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. 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