section is the computation of the
bundle
associated to a vector bundle over a scheme.
Grothendieck groups in [SGA 6 VI]
K-groups of the projective
It generalizes the theorem about
and may be considered as a first step toward a higher
K-theory for schemes (as opposed to the
K'-theory developed in the preceding section).
The method of proof differs from that of [SGA 6] in that it uses the existence of
canonical resolutions for sheaves on projective space which are regular in the sense of
[Mumford, Lecture 14].
method.
We also discuss two variants of this result proved by the same
The first concerns the 'projective llne' over a (not necessarily
ring; it is one of the ingredients for a higher
Theorem' of
the
Bass
K
commutative)
generalization of the 'Fundamental
to be presented in a later paper.
The second is a formula relating
K-groups of a Severi-Brauer scheme with those of the associated Azumaya algebra
and its powers, which was inspired by a calculation of Roberts.
I.
The canonical resolution of a regular sheaf on
(not necessarily noetherian or separated),
S,
and let
X = PE = Proj(SE)
symmetric algebra of
f : X -* S
sheaf of
E
let
E
PE.
Let
S
be a scheme
be a vector bundle of rank
be the associated projective bundle, where
over
0 S.
the structural map.
Let
_~(I)
r
SE
over
is the
be the canonical line bundle on
X
and
We will use the term "X-module" to mean a quasi-coherent
O__x-modules, unless specified otherwise.
The following lemma summarizes some standard facts about the higher direct image
functors
Rqf.
Lemma 1.1.
we will need.
a)
For any
X-module
F,
R~.(F)
is an
S-module which is zero for
q>/r.
b)
For an~¢ X-module
F
and vector bundle
R%.(F) %E,
c)
For any
S-module
=
E'
on
S, one has
R%.(F %E')
N, one has
I
R%.(~(n) %s)
0
=
q~O,r-1
Sn~,®S N
q= 0
(Sr_nE)V®SArE ®S N
where
r - I
,'v" denotes the dual vector bundle.
d)
If
F
affin___~e, then
Parts
is an
F
X-module of finite t~cpe (e. G. a vector bundle), and if
.is a quotient of (~(-1)~n) k
a), c)
Part
b)
finitely many copies of
0 S.
For
Following Mumford, we call an
where as usual,
by
for some
S is
n, k.
result from the standard Cech calculations of the cohomology of projec-
tive space [EGA III 2].
n>~O
q :
F(n) =
is obvious since locally
d), see
X-module
138
is a direct sum of
[EGA II 2.7.10].
F
regular if
_~(I )®nMXF . For example, we have
c).
E'
Rqf.(F(-q)) = O
for
q>O,
_O_x(n)®S/~ is regular for
131
Lemma 1.2. Le__!t O
a) If
b)
If
F(n)
c) If
F'(n+1)
-~ F' --~ F
F'(n) and
F"(n)
and
F" --~ O
are regular, so is
F'(n+1)
F(n+1) and F"(n)
-#
55
be an ~xact sequence qf X-modules.
F(n).
are regular, so is
are regular, and if
F"(n).
f.(F(n))--~ f.(F"(n))
is onto, then
is regular.
Proof.
This
follows immediately from the long exact sequence
Rqf,~'(n-q))---PRqf.(F(n-q))
--~ Rqf.(F"(n-q))
The following two lemmas appear in
;
Rq+If.(F'(n-q)) --~-Rq+If,(F(n-q))
[Mumford, Lecture 14]
and in [SGA 6 XIII 1.3J,
but the proof given here is sliglztly different.
Lemma 1.3. If
Proof.
F
is re~n~lar, then
F(n)
From the canonical epimorphism
(, .4)
~(-i )®~
is regular for all
n~>O.
_~ ®S E -+ _~(I ) one has an epimorphis~
-->
so we get an exact sequence of vector bundles on X
(1.5)
O ~
_~(-r) ~ r E
--~
. . . --~ _Ox(-I) ®S E ---~ _~
by taking the exterior algebra of _O_x(-I) ~S E
1.4. Tensoring with
(1.6)
F
F
Thus if
1.6
with differential the interior product by
we obtain an exact sequence
0 ---~ Z(-r) ~ s A r z - - ~ . . .
Assuming
to be regular, then
--~F(-1) % Z
(F(-p)~PE)(p)
Thus
Proof.
1.2 b)
1.1 b).
--. zp. I --~ o
1.7.
to show by decreasing induction on
l_~f F
p
is regular, then the canoniqal map OX ® ~ . ( F ) - - ~ F
is sur~ective.
From the preceding proof one has an exact sequence
0
map
is seen to be regular using
that Z (p+1) is regular.
P
Zo(1) = F(I ) is regular, so the lemma follows by induction on n.
L~
where
---. Z ---. O .
is split into short exact sequences
o --~ zp --~ z(-p)~sAPz
we can use
> 0
> zI ~
ZI(2 ) is regular.
f.(F(n-1))®~ ~
Thus
F(-I)®sz --~ z - - ~ o
R1f.(Z1(n)) = O
f.(F(n))
for
is surjective for
n~1, so we find that the canonical
n~1.
Hence the canonical map of
SE-modules
is surjectivs.
SZ %f.(Z) ----. I I f,(F(n))
n~>O
The lemma follows by taking associated sheaves.
Suppose now that F
o--.~(-~1)%T
where the T i
is an X-module which admits a resolution
I
are modules on S.
:
.
.--.~%T
°
~F-~o
Breaking this sequence up into short exact sequences
139
132
and applying
56
1.2 b), one sees as in the proof of
1.3
that
F
has to be regular.
Moreover, the above exact sequence can be viewed as a resolution of the zero module by
acyclic objects for the ~-functor
on applying
0
(1.8)
n ~ O.
O
-~
F
" " " --*
---~...
X-module
Z
and
n
Supposing now that
T
F
(1.9)
Zn(n)
0
--~
where the canonical map
we find that
c
Z
f.(F(n)) --~
T n = Tn(F)
as follows.
_O_x®sTn
c
are
X-modules
Z_I = F,
--AoO-(-n)O~Tn~Zn-1"
F.
is surjective by
Zn_1(n ) - - ~
Zn(n÷1 )
is reg%ular,
1.7
O
and the induction hypothesis.
is regular, so the induction works.
= o
and the fact that
f.
By
1.3,
In addition we have
for n~.O
induces an isomorphism after applying
f..
is exact on the category of regular
F ~-~ Tn(F)
X-modules,
is an exact functor from regular
X-modules
S-modules.
We next show that
Zr_ I = O.
Rq-lf.(Zn+q_m(n))
From
~
1.9
we get exact sequences
Rqf,(Zn+q(n)) ~
Rqf.(_~(-q)®#n+q)
which allow one to prove by induction on
q , starting from
1.10, that
for
is regular, since
Rqf.
qtn~O.
1.10
and
This shows that
1.7
we have
Zr_1(r-1)
Zr_l(r-1) = O, so
Combining the exact sequences
sheaf
F
of length
r - I.
Proposition 1.11.
0 ~
where the
F ~-~ Ti(F )
are
Rqf.(Zn+q(n)) = O
is zero for
q~r.
Any regular
By
as was to be shown.
we obtain a canonical resolution of the regular
Thus we have proved the following.
_~(-r÷I)®sTr_I(F) ~
T.l (F)
1.9
Zr_ I = 0
X-module
. . . ~
F
has a resolution of the form
_Ox®s%(F )
~- F ~
O
S-modules determined up to unique isomorphism by
is an exact functor from the category of regular
F.
Moreover
X-modules to the cate6ory
S-modules.
The next three lemmas are concerned with the situation when
on
n
Starting with
be the kernel of the canonical map
n
~
one concludes by induction that
of
T
F, we inductively define a sequence of
f.(Zn(n))
1.9
O
We have an exact sequence
Zn(n+1)
(1.1o)
to
~ O
n
is regular, we show by induction that
n = -I.
From
-'~ f.(F(n))
are additive functors of
this being clear for
c
SnE®s%
~
S-modules
T n = f.(Zn_1(n)), and let
because
Thus
up to canonical isomorphism.
and a sequence of
It is clear that
1.2 c)
is any fixed integer >I O.
which can be used to show recursively that the modules
Conversely, given an
let
~
E ®sT_I
n = O , ..., r-1
Z n = Zn(F )
n
In particular, we have exact sequences
T n -*
determined by
Rqf.(?(n)), where
we get an exact sequence
--~ Sn_r+1E ® S % - 1
for each
for
f.
X.
140
F
is a vector bundle
133
57
~emma 1.12.
Assume
there exists an. integer
S
is quasi,compact.
Then for any vector bundle
no
such that for all
S-modules
N
and
F
~
X,
n ~ n o , one has
~.(F(n)~s~)--o fo__zrq>O
,)
f.(F(n))%~ ~
'.)
f.(F(n))
f.(F(n)%N)
is a vector bundle on
S.
[
'roof.
Because
the lemma when
n
and
k
by
S
S
is the union of finitely many open affines, it suffices to prove
is affine.
1.1 d).
In this case
F
is the quotient of
Thus for any vector bundle
F
on
L = Oy(-n) k
for some
S, there is an exact sequence
of vector bundles
O---@F' --->L
such that the lemma is true for
o
; P'(n)~
,-F--. O
L
by
1.1.
Since
---*.L(n~fi ---,.F(n)~
--+0
is exact, we have an exact sequence
R%.(L(n)~s~) --. R%.(F(n)~)
so part
a)
theorem
[EGA III 2.2.1].
can be proved by decreasing induction on
O --~
for
---,. Rq+lf.(F'(n)~)
n >/ some
Using
a)
q, as in the proof of Serre's
we have a diagram with exact rows
f,(F'(n))®sN
~
f.CL(n))®S~ ~
f , CF(n))~sN ~
O
f,(F'(n)esN)
.-~
f,(L(n)~sN ) ~
f,(F(n)~?)
0
~
n
and all N. Hence u is surjective; applying this to the vector
o
bundle F', we see that u' is surjective, hence u is bijective for n ~> some n
o
and all 4, whence b). By a), f . ( F ( n ) ~ )
is exact as a functor of N for sufficiently large
n, whence using
f.(F(n))
is a quotient of
Applying this to
large
n.
F'
b)
we see
f.(L(n))
we see that
f.(F(n))
for
is a flat
n ~> some
Os-mOdule.
rLo, so
f.(F(n))
On the other hand,
is of finite type.
f.(F(n)) is of finite presentation for all sufficiently
But a flat module of finite presentation is a vector bundle, whence
Lemma 1.13.
n/>0, then
Proof.
If
F
f.(F(n))
is a vector bundle on
The assertion being local on
is a vector bundle on
0 ~
is a vector bundle on
F(n)
S
for large
n
S
F(n)
such that
for all
1.12 c).
~
0 ~
f.(F(n))
Therefore one can show
induction on
f.
>
fo~
n)O.
S
affine, whence
1.5.
For
6"-functor
n>~O, this is a
Rqf. , hence one
one gets an exact sequence
f.(F(n))
f.(F(n+r))®SArE
is a vector bundle for all
n.
141
f.(F(n))
F ( n + r ) ~ A r E v ----~0
with the dual of the sequence
• • • ~
q~>O,
Consider the exact sequence
resolution of the zero module by acyclic modules for the
knows that on applying
Rqf.(F(n)) = 0
S, one can suppose
by
~- F ( n + 1 ) ~ E V - - - ~ . . .
obtained by tensorlng
X
c).
~
0 .
n>~O
by decreasing
134
Lemma 1.14. If
bundle on
S
F
is a regular vector bundle on X, then
for each
i,
using the exact sequences
The ~rojective bundle theorem.
naturally modules over K °
Theorem 2.1. Let
Proj(SE)
E
by
1.8 and the lemma 1.13.
Recall that the K-groups of a scheme are
§3 (I). The following result generalizes
be a vector bundle of rank
the associated projective scheme. If
S
r
over a scheme
[SGA 6 VI 1.1].
S
and X =
is quasi-compact, then one has
isomorphisms
r-1
(KqS)r = .
K X
(ai)0.~i~r
q
where
is a vector
Ti(F)
i.
This follows by induction on
2.
58
zgKX
~
'
Yl zi. f*ai
i=O
is the class of the canonical line bundle
_~(-I)
and
f : X --~S
is the
structural map.
Proof.
F
of
Let
--Pn denote the full aubcategory of
such that Rqf.(F(k))= 0
__P(X) consisting of
F
for
closed under extensions, so its
Lemma 2.2. For all
q ~O
such that
and
F(n)
k~n.
Let
is regular.
Rn
denote the full subcategory
Each of these subcategories is
K-groups are defined.
n, one has isomorphieme:
ina~,e,d b~ the inclueio,~
__P(X) consisting of vector bundles
=Re P C
Kq(Rn)= ~"
Kq(Pn)=~
Kq(P(X))=
=~(x).
To prove the lemma, we consider the exact sequence
(2.3)
>F---*F(1)%E
O,
For each
p>O,
~
F ~-~ F(P)®sAPE
. . .
>F(r)%ArE
is an exact functor from
....... ~ . - 0 .
:n
P
induces a homomorphism
to =Pn-1 ' hence it
Up : Kq(__P
=n) -'> Kq(Pn_ 1 )" From Th. 2, Cor. 3
~-~,p~O(-1)P-lUp is an inverse to the map induced by the inclusion of
Thus we have
P
, so by
Kq(P(X))
K (P .) - ~ K (P) for all n. By 1.12 a),
q =n-I
q =n
~2 (9) we have Kq(P__n)= Kq(P(X)) for all n.
it is clear that
--Pn-1 in Pn "
P(X) is the union of the
=
The proof that Kq(Rn) ~---
is similar, whence the lemma.
Put U n ( N ) = _ ~ ( - n ) ~ for
from
P(S)
=
In view of
to
--Pc by
N in
P(S). For O~_.n<r, U n
is an exact functor
1.1 c), hence it induces a homomorphism
un : K q (P(S))
-* Kq(P o ).
=
2.2, it suffices for the proof of the theorem to show that the homomorphism
r-1
u : Kq(~(S)) r ---~ Kq(P=o)
'
(an)o~n< r ~') ~
Un(an)
is an isomorphism.
From
1.13
we know that Vn(F) = f.(F(n))
is an exact functor from
P
=O
for n~O, hence we have a homomorphism
v : ~q(~o)= -~
where
V
n
KIP(s))=
,
is induced by V . Since
n
142
x ~-~ (Vn(X))o~n<r
.
to
P(S)
=
135
by
1.1 c), it follows that the composition
ones on the diagonal,
Therefore
On the other hand,
Tn
vu
59
vu
is described by a triangular matrix with
is an isomorphism, so u
is an exact functor from =R°
is injective.
to =P(S) by
1.11 and
1.14,
hence we have a homomorphism
t : Kq(Ro)
=
where
tn
is induced by
--~ K (P(S))r
q=
Tn.
see that the composition ut
R
=O
in
P
3.
let
By
=O
t
2.2, ut
,
x
~-~ ((-1)ntn(X))o~n<
Applying Th. 2, Cor. 3
is the map
r
to the exact sequence
1.11, we
Kq(Ro
) = --~ Kq(Po)= induced by the inclusion of
is an isomorphism, so
The ~ro~ective line over a rin~. Let
A
u
is surjective, concluding the proof.
be a (not necessarily commutative) ring
be an indeterminate, and let
A[t]
ii>
¢- i2
Aft,t-l]
denote the canonical homomorphisms.
When
A
A[t-I]
is commutative, a quasi-coherent sheaf on
p l = Proj(A[X ,X.]) may be identified with a triple
0
'
1
M ~Mod(A[t ])
F = (M+,M-,@), where
M+g Mod(A[t]),
+
and
O :
i1(M ) --mx2(M ) iS an Isomorphism of A[t,t J - m o d u l e s .
l o w i n g [Bass XiI §9], we define
Mod(P1)
for A
Fol-
non-commutative to be the abelian
category of such triples, and we define the category of vector bundles on p1 , denoted
I
=P(PA), to be the full subcategory consisting of triples with M+~ __P(ALt~), M-E P(A[t-I/).
Theorem 3.1. Let
h
triple eonsistin~ o f P L t ~
: P(A) --@ P(PIA) be the exact functor sendin~
A[t]IAP ,-P[t-IJ, and multiplication by
t-n
P
~
to the
PLt,t-1].
Then one has isomorphisms
(KqA)2
~
Kq(p(p1))
,
(x,y) ~-~(ho).(x ) + (hl).(y)
and the relations
(3.2)
( ~ ) . - 2(hn_I ) . + (~_2). = O
for all
n.
When
module
A
is commutative, this follows from
0x(n)®sP . For the non-commutative case, one modifies the proof of
straightforward way. For example, if
Xo,X I : F(n-1)--> F(n)
XI
= t
2.1, once one notices that
on
M+
and
O
corresponding to
I
F = (M+,M-,8), we put
be the homomorphisms given by X ° = 1
on M-).
r- F(n-2)
hn(P)
2.1
is the
in a
F(n) = (M+,M-,t-n@), and let
on M +
and
t-I on M-,
Then we have an exact sequence
XoPrl
(XI ''X°) ~ F(n-l)2
1.6, which leads to the relations
, , , F(n)
+ XlPr2
~O
3.2. Also using the fact that
can be computed by means to the standard open affine covering of
Rqf.
p1, we can define
R~.(F)
in the non-commutative case to be the homology of the complex concentrated in
degrees
0 , I given by the map
d : M + x M-
~
i~(M-) , d(x,y) = @(1~x) - m~y .
One therefore has available all of the tools used in the proof of
2.1
tative case; the rest is straightforward checking which will be omitted.
143
in the non-commu-
136
4.
60
Severi-Brauer
schemes and Azuma[a algebras.
,i|N i
|
i
a Severi-Brauer scheme over
S
Let
of relative dimension
S-scheme locally isomorphic to the projective space
(see
r2
r-1.
PSr-1
be a scheme and let
By definition
X
X
be
is an
for the etale topology on
S.
[GrothendieckJ ), and it is essentially the same thing as an Azumaya algebra of rank
over
S.
We propose now to generalize
When there exists a line bundle
geometric fibre, one has
the etale topology on
X.
bundle of rank
X
r
on
PGLr, S
=
PGLr, S
E
Y
associated to a toreor
GLr, S
act on
X
Since
Y
Gin,S
if
Y
over
g
factors through the projective group
T
for the etale topology on
under
PGLr, S
of rank
X = PE.
X
S
and
X' = PE
Oy(-m)~0S ,
PGLr, S
is the bundle over
Y.
Oy(-1)~O S
on
Y
As
is the group
S
with fibre
locally trivial for the etale topology.
Y
Thus
gives rise to a vector
r.
J
is compatible with base change, and that
In the general case there is a cartesian square
)If
g'
~X
S'
g
; S
I
is faithfully flat (e.g. an etale surjective map over which
such that
r-1
Y = PS
=
in the standard way, and put
S, one knows that
I
where
S, f : X -, S
exists only locally for
acts trivially on the vector bundle
It is clear that the construction of
J = _~(-I)®s E
f,LV on
L
O(-I)r on each geometric fibre.
O~
by faithfully flat descent, the bundle
on
is the vector bundle
0(-I) on each
However, we shall now show that there is a canonical vector
is locally isomorphic to
J
to this situation.
which restricts to
operates on this vector bundle compatibly with its action on
of automorphisms of
bundle
X
In general such a line bundle
The induced action on
GLr,s/Gm, S .
the group
X.
2.1
on
which restricts to
Let the group scheme
Proj(S(_O~)).
L
X = PE, where
being the structural map of
X
S
for some vector bundle
E
of rank
r
on
T
becomes trivial)
S', and further
g,*(j) _- ~,!-,)%,~.
Let
A
he the sheaf of (non-commutative)
A
where
=
_0_s-algebras given by
f.(_~(j))op
'op' denotes the opposed ring structure.
As
g
is flat, we have
g*f. = f~g'* .
Rence we have
~(A)°P=
hence
A
f'.(~_~.(~.(-,)%,E))
is an Azumaya algebra of rank
f'A
=
as one verifies by pulling back to
Let
that
An
Jn (resp. An)
be the
r2
= f'.(Ox,~s,~n__%,(E>) _- _E~S,(E) ,
over
S.
Moreover one has
~(j)Op
X'.
n-fold tensor product of
is an Azumaya algebra of rank
(rn) 2
144
such that
J
on
X (resp.
A
on
S), so
137
An
Let
=P(An)
Since
Jn
=
f.(_En_~(Jn))°P
61
, f*(An)
=
denote the category of vector bundles on
is a right
E
n~(Jn)°P
S
f*(An)-module , which locally on
which are left modules for
An
is a direct summand of
f*(An),
X
we have an exact functor
JaVA ?
n
: P(An)
=
and hence an induced map of
Theorem 4.1.
~
P(X)
=
,
M
f*(M) .
K-groups.
I_~f S is quasi-compact, one has isomorDhisms
r-1
r-1
[ I Ki(A) ~
Ki(X)
' (Xn) ~'@ Z
( J n ~ C ) * ( x n)
n=O
n=O
This is actually a generalization of
equivalent, and hence have isomorphic
2.1
because if two Azumaya algebras
K-groups.
S, then the categories
Thus
Ki(P(An) ) = Ki(S)
A , B
P(A), =P(B) are
=
for all
n
if
is the projective bundle associated to some vector bundle.
The proof of
F
Jn ~
f*(An)
represent the same element of the Brauer group of
X
~
4.1
is a modification of the proof of
to be regular if its inverse image on
X' = PE
2.1. One defines an X-module
is regular.
For a regular
F
one
constructs a sequence
(4.2)
0
---~Jr_I®Ar_ITr_I(F)
~
. • . ~
O__x~To(F) ~
F
~
0
recursively by
Tn(F )
=
f.(H~(Jn,Zn_1(F)) ) ; Zn(F ) =
Ker IJn~A Tn(F) --* Zn_I(F) )
n
starting with
Z I(F) = Fo
It is easy to see this sequence when lifted to
the the canonical resolution
faithfully flat over
X,
1.11 for the inverse image of
4.2
is a resolution of
F
on
X'.
X'
Since
coincides
X'
is
F.
We note also t~mt there is a canonical epimorphism
J - ~ _~ obtained by descending
1.4, and hence a canonical vector bundle exact sequence
o--~A%
on
X
~
corresponding to
the proof of
proof of
4.1
Example:
2.1
~, a - - . ~ - - + o
...
1.5. Therefore it should be clear that all of the tools used in
are available in the situation under consideration; the rest of the
will be left to the reader.
Let
X
be a complete non-singular curve of genus zero over the field
k = H°(X,O__x) , and suppose
X
has no rational point. Then
over
k
of relative dimension one, and
rank
2
over
X
with degree
K.(X)I =
where
A
-2.
J
X
is a Severi-Brauer scheme
is the unique indecomposable vector bundle of
The above theorem says
Ki(k) ~Ki(A)
is the skew-field of endomorphisms of
been proved by Leslie Roberts ([Roberts]).
145
J.
This formula in low dimensions has
138
62
References
H. Bass:
Algebraic
S. Bloch:
K2
K-theory, Benjamin 1968.
and algebraic cycles, these procedings.
K. Brown and S. Gersten:
Algebraic
K-theory as generalized sheaf cohomology, these
prooedings.
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---
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~--
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----- 2:
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....
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147