April 8, 1*
*992
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 symmetricp owers on any suitable exact category,by construc*
*ting 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 pro duct M N, sym*
*metric
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 may*
* fix a
group and a commutative ring Rand take Pto be the category P(R; ) of represent*
*ations
of on projective finitely generated R-modules.We impose certain exactness requ*
*irements
on these functors,so that in particular the tensor product is required to be bi*
*-exact,and
this prevents us from taking forP a categorysuch as the category M(R) of finit*
*ely
generated R-modules.
In a previous paper [5] I showed how to usethe exterior power operations on m*
*odules
to construct the lambda operations kon the higher K-groups as a map of spaces k*
* :
jGPj ! jGkPj in a combinatorial fashion. Here the simplicial set GP,due to Gill*
*et 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 ss0and is*
* not
divorced from the higher K-groups. TheQ-construction of Quillen and the S-const*
*ruction
of Waldhausen are the original definitions for the K-groups of P, but involve a*
* shift in
degree, so that Ki(P) = ssi+1jS:Pj = ssi+1jQPj; since the lambda operations are*
* not
additive on K0,but any function on ss1 arising from a map wouldb e ahomomorphis*
*m,
neither of these two spaces could be used to define lambda operations combinato*
*rially.
The Adams operation k is derived fromthe lambda operation k by a naturalpro *
*cedure
which makes k additive on K0. Thus there is no apparent obstruction to the pre*
*sence
of a combinatorial description for kthat involves jS:Pj or jQPj. The purpose *
*of this
paper!is!to present such a combinatorial construction of the Adams operation as*
* a map of
spaces! k: jS:Pj ! jS:G"(k)Pj. The map works by considering symmetric powers of*
* acyclic
complexes!of!length one,and by introducing a sort of symmetric product of the m*
*embers
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, 19D9*
*9.
Supported by NSF grant DMS 90-02715
2 DANIEL R. GRAYSON
derivable from the lambda operationmaps. I don't know whether further deloopin*
*g is
possible without inverting some integers, and Isuspect that this one-fold deloo*
*ping is new,
even on the level of Z BU. (One may refer to [12] for methods that can be used*
* to
transfer these results to topological K-theory.)
The construction G"(k)appearing in the target of the map is a (k 1)-dimension*
*al
cube of exact categories, each of which involves acyclic complexes of length ka*
*s well as
total complexes of multi-dimensionalcomplexes that are acyclic in two direction*
*s. 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
jS:Pj ! jS:"G(k)Pj.
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, whereaswe deloop only once in this paper. Alexander Nenashe*
*v will
write a paper in which he constructs lambda operations based on techniques in [*
*5], but
using long exact sequences instead of cub es,as suggested in [3]. For other dis*
*cussions of
lambda-operations and Adams operations on algebraic K -theory, the reader may w*
*ish to
refer to [7], [8], [9], [4], and [10].
I thank Henri Gillet for useful discussions and the idea of using the seconda*
*ry Euler
characteristic. Ithank David Benson for the definition ofthe symmetric power of*
* a complex
that I use; theone I was originally using was based on the theory of non-additi*
*vederived
functors of Dold and Puppe, [2]. Ithank Pierre Deligne and Jens Franke, who exp*
*lained 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 relation*
*shipb etween
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 o*
*f exposi-
tion below, butit will be apparent that any of the constructions we use will wo*
*rkequally
well for locally free sheaves offinite type (vector bundles) on a scheme X, or *
*for represen-
tations of a group G in finitely generated pro jective R modules. All tensor pr*
*oducts will
be over R.
If R is a commutative ring and Mis an R-module, then the k-th symmetric power*
* SkM
of M is defined to be the quotient of M k 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