April 8, *
*1992
ADAMS OPERATIONS ON HIGHER K-THEORY
Daniel R. Grayson
University of Illinois at Urbana-Champaign
Abstract. We construct Adams operations on higher algebraic K-groups induc*
*ed by oper-
ations such as symmetric powers on any suitable exact category, by constru*
*cting an explicit
map of spaces, combinatorially defined. The map uses the S-construction of*
* Waldhausen,
and deloops (once) earlier constructions of the map.
1. Introduction.
Let P be an exact category with a suitable notion of tensor product M N, sy*
*mmetric
power SkM, and exterior power kM. For example, we may take P to be the category
P(X) of vector bundles on some scheme X. Or we may take P to be the category P(*
*R) of
finitely generated projective R-modules, where R is a commutative ring. Or we m*
*ay fix a
group and a commutative ring R and take P to be the category P(R; ) of represe*
*ntations
of on projective finitely generated R-modules. We impose certain exactness req*
*uirements
on these functors, so that in particular the tensor product is required to be b*
*i-exact, and
this prevents us from taking for P a category such as the category M(R) of fini*
*tely
generated R-modules.
In a previous paper [5] I showed how to use the exterior power operations on*
* modules
to construct the lambda operations k on the higher K-groups as a map of spaces *
*k :
|GP| ! |GkP| in a combinatorial fashion. Here the simplicial set GP, due to Gil*
*let and
me [3], provides an alternate definition for the K-groups of any exact category*
*, KiP =
ssiGP, which has the advantage that the Grothendieck group appears as ss0 and i*
*s not
divorced from the higher K-groups. The Q-construction of Quillen and the S-cons*
*truction
of Waldhausen are the original definitions for the K-groups of P, but involve a*
* shift in
degree, so that Ki(P) = ssi+1|S:P| = ssi+1|QP|; since the lambda operations are*
* not
additive on K0, but any function on ss1 arising from a map would be a homomorph*
*ism,
neither of these two spaces could be used to define lambda operations combinato*
*rially.
The Adams operation k is derived from the lambda operation k by a natural p*
*rocedure
which makes k additive on K0. Thus there is no apparent obstruction to the pr*
*esence
of a combinatorial description for k that involves |S:P| or |QP|. The purpose*
* of this
paper is to present such a combinatorial construction of the Adams operation as*
* a map of
spaces k : |S:P| ! |S:G"(k)P|. The map works by considering symmetric powers o*
*f acyclic
complexes of length one, and by introducing a sort of symmetric product of the *
*members
of a filtration of acyclic complexes. The map is a delooping of the Adams opera*
*tion map
_____________
1991 Mathematics Subject Classification. Primary 18F25; Secondary 19C30, 19D*
*99.
Supported by NSF grant DMS 90-02715
1
2 DANIEL R. GRAYSON
derivable from the lambda operation maps. I don't know whether further deloopi*
*ng is
possible without inverting some integers, and I suspect that this one-fold delo*
*oping is new,
even on the level of Z x BU. (One may refer to [12] for methods that can be us*
*ed to
transfer these results to topological K-theory.)
The construction G"(k)appearing in the target of the map is a (k - 1)-dimens*
*ional
cube of exact categories, each of which involves acyclic complexes of length k *
*as well as
total complexes of multi-dimensional complexes that are acyclic in two directio*
*ns. It is
arranged so that the target of the map is yet another space whose homotopy grou*
*ps are the
K-groups, and, in fact, there is a natural, combinatorially defined, homotopy e*
*quivalence
|S:P| ! |S:G"(k)P|.
In [13] Schechtmann gives a construction of operations analogous to the one *
*I present
here, but it yields a homotopy class of maps rather than a single explicit map;*
* at the
expense of tensoring with the rational numbers, he shows that the Adams operati*
*ons are
infinite loop maps, whereas we deloop only once in this paper. Alexander Nenash*
*ev will
write a paper in which he constructs lambda operations based on techniques in [*
*5], but
using long exact sequences instead of cubes, as suggested in [3]. For other dis*
*cussions of
lambda-operations and Adams operations on algebraic K-theory, the reader may wi*
*sh to
refer to [7], [8], [9], [4], and [10].
I thank Henri Gillet for useful discussions and the idea of using the second*
*ary Euler
characteristic. I thank David Benson for the definition of the symmetric power *
*of a complex
that I use; the one I was originally using was based on the theory of non-addit*
*ive derived
functors of Dold and Puppe, [2]. I thank Pierre Deligne and Jens Franke, who ex*
*plained to
me that it ought to be possible to realize the eigenspaces of the Adams operati*
*ons on the
rational K-groups as the rational homotopy groups of spaces; perhaps the constr*
*uction of
this paper is a step in that direction, and thus might help analyze the relatio*
*nship between
K-theory and motivic cohomology.
2. Symmetric powers of complexes and symmetric products of filtered com-
plexes.
We will write about finitely generated projective R-modules for convenience *
*of exposi-
tion below, but it will be apparent that any of the constructions we use will w*
*ork equally
well for locally free sheaves of finite type (vector bundles) on a scheme X, or*
* for represen-
tations of a group G in finitely generated projective R modules. All tensor pro*
*ducts will
be over R.
If R is a commutative ring and M is an R-module, then the k-th symmetric pow*
*er SkM
of M is defined to be the quotient of Mk by the relations
x1 . . .xi xi+1 . . .xk ~ x1 . . .xi+1 xi . . .xk:
Similarly, the k-th exterior power kM of M is defined to be the quotient of Mk *
* by the
relations
x1 . . .xi xi+1 . . .xk ~ -x1 . . .xi+1 xi . . .xk:
and
x1 . . .xi xi+1 . . .xk ~ 0
ADAMS OPERATIONS ON HIGHER K-THEORY 3
if xi= xi+1. The first of these two relations follows easily from the second.
Now let M be a Z-graded R-module, (or even a Z=2Z-graded R-module). If x 2 M*
*p, then
we say that x is a homogeneous element of M and that degx = p. We may mix the r*
*elations
for symmetric and exterior powers mentioned above, and define the k-th symmetri*
*c power
SkM of M to be the quotient of Mk by the relations among homogeneous elements*
* xi
of M,
x1 . . .xi xi+1 . . .xk ~ (-1)degxi.degxi+1x1 . . .xi+1 xi . . .xk;
and
x1 . . .xi xi+1 . . .xk ~ 0
whenever xi= xi+1 and degxi is odd. We let x1 . x2 . : :.:xk denote the image i*
*n SkM of
x1 . . .xk.
If M is concentrated in even degrees, then SkM is the k-th symmetric power o*
*f the
underlying module, and if M is concentrated in odd degrees, then SkM is the k-t*
*h exterior
power.
The module SkM is itself a graded module, with
deg (x1 . : :.:xk) = degx1 + . .+.degxk
If the graded module M is free, (which we take to mean that each component M*
*p is
free), then we may take a basis {ej} for it that consists of homogeneous elemen*
*ts. We say
that a tensor product ei1 . . .eik or its image ei1. : :.:eik in SkM is a monom*
*ial. We
may write SkM as the quotient of Mk by the following monomial relations:
ej1 . . .eji eji+1 . . .ejk~ (-1)degeji.degeji+1ej1 . . .eji+1 eji . . .ejk;
and
ej1 . . .eji eji+1 . . .ejk~ 0
whenever eji= eji+1and degejiis odd. Repeated application of the first of these*
* types of
relations to a monomial will accumulate a sign which is the sign of the permuta*
*tion affecting
the factors of odd degree; if we are ever led thereby to a relation of form ej1*
*. : :.:ejk ~
-ej1.: :.:ejk, then we must have a repeated factor of odd degree, so that ej1.:*
* :.:ejk~ 0 is
a consequence of the second relation. These remarks make it clear that sorting *
*the factors
in a monomial modulo the two relations is a well-defined operation, so that SkM*
* is a free
R-module, with a basis consisting of those monomials ej1. : :.:ejk such that j1*
* . . .jk,
and ji= ji+1 only if degejiis even.
Now suppose that M is a chain complex of R-modules, so it is a Z-graded modu*
*le with
a differential d of degree -1. We define a differential d on Mk by means of *
*the usual
Leibniz rule
Xk
d(x1 . . .xk) = (-1)degx1+...+degxi-1x1 . . .dxi . . .xk
i=1
4 DANIEL R. GRAYSON
and observe that this respects the relations defining the quotient SkM, thereby*
* defining a
differential on SkM and making it into a chain complex.
An important special case arises when M is the mapping cone CN = C1N of the *
*identity
map on a finitely generated projective R-module N, so that M is an acyclic chai*
*n complex
of length 1, with a copy of N in degrees 0 and 1. In this case one sees that Sk*
*CN is the
usual Koszul complex of N, in which (SkM)p = Sk-pM0 pM1 = Sk-pN pN. It
is known [1, p. 528] that the Koszul complex SkCN is acyclic when k > 0, and i*
*s the
ring R concentrated in degree 0 when k = 0; a simple proof can be given based o*
*n the
multilinearity property below (2.1), by induction on k and the rank of N.
We remark that if M is an acyclic free complex concentrated in degrees 1 and*
* 2, then
SkM is not in general acyclic. For example, with k = 2, one gets a complex S2N*
* !
N N ! 2N which fails to be exact in the middle because of elements like x x w*
*hich
are not hit.
We proceed now to the next generalization. We will overload the subscript no*
*tation a
bit, and use subscripts to denote both the members of a filtration and the comp*
*onents of
a graded module. Let M be a filtered complex with k steps, so that we have comp*
*lexes
M1 . . . Mk = M. If we need it, we will refer to the degree p component of the
complex Mi as Mip. We define the symmetric product M1 . : :.:Mk of M to be the *
*image
of M1 . . .Mk in SkMk.
We will always assume that M is an admissible filtered complex of finitely g*
*enerated
projective R-modules, so that every module Mip in it is a finitely generated pr*
*ojective
module, and so that each inclusion Mi-1;p Mip is admissible in the sense that i*
*ts
cokernel is projective. We say that M is free if every Mip is free, and every *
*quotient
Mip=Mi-1;pis free. A basis for a free admissible filtered complex M will be a c*
*ollection
of bases for each Mipthat are upward compatible, and thus induce bases on the q*
*uotients
Mip=Mi-1;p. We remark that an admissible filtered complex M is locally free.
The symmetric product of an admissible filtered complex M can also be define*
*d by gen-
erators and relations (and this might be a preferable definition when M is not *
*admissible,
or does not consist of projective modules). It is the quotient of M1 . . .Mk b*
*y those
relations used before where the i-th factor in the tensor is required to lie in*
* Mi. To be
precise, the relations among tensor products of homogeneous elements xi of M are
x1 . . .xi xi+1 . . .xk ~ (-1)degxi.degxi+1x1 . . .xi+1 xi . . .xk
whenever xj 2 Mj for all j, and moreover xi+1 2 Mi, and
x1 . . .xi xi+1 . . .xk ~ 0
whenever xj 2 Mj for all j, xi = xi+1 and degxi is odd. To prove this assertion*
*, we may
localize sufficiently to ensure that M is free, and then we may pick a basis {e*
*j} for M and
order it in such a way that the basis elements for M1 come first, and then come*
* some more
elements to complete a basis for M2, and so on. The relations mentioned suffice*
* to sort
the factors of any monomial drawn from M1 . . .Mk, and allow us to write down *
*an
explicit basis for the quotient, consisting of those monomials ej1 . . .ejk whe*
*re eji2 Mi
for each i, j1 . . .jk, and ji= ji+1 only if degejiis even. Since these monomi*
*als are a
ADAMS OPERATIONS ON HIGHER K-THEORY 5
subset of the monomials that serve as basis for SkMk, and are the same monomial*
*s that
span the image of M1 . . .Mk in SkMk, we have proved our assertion.
The main fact about symmetric products of admissible filtered complexes gove*
*rns what
happens when one of the terms in the filtration is perturbed slightly, and is a*
* property we
will call multilinearity. Suppose M is an admissible filtered complex, and supp*
*ose M0j+1
is an alternative for the step Mj+1 in the filtration M, which we take to mean *
*that
M1 . . .Mj M0j+1 Mj+1 . . .Mk
is an admissible filtrations of complexes. By localizing sufficiently to make e*
*verything free,
one sees that M1.: :.:M0j+1.: :.:Mk is an admissible subcomplex of M1.: :.:Mj+1*
*.: :.:Mk.
The multilinearity property identifies the quotient via a certain natural isomo*
*rphism:
Mj+1 Mk
(2.1) M1_._:_:.:Mj+1_._:_:.:Mk_M~=M1 . : :0:. Mj _____0. : :.:_____0:
1 . : :.:Mj+1 . : :.:MkMj+1 Mj+1
Indeed, both sides of this isomorphism are quotients of M1 . . .Mk by various *
*explicit
relations, and all one has to do is to check that the two sets of relations are*
* equivalent; this
can be done. Another way is to localize sufficiently so that all the everything*
* is free, pick
an ordered basis {ej} for M compatible with the filtration as we did above, and*
* observe
that the same set of monomials gives a basis for both sides.
Here is an important corollary of the multilinearity of symmetric products. *
* Suppose
M is an acyclic admissible filtered complex of length 1, which we take to mean *
*that (in
addition to begin admissible) each step Mi in the filtration is an acyclic comp*
*lex of length
1. I claim that the symmetric product M1 . : :.:Mk is an acyclic complex. The p*
*roof goes
by induction on k; making use of multilinearity and the fact that a tensor prod*
*uct of two
acyclic complexes is acyclic allows us to modify M2; : :;:Mk successively so th*
*at they all
equal M1, reducing us to the previously mentioned result about Koszul complexes*
* being
acyclic.
Here is an example of the symmetric product. In the case where k = 2 and M =*
* CN
is the mapping cone of a admissible filtered module N1 N2 we find that M1 . M2*
* is the
acyclic complex
0 ! N1 ^ N2 ! (N1 N2) + (N2 N1) ! N1 . N2 ! 0
which sits as an admissible subcomplex of the Koszul complex of N2:
0 ! 2N2 ! N2 N2 ! S2N2 ! 0:
Here we use N1 ^ N2 to denote the image of N1 N2 in 2N2, and N1 . N2 to denote*
* the
image of N1 N2 in S2N2.
We have seen that the symmetric product of an admissible filtered acyclic co*
*mplex of
length one is a natural generalization of the Koszul complex. There is another *
*conceivable
generalization of the Koszul complex that also turns out to be acyclic, but whi*
*ch we do
not need in the sequel; uninterested readers may skip to the beginning of the n*
*ext section
6 DANIEL R. GRAYSON
now. For an admissible filtration N1 . . .Nk of finitely generated projective *
*modules
it looks like
0 ! kN1 ! . .!.N1 . : :.:Nk-p pNk-p+1 ! . .!.N1 . : :.:Nk ! 0:
It can be constructed from the symmetric product CN1.: :.:CNk by an interesting*
* pruning
procedure, which I describe now.
Suppose that a complex M of length k has a filtration 0 = M-1 . . .Mk = M
with the property that each quotient Mp=Mp-1 is a complex of length p whose hom*
*ology
vanishes except in degree p. A new complex M", also of length k, can be define*
*d by
setting M"p = Hp(Mp=Mp-1). A straightforward diagram chase defines the differe*
*ntials
in M", shows that M" is a complex, constructs a map M" ! M, and shows that the *
*map
M" ! M is a quasi-isomorphism. (This is related to the way that the skeletal fi*
*ltration of
a cell-complex leads to the complex of cellular chains from the complex of sing*
*ular chains.)
Instead of doing the diagram chase, one could regard the spectral sequence asso*
*ciated to
M, and take M" to be the nonvanishing row of the E1 term. We say that M" is obt*
*ained
from M by pruning.
We may prune the symmetric product W = CN1 . : :.:CNk by means of the filtra*
*tion
whose p-th step is Wp = N1 . : :.:Nk-p . CNk-p+1 . : :.:CNk. Here we regard eac*
*h module
Ni as a complex by concentrating it in degree 0; in this way it is a subcomplex*
* of CNi.
By multilinearity (2.1) the quotient Wp=Wp-1 is
N1 . : :.:Nk-p CNk-p+1__N. : :.:_CNk___:
k-p+1 Nk-p+1
We may modify the latter complex so that Nk-p+2; : :;:Nk are successively repla*
*ced by
Nk-p+1, without changing the quasi-isomorphism class, by using the multilineari*
*ty prop-
erty with the acyclicity of complexes of the form
CN`+1__. : :.:CNk__:
CN` CN`
The result, after the modifications, is
N1 . : :.:Nk-p CNk-p+1__N. : :.:CNk-p+1__= N1 . : :.:Nk-p pNk-p+1[-p]:
k-p+1 Nk-p+1
We conclude that Wp=Wp-1 has homology only in degree p, and that pruning W lead*
*s to
the complex announced above.
3. The Adams operation as the secondary Euler characteristic of the Koszul
complex.
Use the symbol [N] to denote the class of a finitely generated projective R-*
*module N
in the Grothendieck group K0R, or the class of a vector bundle N on a scheme X *
*in the
Grothendieck group K0X. All complexes below will be bounded chain complexes. Le*
*t M
be a complex of finitely generated projectivePR-modules with differential dp : *
*Mp+1 ! Mp,
and recall the Euler characteristic O(M) = p(-1)p[Mp]. If M is acyclic, then *
*O(M) = 0,
ADAMS OPERATIONS ON HIGHER K-THEORY 7
P
and the secondaryPEuler characteristic may be defined as O0(M) = p(-1)p+1p[Mp*
*] or as
O0(M) = p(-1)p[im dp]. If 0 ! M0 ! M ! M00! 0 is a short exact sequence of ac*
*yclic
complexes, then O0(M) = O0(M0) + O0(M00).
We say that a bicomplex is doubly acyclic if each row and each column are ac*
*yclic. The
tensor product of two acyclic complexes of projective modules (regarded as a bi*
*complex)
is doubly acyclic. If M is a doubly acyclic bicomplex, and TotM is its total co*
*mplex, then
O0(Tot M) = 0; one proves this by considering the filtration on Tot M arising f*
*rom the
canonical filtration with respect to the columns and using the additivity of O0*
*to show that
O0(Tot M) is the alternating sum of O0 of the columns of M, which is then zero *
*because
the columns of M fit into a long exact sequence. Even more is true: if d = d0*
*+ d00is
the differential on TotM, where d0 and d00are the horizontal and vertical diffe*
*rentials on
M, then the projective modules im dp may be assembled into an acyclic complex b*
*y using
the maps induced by d0 (or by d00) as differential. The proof (for the ring cas*
*e) goes by
filtering M in both directions in such a way that the successive quotients are *
*doubly acyclic
bicomplexes of size 1 by 1, in which case the statement can be checked easily.
Let k denote the k-th Adams operation on K0R or K0X. I claim that for any N*
* as
above the following formula holds.
(3.1) k[N] = O0(SkCN)
We prove this by verifying, for the right hand side of the equation, the two pr*
*operties
that (according to the splitting principle) characterize k. Firstly, when rank*
*N = 1 the
Koszul complex SkCN is just C(Nk ), so O0(SkCN) = Nk . Secondly, if 0 ! N0 !
N ! N00! 0 is a short exact sequence, then we can verify the additivity O0(SkCN*
*) =
O0(SkCN0)+O0(SkCN00) of the right hand side by making use of multilinearity (2.*
*1). From
the filtration
SkCN0 = CN0 . : :.:CN0 . CN0
CN0 . : :.:CN0 . CN
CN0 . : :.:CN . CN : : :
CN . : :.:CN . CN = SkCN
we deduce that
k-1X
O0(SkCN) = O0(SkCN00) + O0(SkCN0) + O0(SiCN00 Sk-iCN0)
i=1
= O0(SkCN00) + O0(SkCN0):
The cross-terms drop out because the secondary Euler characteristic of a tensor*
* product
of acyclic complexes is zero.
As an example, we may compute 2[N]. In this case, the complex S2CN is 0 ! 2*
*N !
N N ! S2N ! 0, and O0(S2CN) = [S2N] - [2N].
If we let LkpN denote the image of dp in the Koszul complex SkCN. The functo*
*r LkpN
is the Schur functorPcorresponding to the Young diagram (k - p; 1; : :;:1) of h*
*ook type.
We see that k[N] = (-1)p[LpN].
8 DANIEL R. GRAYSON
We remark that formula (3.1) is like the nonstandard definition of the diffe*
*rential of a
C1 -map f : M ! N of manifolds. If we think of M and N as being embedded manifo*
*lds
containing the origin, the differential of f at the origin can be written as (d*
*f)0(v) =
standard part of1_fflf(fflv) , where v is a vector tangent to M at 0, and ffl i*
*s an infinitesimal
number. Comparing with (3.1) we see that multiplication of v by ffl is analogou*
*s to forming
the mapping cone of the identity map on N. This suggests that we regard acyclic*
* complexes
as being infinitesimal in size when compared to arbitrary complexes, and that w*
*e regard
the category of complexes as being an enlargement of the category of modules. W*
*e also
see that the final step of dividing by ffl and taking the standard part is anal*
*ogous to taking
the secondary Euler characteristic of an acyclic complex. The fact that terms *
*in the
expansion of f(fflv) involving ffl2 drop out when we divide by ffl and take the*
* standard part
corresponds to the fact that doubly acyclic complexes yield 0 when we take the *
*secondary
Euler characteristic, and the two facts arise in the same way in the proof of a*
*dditivity. This
suggests that we regard doubly acyclic complexes as being doubly infinitesimal *
*in size when
compared to arbitrary complexes. It also suggests that we regard the Adams oper*
*ation k
as being the differential of the functor N 7! SkN from the category of finitely*
* generated
projective modules to itself; the differential is formed by first extending the*
* domain of
the functor from modules to complexes of modules, which is somehow analogous to*
* first
extending the domain of f from M to a nonstandard model of M.
4. The multi-relative S.-construction.
We let [1] denote the ordered set {0 < 1} regarded as a category. By an n-di*
*mensional
cube M of (exact) categories we will mean a functor from [1]n to the category o*
*f (exact)
categories.
In this section we show how, given an n-dimensional cube M of exact categori*
*es, we may
construct a certain n-fold multisimplicial exact category called CM to serve as*
* the mapping
cone of the cube. In the case n = 1, it will be the same as a construction of W*
*aldhausen
[15, p. 343] denoted S:(M0 ! M1); in [14, p. 182-184] the same construction is *
*called
F:(M0 ! M1).
If M and M0are n-dimensional cubes of exact categories, we let Exact(M; M0) *
*denote
the set of natural transformations M ! M0.
If M is an exact category, we let [M] denote the corresponding 0-dimensional*
* cube of
exact categories. We will often simply identify M with [M].
Given n-dimensional cubes M and M0 of exact categories, and an exact functor*
* g 2
Exact(M; M0), we may assemble M0 and M into an n + 1-dimensional cube of exact
categories; we will use the symbol [M0 ! M] to denote it. We will also use squ*
*are
brackets enclosing a commutative square of n-dimensional cubes of exact categor*
*ies to
denote the corresponding n + 2 dimensional cube.
If M is an n-dimensional cube of categories and M0 is an n0-dimensional cube*
* of cat-
egories, then we let M M0 denote the evident n + n0-dimensional cube of catego*
*ries
defined by
(M M0)(ffl1; : :;:ffln+n0) = M(ffl1; : :;:ffln) x M0(ffln+1; : :;:f*
*fln+n0):
We let denote the category of finite nonempty totally ordered sets. If C is*
* a category,
let Ar(C) denote the category of arrows in C, where an arrow in this category i*
*s a commu-
ADAMS OPERATIONS ON HIGHER K-THEORY 9
tative square. If A is an ordered set regarded as a category, we will use j=i t*
*o denote the
arrow from i to j in A, if i j.
Given an exact category M with a chosen zero object 0 and an ordered set A, *
*we call
a functor F : Ar(A) ! M exact if F (i=i) = 0 for all i, and 0 ! F (j=i) ! F (k=*
*i) !
F (k=j) ! 0 is exact for all i j k. The set of such exact functors is denote*
*d by
Exact(Ar (A); M). Given ordered sets A1; : :;:An, we let Exact(Ar (A1)x. .x.Ar(*
*An); M)
denote the set of multi-exact functors, i.e., functors that are exact in each v*
*ariable.
Given A; B 2 let AB denote the totally ordered set constructed from A and B*
* by
concatenation, i.e., as the disjoint union of A and B with every element of A d*
*eclared to
be less than every element of B.
Now let L be a symbol, and consider {L} to be an ordered set. Given an n-dim*
*ensional
cube of exact categories M, we define an n-fold multisimplicial exact category *
*CM as a
functor from (n)op to the category of exact categories by letting CM(A1; : :;:A*
*n) be the
set
Exact([Ar (A1) ! Ar({L}A1)] . . .[Ar (An) ! Ar({L}An)]; M)
of multi-exact natural transformations. When n = 0, we may identify CM with M.
We define S:M to be S:CM, the result of applying the S: construction of Waldhau*
*sen
degreewise. The construction S:M is a n + 1-fold multisimplicial set; to make t*
*hat explicit
we write the new argument to the left of the other ones, and see that
S:M(A0; A1; : :;:An) = Exact([Ar (A0)][Ar (A1) ! Ar({L}A1)]. .[.Ar(An) ! Ar({L}*
*An)]; M)
for A0; : :;:An 2 .
Lemma 4.1. Suppose we are given M0! M as above.
(a) There is a fibration sequence S:[0 ! M] ! S:[M0! M] ! S:[M0! 0].
(b) In the case where g is the identity map, the space S:[M ! M] is contract*
*ible.
(c) S:[0 ! M] is homotopy equivalent to S:M.
(d) S:[M ! 0] is a delooping of S:M.
(a) There is a fibration sequence S:M0! S:M ! S:[M0! M].
Proof. One uses the additivity theorem of Waldhausen, just as in [15, p. 343] o*
*r [14, p.
182-184].
We remark that if
R ----! S
?? ?
y ?y
T ----! U
is a square of commutative rings, then tensor product of projective modules lea*
*ds imme-
diately to a map
fifi 2 3fi
fifi 6P(R)? ----! P(S)?7f*
*ififi
|S:[P(R) ! P(S)]| ^ |S:[P(R) ! P(T )]| ! fifiS:S: 64 ?y ?y 75*
*fifi
fifi fi
P(T ) ----! P(U) fi
which can be used to define products on relative K-groups.
10 DANIEL R. GRAYSON
5. The construction of the Adams operations.
For the construction of the Adams operation k on the K-groups of an exact ca*
*tegory
P we will need to consider k-dimensional multi-complexes N of length one in eac*
*h direc-
tion, and to take total complexes of them in a certain partial way. These "part*
*ial" total
complexes will have fewer dimensions than N has, and their lengths will be grea*
*ter; the
"total" total complex of N will be of dimension 1 and length k.
We describe now what sort of "partial" total complexes we have in mind.
An equivalence relation ' on a totally ordered set A 2 is compatible with t*
*he ordering
if the quotient set A=' inherits an ordering from A so that the quotient map is*
* order-
preserving. If we denote the equivalence classes by A1; : :;:At, then we may w*
*rite A as
the concatenation A = A1A2 . .A.t, and the quotient A=' as the ordered set A=' =
{A1; : :;:At}.
Let N be a multi-complex whose directions are indexed by the elements of A. *
* We
assume that the differentials anti-commute with each other, i.e., @i@j + @j@i =*
* 0 ; this
ensures that when taking total complexes, the sum of the differentials immediat*
*ely provides
a differential. A homogeneous element x 2 N has a multi-degree p : A ! Z which*
* is a
sequence of integers indexed by the set A, and we let Np denote the set of homo*
*geneous
elements of N of multi-degree p, together with 0. Define B = A=', and let ss :*
* A ! B
be the quotient map. We may define a multi-complex N0 = Tot' N whose directions*
* are
indexed by B by specifying ss*p =Pq : B ! Z, the degree of x as an element of N*
*0. It will
be given by the formula q(b) = p(a). This corresponds to setting
a2'-1(b)
X
N0q= Np:
ss*(p)=q
Let's use [1; k] as notation for the ordered set {1; 2; : :;:k}. The number *
*of equivalence
relations on [1; k] compatible with the ordering is 2k-1, as such relations are*
* freely and
completely specified by the truth or falsity of the statements i ' i + 1 for i*
* = 1; : :;:k - 1.
We may consider the set of equivalence relations on [1; k] compatible with the *
*ordering to
be a set of subsets of [1; k] x [1; k], and order it by inclusion. It is isomor*
*phic, as a partially
ordered set, to [1]k-1. We use the isomorphism that associates (ffl1; : :;:fflk*
*-1) to ', where
ae0 if i 6 i + 1
ffli= '
1 if i ' i + 1.
For each equivalence relation ' on [1; k] compatible with the ordering, we l*
*et `1; : :;:`t
denote the cardinalities of the equivalence classes, in sequence. Consider the *
*category M'
of t-dimensional chain-complexes that are, for each i, of length `i in the i-th*
* direction,
and that are acyclic in direction 1 and in direction t. There is a total-compl*
*ex functor
M' ! M if ' , because the lengths add when total complexes are constructed, *
*and
because the total complex of a multi-complex that is acyclic in one direction i*
*s acyclic.
Using these total-complex functors we may assemble the categories M' into a k -*
* 1
dimensional cube G"(k)P = M of exact categories which will serve as the target *
*for our
Adams operation map k.
ADAMS OPERATIONS ON HIGHER K-THEORY 11
Actually, there is a little problem with getting G"(k)P to be a functor from*
* [1]k-1 to
the category of exact categories, because the composition of two total-complex *
*functors
is perhaps only isomorphic to the combined total-complex functor; this is somet*
*hing like
failure of strict associativity for direct sums, and can be cured with an easy *
*set-theoretic
trick, or by considering G"(k)P instead to be a category cofibered over [1]k-1 *
*in exact
categories.
We remark that there is homotopy equivalence G"(2)P ! GP that associates to *
*an
acyclic complex of length 2, the images of the two differentials in it. The ma*
*p 2 can
be viewed as a map |S:P| ! |S:GP|, and it was this version which was found firs*
*t, and
motivated the more general construction described in this paper.
Lemma 5.1. S:G"(k)P is homotopy equivalent to S:P
Proof. Consider the edges of the cube "G(k)P that lie in direction 1. These edg*
*es are total
complex functors M' ! M where the only difference between ' and is that 1 6 *
*'2
and 1 2.
Consider first the case where 2 ' 3 ' . . .'k. The category M' is the ca*
*tegory
of bicomplexes of length 1 in direction 1, of length k - 1 in direction 2, and *
*acyclic in
both directions. It is equivalent to the category of acyclic complexes of lengt*
*h k - 1, and
the functor M' ! Pk-2 that assigns to an acyclic complex the collection of imag*
*es of its
differentials yields a homotopy equivalence on K-theory, by the additivity theo*
*rem. The
category M is the category of acyclic complexes of length k. The functor M ! *
*Pk-1
that assigns to an acyclic complex the collection of images of its differential*
*s yields a
homotopy equivalence on K-theory, by the additivity theorem. Let C : P ! M be *
*the
functor that assigns CP to P 2 P, regarded as an acyclic complex of length k. T*
*hen the
map S:[0 ! P] ! S:[M' ! M ] is a homotopy equivalence.
Consider now the other case, where there exists j 2 so that j 6 'j + 1; we *
*claim that
S:M' ! S:M is a homotopy equivalence. This again is a straightforward applicat*
*ion of
the additivity theorem, just as in the previous paragraph. It is enlightening t*
*o regard the
additivity theorem itself as a statement something like the one at hand: it say*
*s that the
total complex functor from the category of one-by-one bicomplexes, acyclic in d*
*irection 1,
to the category of acyclic complexes of length 2, gives a homotopy equivalence *
*on K-theory.
Combining both cases, we see that we have a map S:P ! S:G"(k)P, obtained by *
*adding
additional trivial simplicial directions to S:P, which is a homotopy equivalenc*
*e.
6. The construction of the map.
In this section we give the formula for the combinatorial Adams operation map
k : Subk S:P ! S:G"(k)P:
Here Subk is the k-fold subdivision introduced in [5]: if X is a simplicial set*
*, then Subk X
is the k-fold multisimplicial set defined by
Subk X(A1; : :;:Ak) = X(A1 . .A.k):
There is a natural homeomorphism |X| ' | SubkX|, presented in [5]. Here is *
*a way
to see how that homeomorphism works. Let V be an affine space of dimension n (t*
*orsor
12 DANIEL R. GRAYSON
under Rn). Given points v1; : :;:vk 2 V define their barycenter v1* . .*.vk to *
*be the point
(v1 + . .+.vk)=k. If S1; : :;:Sk are subsets of V , then we let S1 * . .*.Sk de*
*note the set
{v1 * . .*.vk | vi2 Si}. __
If A is a set {v0; : :;:vp} _V_, let A denote the convex hull of A. If the v*
*ectors in A
are affinely independent, then A is a p-simplex. Let B and C be subsets of A; w*
*e write
B_|_C if i__j_for all vi 2 B and all vj 2 C. Given subsets B1 |_._.|.Bk of A, *
*the set
B1 * . .*.Bk is a product of simplices, and such sets subdivide A in exactly th*
*e same way
that | SubkX| subdivides each simplex of |X|.
Given
M 2 Subk S:P(A1; : :;:Ak) = Exact(Ar (A1 . .A.k); P)
we define
kM 2 Exact([Ar (A1)] [Ar (A2) ! Ar({L}A2)] . . .[Ar (Ak) ! Ar({L}Ak)]; "G(k)*
*P)
by the formula
(6.1) (kM)(i1=j1; : :;:ik=jk) = CM(i1=`1) *2CM(i2=`2) *3. .*.kCM(ik=`k):
Here i1=j1 2 Ar(A1), and ir=jr 2 Ar({L}Ar) for 2 r k. We define
ae. if j = L
*r= r
if jr 6= L
and ae
`r-1 if jr =2Ar and r > 1
`r =
jr if jr 2 Ar or r = 1
We spell out the needless conditions concening r = 1 and r > 1 so the same defi*
*nition will
work below, in a context where j1 =2A1 is possible. Notice that jr =2Ar is equi*
*valent to
j = L, for r > 1. Finally, one must interpret the symbols arising in (6.1) as *
*instances
of the symbol *r correctly: they are tensor products of acyclic complexes, but *
*are to be
interpreted as yielding bicomplexes if we are looking at Ar(Ar), or as yielding*
* complexes
if we are looking at Ar({L}Ar); this builds into the notation the business with*
* all the
total-complex functors. One checks that kM is exact in the variables ir=jr usi*
*ng the
multilinearity property (2.1), just as in [5].
On the level of the Grothendieck group, the secondary Euler characteristic g*
*ives the
inverse to the isomorphism K0P ! K0G"(k)P. Combining this with the formula (3.1*
*) we
see that our map k agrees with the usual Adams operation on K0P.
We now check that our Adams operations agree with the usual ones on the high*
*er
K-groups of a ring R. Consider the fibration sequence
G:P ! P:P ! S:P
from [3] which holds for any exact category P. For reference, we state the def*
*initions,
where A 2 .
G:P(A) = Exact(Ar ({L; R}A); P)
P:P(A) = Exact(Ar ({L}A); P)
S:P(A) = Exact(Ar (A); P)
ADAMS OPERATIONS ON HIGHER K-THEORY 13
Here we regard {L; R} as an partially ordered set where L and R are incomparabl*
*e symbols,
and interpret the concatenation {L; R}A for A 2 as a concatenation of partiall*
*y ordered
sets, yielding a partially ordered set; it was called fl(A) in [5]. The definit*
*ion of k given
in (6.1) applies unchanged to each term of this fibration sequence, except that*
* now j1 =2A1
becomes a possibility, for we may have j1 = L or j1 = R. The result is the foll*
*owing map
of fibrations.
SubkG:P ----! Sub kP:P ----! SubkS:P
?? ?? ??
k y k y k y
G:G"(k)P ----! P:G"(k)P ----! S:G"(k)P
Having transferred our construction of k to the level of the G-construction, on*
*e may
use methods just like those of [5] to prove that our Adams operation agrees wit*
*h the one
defined by Quillen in [8], or those defined in [9].
One should be able to show directly, for any exact category P with suitable *
*tensor
products and exterior power operations, that the Adams operations on the K-grou*
*ps con-
structed here agree with those deduced from the lambda operations constructed i*
*n [5].
One striking feature of the construction of k is the definition of the categ*
*ory "G(k)P, in
which the multi-dimensional complexes are required only to be acyclic in the fi*
*rst direction
and the last direction. The map k, on the other hand, involves only tensor pro*
*ducts
of generalized Koszul complexes, so the multi-dimensional complexes occurring i*
*n it are
acyclic in every direction. One might imagine refining the map by changing the *
*definition
of "G(k)P accordingly. This would lead to a space which contains various deloop*
*ings of the
K-theory space for P, and thus might lead to Adams operations maps that decreas*
*e the
degree, KiP ! Ki-jP. I think that all such maps might well be zero, and so thes*
*e maps
will turn out to be simply spurious.
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