THE MCCORD MODEL FOR THE TENSOR PRODUCT OF A SPACE AND A COMMUTATIVE RING SPECTRUM NICHOLAS J. KUHN Abstract. We begin this paper by noting that, in a 1969 paper in the Transactions, M.C.McCord introduced a construction that can be interpreted as a model for the categorical tensor product of a based space and a topological abelian group. This can be adapted to Segal's very special -spaces - indeed this is roughly what Segal did - and then to a more modern situation: K R where K is a based space and R is a unital, augmented, commutative, associative S-algebra. The model comes with an easy-to-describe filtration. If one lets K = Sn, and then stabilize with respect to n, one gets a filtered model for the Topological Andr'e-Quillen Homology of R. When R = 1 1 X, one arrives at a filtered model for the connective cover of a spectrum X, constructed from its 0thspace. Another example comes by letting K be a finite complex, and R the S-dual of a finite complex Z. Dualizing again, one arrives at G. Arone's model for the Goodwillie tower of the functor sending Z to 1 MapT (K, Z). Applying cohomology with field coefficients, one gets various spec- tral sequences for deloopings with known E1-terms. A few nontrivial examples are given. In an appendix, we describe the construction for unital, commutative, associative S-algebras not necessarily augmented. 1. Introduction The point of this paper is to tell a story that begins with a 1969 paper of M.C.McCord [McC ], and ends with various disparate objects of current in- terest, e.g. Goodwillie towers, Topological Hochschild Homology, and Topo- logical Andr'e-Quillen Homology. Line by line, I think most of this story is known. However, taken as a whole, I think the older work sheds some light on the newer. Moreover, in this era of tremendous activity in homotopical algebra of various sorts, it seems important to remind ourselves that the genesis of many of the most useful ideas lies way back in the literature. Conceptually, we feature the following categorical notion. Let T be the category of based topological spaces. If C is a category enriched over T , there is the notion of the tensor product of a space K 2 T with an object ____________ Date: January 28, 2002. 2000 Mathematics Subject Classification. Primary 55P43; Secondary 18G55. This research was partially supported by a grant from the National Science F* *oundation. 1 2 KUHN X 2 C: this is an object K X 2 C satisfying Map C(K X, Y ) = Map T(K, Map C(X, Y )) for all X, Y 2 C. We give an overview of the paper. We consider various topological categories of structured objects. Let Ab be the category of abelian topological monoids. Let T be G.Segal's category of -spaces [Se]: functors X : ! T , where is the category of finite based sets. Let Alg be the category of commutative, associative, augmented S- algebras, where S is the sphere spectrum. These categories are closely related. First of all, an abelian topological monoid A defines in a natural way Ax 2 T . In Segal's terminology, Ax is an example of a `special' object, and the abelian topological groups define `very special objects'. A very special -space is roughly the same thing as an infinite loopspace, and, in this introduction, we will tempt fate and often identify these two notions. Now we note that if X is either an abelian topological monoid or an infinite loopspace, then 1 X+ is in the category Alg. With C equal to any of these three categories, McCord's construction yields a functor1 SP 1 : T x C ! C. His construction generalizes the infinite symmetric product construction studied by Dold and Thom in the 1950's: with N denoting the natural numbers, SP 1(K, N) = SP1 (K). If C is either Ab or Alg, there is an isomorphism (1.1) SP 1 (K, X) = K X. See Proposition 2.2(1), and Proposition 4.8. The latter proposition (or at least its proof) seems to be new. Equation (1.1) is almost true if C = T . S. Schwede [S], following Bousfield and Friedlander [BF ], defines a model category structure on T , having the very special -spaces as the fibrant objects, and such that ho(T ) is equivalent to the homotopy category of connective spectra. With this structure, SP 1 (K, ) preserves fibrant objects and the natural map K X ! SP1 (K, X) is a weak equivalence. McCord's interest stemmed from the following fundamental property: (1.2) for X 2 Ab, SP 1(S1, X) is a classifying space for X. Suitably interpreted, the same result is true if X is a special object of T . For R 2 Alg, SP 1(S1, R) is also of interest: SP 1 (S1, R) = S1 R equals T HH(R; S), the Topological Hochschild Homology of R with coefficients in the bimodule S. See Proposition 7.1; this is deduced from a similar result due to J. McClure, R. Schwänzl, and R. Vogt [MSV ]. ____________ 1McCord uses the notation B(A, K) where we use SP1 (K, A); I have borrowed my notation from [FF , Chapter 9]. A MODEL FOR A SPACE TENSORED WITH A RING SPECTRUM 3 The construction has two other basic properties, both discussed by Mc- Cord when C = Ab. Firstly, there are natural isomorphisms (1.3) SP 1(K ^ L, X) = SP1 (K, SP1 (L, X)). From this and (1.2) one quickly deduces that SP 1(Sn, X) is an n-fold de- looping of X for X 2 Ab , or for very special X 2 T . For R 2 Alg, SP1 (Sn, R) can be interpreted as `higher' Topological Hochshild Homology of R. Secondly, SP 1(K, X) comes with a nice increasing filtration. When C = Ab, one easily sees that there is an equivalence (1.4) FdSP 1 (K, X)=Fd-1SP 1 (K, X) ' K(d)^ d X^d, and little variants of this hold when C = T or Alg. Here K(d)denotes the d-space obtained from the d-fold smash product K^d by collapsing the the fat diagonal to a point.2 This much of the story will be fleshed out in sections 2, 3, and 4, with the statement about T HH appearing in x7. In [B ], it is observed that Alg is Quillen equivalent to the category Alg0, of commutative, nonunital S-algebras. In x5, we discuss the corresponding filtered SP1 construction, which again agrees with the tensor product. This reduced construction is `smaller' than what is done in Alg, and the category E of finite sets and epimorphisms replaces . The quotients of the filtration, and the use of E, may look vaguely familiar to readers of [Ar ], and this is what we explain in x6. There is a contravariant functor from T to Alg0 sending a based space Z to the ring spectrum R = D(Z), where D denotes the S-dual, and the multiplication on R is induced by the diagonal on Z. We note that there is a natural map in Alg0 K D(Z) ! D(MapT (K, Z)) and reinterprete the main convergence theorem of [Ar ] as saying that the adjoint of this, 1 MapT (K, Z) ! D(K D(Z)), is an equivalence if both Z and K are finite dimensional complexes, and the dimension of K is less than the connectivity of Z. See Theorem 6.6. Since K D(Z) has a nice increasing filtration, the S-dual is a tower of fibrations. This tower is visibly equivalent to the tower found by Arone, and is thus the Goodwillie tower associated to the functor sending a space X to the spectrum 1 MapT (K, Z). In x7, we discuss the following construction. Given R 2 Alg, nice prop- erties of SP 1(K, R) as a functor of K allow us to define a filtered spectrum T AQ(R) by T AQ(R) = hocolimn!1 nSP 1 (Sn, R). ____________ 2This notation, which the author likes, comes from [Ar]. 4 KUHN This is one of various equivalent definitions of the Topological Andr'e-Quillen spectrum of R. When R = 1 X+ , with X an infinite loopspace, T AQ(R) is the connective delooping of X. When R = D(Z+ ), the filtered spectrum T AQ(R) is related to constructions studied by the author in [K3 ]. Using (1.4), one can identify the quotients of the filtration of T AQ(R): (1.5) FdT AQ(R)=Fd-1T AQ(R) ' Kd ^h d (R=S)^d, where Kd is the dth partition complex which arose in the work of Arone and Mahowald on the Goodwillie tower of the identity [AM ]. In x8 we note that applying ordinary cohomology with field coefficients F to the filtered spectra SP 1(Sn, R) and T AQ(R) yield spectral sequences converging to H*(Sn R; F) and H*(T AQ(R); F), and having E1 terms isomorphic to known functors of H*(R; F): see Theorem 8.1. These spectral sequences appear to be unexplored, even in the case when R = 1 X+ , with X an infinite loop space, so that, e.g., the T AQ(R) spectral sequence is calculating the cohomology of a connective spectrum from knowledge of the cohomology of its 0th-space. As examples, we use results of ours from [K3 ] to explain how the spectral sequence works, when F has characteristic 2, in the cases R = 1 (Z=2+ ), 1 (S1+), and, most interestingly, D(S1+). In the Appendix, we note how our models need to be slightly tweaked when one considers commutative unital S-algebras not necessarily aug- mented. Very influential to me in my understanding of the older work surveyed in this paper was Chapter of 9 of the unpublished book The actions of the classical small categories of topology by Bill and Ed Floyd [FF ]. Writing this book was Ed's project in the late 1980's, after returning to ordinary academic life after finishing a term as provost of the University of Virginia. I have also benefited from conversations with Mike Mandell, Bill Dwyer, and Greg Arone. Versions of this work were presented at talks at the Johns Hopkins topol- ogy conference in the spring of 2000, and at the Union College topology and category theory conference of fall 2001. 2. McCord's construction Let K be a based space with basepoint *, and let A be an abelian topo- logical monoid. Imagine using (1.1) to guide the construction SP 1 (K, A). As a first experiment, suppose A = N. In this case (1.1) tells us that, for all A 2 Ab, we should have Map Ab(SP 1 (K, N), A) = Map T(K, Map Ab(N, A)). But, since N is the free abelian (topological) monoid on one generator, we can identify Map Ab (N, A) with A. Thus we are asking that SP 1 (K, N) satisfy Map Ab(SP 1 (K, N), A) = Map T(K, A). A MODEL FOR A SPACE TENSORED WITH A RING SPECTRUM 5 In other words, SP 1(K, N) should be the free topological abelian monoid generated by K, a.k.a. SP 1 (K). Note that elements in SP 1 (K) are words of the form kn11. .k.ndd, with ki2 K and ni2 N. This suggests a definition. Definition 2.1. Let SP 1 (K, A) be the abelian topological monoid with generators ka with k 2 K and a 2 A subject to the relations: (i) *a = * for all a 2 A, (ii) k0 = * for all k 2 K, (iii) ka1ka2 = ka1+a2 for all a1, a2 2 A. Viewing ka as an element`in K xA, SP1 (K, A) is topologized as the evident quotient space of 1d=0(K x A)d. Note that the abelian topological monoid satisfying only the relations of type (i) and (ii) is SP1 (K ^ A). If B is another abelian topological monoid, a monoid map SP 1(K ^ A) ! B corresponds to a map of topologial spaces OE : K ^ A ! B which itself corresponds to a map K ! MapT (A, B). This latter map takes values in Map Ab(A, B) exactly when OE(k, a1 + a2) = OE(k, a1)OE(k, a2). Thus we see that the quotient of SP 1(K ^ A) having type (iii) relations imposed satisfies the universal property of K A. (Compare with [FF , p.164].) We have checked the first part of the next proposition. Proposition 2.2. There are the following natural identifications in Ab . (1) SP 1(K, A) = K A. (2) SP 1(S0, A) = A. (3) SP 1(K ^ L, A) = SP1 (K, SP1 (L, A)). (4) SP 1(K _ L, A) = SP1 (K, A) x SP1 (L, A). The last three parts of this proposition follow formally from statement (1). For example, (3) follows by manipulating adjunctions: Map Ab(SP 1 (K ^ L, A), B)= Map T(K ^ L, Map Ab(A, B)) = Map T(K, Map T(L, Map Ab(A, B))) = Map T(K, Map Ab(SP 1 (L, A), B)) = Map Ab(SP 1 (K, SP1 (L, A)), B). For statement (4), one needs to also note that the coproduct in Ab is the product. (In the next section, we will see that there are also reasonable direct proofs of (3) and (4).) 6 KUHN In the introduction to [McC ], McCord comments that SP 1( , A) äh s a tendancy to convert cofibrations . . . to quasifibrations", and proves this in various cases [McC , Thm.8.8]. Note that statement (4) of the last proposi- tion nicely illustrates his statement. In particular, when applied to the cofibration S0 ,! I ! S1, his ob- servations suggest that SP 1(I, A) ! SP 1(S1, A) is a quasifibration with homotopy fiber A = SP1 (S0, A). One has Proposition 2.3. If A has a nondegenerate base point, then SP 1(S1, A) is a classifying space for A. Combined with statement (3) of the previous proposition, this implies Corollary 2.4. In this case, SP 1(Sn, A) is an n-fold classifying space of A. With such a mild point set hypothesis, this proposition and corollary occur as [FF , Cor.9.16]. We end this section by noting that SP 1(K, A) is filtered by letting the dth`filtration, which we denote FdSP 1 (K, A), be the evident quotient of d r ^d r=0(K x A) . Let i : (K, d) ,! K denote the inclusion of the fat diagonal into the d-fold smash product. Under reasonable conditions, e.g. if K is a based C.W. complex, the inclusion i will be a d-equivariant cofibration, and we let K(d)= K^d= (K, d). Assuming the basepoint in A is also nondegenerate, the inclusion Fd-1SP 1 (K, A) ,! FdSP 1 (K, A) will be a cofibration, and there is a homeomorphism (2.1) FdSP 1 (K, A)=Fd-1SP 1 (K, A) ' K(d)^ d A^d. Statements similar to this appear in [McC , x6]. Remark 2.5. One should note that specializing the filtration on SP 1(K, A) to SP1 (K, N) = SP1 (K) does not yield the standard filtration on SP1 (K). For example, in this paper an element k2 2 SP 1(K) would be in filtration 1. 3. -spaces and Segal's theorem Our first goal in this section is to rewrite our construction SP 1(K, A) in a way allowing for generalization. We begin by defining more precisely the category . Definition 3.1. Let be the category with objects the based finite sets 0 = ;+ and n = {1, 2, . .,.n}+ , n 1, and with all based functions as morphisms. Note that 0 is both an initial and terminal object. We remark that, unfortunately, it was the opposite of this category that was called in [Se], and the literature is strewn with inconsistent notation. A -space is then defined to be a covariant functor X : ! T that is `based' in the sense that it sends 0 to the one point space. These are A MODEL FOR A SPACE TENSORED WITH A RING SPECTRUM 7 the objects of a category T having the natural transformations as mor- phisms. This is a category enriched over T : the set of morphisms between two -spaces, Map (X, X0), has a natural topology. Similarly, a based con- travariant functor Y : op ! T willobepcalled a op-space, and these are objects in a topological category T . Example 3.2. If A is an abelian topological monoid, there is an associated -space Ax defined as follows. Firstly, Ax (n) = An. Then, givenQff : n ! m, the ith component of ff* : An ! Am sends (a1, . .,.an) to aj, with the product running over j such that ff(j) = i. This product is interpreted to be the unit of A if there are no such j. Example 3.3. Since, for any K 2 T , Kn = MapT (n, K), there is the evident op-space Kx with Kx (n) = Kn. Note that theoconstructionspin these last examples embed Ab into T , and T into T , as a full subcategories. We now recall the coend construction. If X is a -space and Y is a op-space, we let Y ^ X 2 T denote the quotient space ` Y (n) ^ X(n)=(~), n where ff*(y) ^ x ~ y ^ ff*(x) generates the equivalence relation. It is useful to observe that, because X(0) = * = Y (0), Y ^ X 2 T = Y x X, where Y x X is the quotient space a Y (n) x X(n)=(~), n where ff*(y) ^ x ~ y ^ ff*(x) generates the equivalence relation. By inspection, one observes Lemma 3.4. SP 1 (K, A) = Kx ^ Ax . This suggests a generalization of our construction. Definition 3.5. Given K 2 T and X 2 T , let SP 1(K, X) = Kx ^ X. A couple of remarks are now in order. Firstly, a Yoneda's lemma type argument shows that SP 1 (n, X) = X(n). Thus, as a functor of K, SP 1(K, X) extends X to T ; more precisely, this is the left Kan extension [MacL , Chap.X]. Secondly, we can extend our construction to SP 1 : T x T ! T by first letting Xn(m) = X(nm ), and then by defining SP 1 (K, X)(n) = SP1 (K, Xn). 8 KUHN It is natural to wonder if SP1 (K, X) can be then be interpreted as a tensor product in T . Alas, this is not the case: the simple minded construction K ^ X defined by (K ^ X)(n) = K ^ X(n) is easily seen to play this role3. Since SP1 (K, X) is not the tensor product in T , formal arguments used to prove the last three statements in Proposition 2.2 don't apply. However, we still can prove suitable versions of these. First of all, the two remarks above (along with the observation that 1 = S0) combine to show that SP 1(S0, X) = X. Less obvious are the other two. Statement (3) of Proposition 2.2 is un- changed in our greater generality. Proposition 3.6. SP 1 (K ^ L, X) = SP1 (K, SP1 (L, X)). To generalize statement (4), let a : x ! be the functor sending (m, n) to m + n . Pulling back by a defines a* : T ! T x . Proposition 3.7. SP 1 (K _ L, X) = (Kx x Lx ) x x a*(X). Proposition 2.2(4) follows from this, once one observes that a*(Ax ) = Ax x Ax , so that SP 1(K _ L, Ax )= (Kx x Lx ) x x (Ax x Ax ) = (Kx x Ax ) x (Lx x Ax ) = SP1 (K, Ax ) x SP1 (L, Ax ). Let m :op xop ! obepthe functorosendingp(m,on)ptoomnp 4. We let a* : T x ! T and m* : T x ! T respectively denote the left adjoints to pulling back by a and m. We have two fundamental lemmas. Lemma 3.8. m*(Kx x Lx ) = (K ^ L)x . Lemma 3.9. a*(Kx x Lx ) = (K _ L)x . Assuming these for the moment, Proposition 3.6 and Proposition 3.7 fol- low. For example, using Lemma 3.8, we have identifications SP 1 (K, SP1 (L, X))= Kx x (Lx x X*) = (Kx x Lx ) x x m*(X) = m*(Kx x Lx ) x X = (K ^ L)x x X = SP1 (K ^ L, X), and Proposition 3.6 follows. The proof of Proposition 3.7 is similar. ____________ 3One can then formally deduce that, given A, B 2 Ab, the natural map K ^ Ax ! SP1 (K, Ax) induces a homeomorphism Map (SP1 (K, Ax), Bx ) = Map (K ^ Ax, Bx ). 4More precisely, m is the smash product followed by the lexicographic identi* *fication of m ^ n with mn . A MODEL FOR A SPACE TENSORED WITH A RING SPECTRUM 9 Proof of Lemma 3.8. Given a opx op-space Y , m*(Y ) is explicitly given by m*(Y )(c) = colimY c#m where c # m is the category with objects all triples (a, b, fl) with fl : c ! ab, and morphisms given by pairs (ff : a ! a0, fi : b ! b0) making an appropriate diagram commute. One such triple is (c, c, ), where : c ! cc is the diagonal, and there is a canonical morphism from this triple to any other triple (a, b, fl) given by the two components of fl : c ! ab . It follows that the canonical map Y (c, c) ! colimc#mY is a quotient map. Now we specialize to Y = Kx xLx . Both colimc#mKx xLx and (K ^L)c are quotients of Kc x Lc, thus we just need to verify that each maps to the other, as quotient spaces of Kc x Lc. To construct a map from the former to the latter, we observe that that the smash product construction, MapT (A, K) x MapT (B, L) ! MapT (A ^ B, K ^ L), specializes to give natural maps ^ : Ka ^ Lb ! (K ^ L)ab. Thus, associated to a triple (a, b, fl), there is a canonical map * Ka ^ Lb ^-!(K ^ L)ab fl-!(K ^ L)c, and these induce the needed map colimc#mKx x Lx ! (K ^ L)c. To construct a map in the other direction, we observe that (K ^ L)c is the quotient of Kc x Lc obtained by collapsing to a point the subspace {(k1, . .,.kc, l1, . .,.lc) | for all i, either ki= * or li=}*. One checks easily that this subspace is precisely the union of the images of maps ff* x fi* : Ka x Lb ! Kc x Lc such that the composite c -! cc -fffi-!abis the constant map 0. Thus the subspace maps to the basepoint in colimc#m Kx x Lx , i.e. the projection Kc x Lc ! colimc#mKx x Lx factors through (K ^ L)c. Sketch proof of Lemma 3.9.This follows easily from the observation that there are canonical decompositions ` (K _ L)c = Ka _ Lb, with the wedge running over bijections fl : c ! (a + b) which are order preserving when restricted to fl-1 ((a + b) - a) and fl-1 ((a + b) - b). 10 KUHN Remarks 3.10. The two lemmas include the statements that wedge and smash are the left Kan extensions to T xT of the composites x a-! ,! T and x m-! ,! T . We suspect that Lemma 3.9 has been observed by others. Lemma 3.8 seems less familiar. (Compare our proof of Proposition 3.6 to the proof of [Se, Lemma 3.7].) We note that T x T ^-!T x -m*-!T is the smash product of [L ]. The category of -spaces admits products in the obvious way: if X and Y are -spaces, one lets (X x Y )(n) = X(n) x Y (n). We have Proposition 3.11. SP 1 (K, X x Y ) = SP1 (K, X) x SP1 (K, Y ). To prove this, we first note that X xY = *(X xY ), where * : T x ! T is induced by the diagonal : ! x . Thus we have identifications SP 1 (K, X x Y )= Kx x *(X x Y ) = *(Kx ) x x (X x Y ) = (Kx x Kx ) x x (X x Y ) = (Kx x X) x (Kx x Y ) = SP1 (K, X) x SP1 (K, Y ), where we have used the next lemma. Lemma 3.12. *(Kx ) = Kx x Kx . Proof.This can be proved in various ways. Perhaps the slickest proof is to first note that is right adjoint to a. That * = a* formally follows. Finally, it is evident that a*(Kx ) = Kx x Kx . It remains, in this section, to discuss how the SP 1 construction interacts with the homotopy theory of -spaces. Define ßs*(X) = colimn!1 ß*+n(SP 1 (Sn, X)). The colimit here arises from maps S1 ^ SP1 (Sn-1 , X) ! SP1 (Sn, X) which themselves are special cases (K = S1 and Y = SP1 (Sn-1 , X)) of the natural transformation K ^ Y ! SP1 (K, Y ). If we define weak equivalences to be maps f : X ! Y with ßs*(f) an iso- morphism, then Bousfield and Friedlander [BF ], following Segal [Se], showed that that the localized category T [weq-1 ] is equivalent to the homotopy category of connective spectra. Even more, this equivalence is induced by a Quillen equivalence between appropriate model categories. Schwede [S] modifies the cofibration and fibrations slightly. All these authors work with simplicial sets rather than topological spaces, but [S, Appendix B] allows for some translation into our setting. The upshot is roughly the following. Cofibrations are maps f : X ! Y where Y is obtained from X by successively attaching appropriate sorts of A MODEL FOR A SPACE TENSORED WITH A RING SPECTRUM 11 free -spaces. Fibrant objects agree with Segal's notion of a very special -space, where X is very special means that each map (3.1) X(a + b) ! X(a) x X(b) is a weak equivalence of spaces, and also (3.2) the monoid ß0(X(1)) is a group. Proposition 3.13. If K is a C.W. complex, then SP 1 (K, ) preserves cofibrations and acyclic cofibrations. Proposition 3.14. If K is a C.W. complex, and X is cofibrant, then the natural map K ^ X ! SP1 (K, X) is a weak equivalence. Proposition 3.15. If K is a C.W. complex, and X is cofibrant and very special, then SP 1(K, X) is again very special. Theorem 3.16. If X is cofibrant and very special, then SP 1( , X) takes cofibration sequences of C.W. complexes to a homotopy fibration sequence. In particular, there are weak equivalences of spaces X(1) ~-! SP 1 (S1, X)(1) ~-! 2SP 1 (S2, X)(1) ~-!. ... We briefly indicate why the propositions hold. Firstly, under the cofi- brancy hypotheses, SP 1(K, X) will be nicely filtered, and satisfy (3.3) FdSP 1 (K, X)=Fd-1SP 1 (K, X) = K(d)^ d (X(d)=Xsing(d)) where Xsing(d) denotes the union of all the images of maps X(c) ! X(d) with c < d. It follows then that then K ^ X(1) ! SP1 (K, X) is a weak equivalence through a stable range, and the first two of the propositions easily can be deduced. For the next proposition, we note that, if (3.1) holds, then Xa+b ! Xa x Xb is a strict equivalence of -spaces, where a map is a strict equivalence if evaluating on any n yields a weak equivalence of spaces. Then we have equivalences SP 1 (K, X)(a + b)= SP1 (K, Xa+b) -~!SP 1 (K, X a x Xb) = SP1 (K, Xa) x SP1 (K, Xb) = SP1 (K, X)(a) x SP1 (K, X)(b), showing that SP 1(K, X) again satisfies (3.1). For the theorem, see [Se, Prop.3.2] and [BF , Lemma 4.3]. It follows that if X is cofibrant and very special, then X(1) is canonically weakly equivalent 12 KUHN to an infinite loop space. Furthermore, for any C.W. complex K, there are weak equivalences (3.4) FdSP 1 (K, X)=Fd-1SP 1 (K, X) ' K(d)^ d X(1)^d. 4. Commutative ring spectra We now show that the results of the previous sections extend nicely to the world of structured ring spectra. We work within the category S, the category of S-modules studied in [EKMM ]. Given K 2 T and X 2 S, 1 K, K ^ X, and Map (K, X) will denote the usual S-modules5. Let Alg be the category of unital, commutative, associative, augmented S-algebras. Thus an object in Alg is an S-module R, together with multi- plication ~ : R ^ R ! R, unit j : S ! R and counit ffl : R ! S satisfying the usual identities. Morphisms preserve all structure. This category is enriched over T : given R, Q 2 Alg, the morphism space MapAlg (R, Q) is based with basepoint R -ffl!S -''!Q. We also note that the coproduct in Alg of R and Q is R ^ Q. As observed in [B , x1], results in [EKMM ] show that Alg has a topological model category structure in which weak equivalences are morphisms that are weak equivalences as maps of S-modules6. We have two important sources of examples. Example 4.1. If A 2 Ab , then 1 A+ 2 Alg. More generally, if X is an E1 -space (e.g. an infinite loop space), then 1 X+ is naturally an object in Alg. (See [M , Ex.IV.1.10] and [EKMM , xII.4].) Example 4.2. Given a based space Z, let D(Z+ ) denote Map (Z+ , S). This is an object in Alg : the unit and the counit are respectively induced by Z+ ! S0 and S0 ! Z+ , and the diagonal : Z ! Z x Z induces the multiplication * D(Z+ ) ^ D(Z+ ) ! D(Z+ ^ Z+ ) --! D(Z+ ). Given R 2 Alg, we let R^ : ! S denote the functor with R^(n) = R^n analogous to Example 3.2. Definition 4.3. Given K 2 T and R 2 Alg, let SP 1(K, R) = Kx ^ R^. We will momentarily see that SP 1(K, R) is again an object in Alg. Proofs from x3 extend immediately to prove the next proposition. Proposition 4.4. There are the following natural identifications. (1) SP 1(S0, R) = R. (2) SP 1(K _ L, R) = SP1 (K, R) ^ SP1 (L, R). ____________ 5What we are calling Map(K, X) is FS( 1 K, X) in [EKMM ]. 6In Basterra's notation, Algis denoted CS=S. A MODEL FOR A SPACE TENSORED WITH A RING SPECTRUM 13 (3) SP 1(K, R ^ Q) = SP1 (K, R) ^ SP1 (K, Q). A consequence of this proposition is that SP 1(K, R) takes values in Alg, with multiplication given by the composite SP1 (K,~) 1 SP1 (K, R) ^ SP1 (K, R) = SP1 (K, R ^ R) ------! SP (K, R). We note that this multiplication agrees with the composite SP1 (r,R) 1 SP1 (K, R) ^ SP1 (K, R) = SP1 (K _ K, R) -------! SP (K, R), where r : K _ K ! K is the fold map. With this structure, all the identifications in the last proposition are as objects in Alg, and we also have the next proposition, whose proof follows from the arguments of the last section. Proposition 4.5. SP 1 (K ^ L, R) = SP1 (K, SP1 (L, R)). Now we check that SP 1(K, R) is the categorical tensor product in Alg. The following lemmas are easily verified, where we use the following nota- tion: with C either T or S, and X and Y functors from to C, Map C(X, Y ) denotes the space of natural transformations from X to Y . Lemma 4.6. For all K, L 2 T , Map T (Kx , Lx ) = MapT (K, L). Lemma 4.7. For all R, Q 2 Alg, Map S(R^, Q^) = MapAlg (R, Q). Proposition 4.8. For all K 2 T and R 2 Alg , SP 1 (K, R) is naturally isomorphic to K R. Proof.We check that SP1 (K, R) satisfies the universal property of the ten- sor. Given K 2 T , and R, Q 2 Alg, we have MapAlg (SP 1 (K, R), Q)= Map S(SP 1 (K, R)^, Q^) = Map S(SP 1 (K, R^), Q^) = Map S(Kx ^ m*(R^), Q^) = Map T(Kx , Map S(m*(R^), Q^)) = Map T(Kx , MapAlg(R^, Q)) = Map T(Kx , MapAlg(R, Q)x ) = Map T(K, MapAlg(R, Q)). Here m : x ! is multiplication as in the last section. As before, SP 1(K, R) is naturally filtered. Let R=S denote the cofiber of j : S ! R. If K is a C.W. complex, and j is a cofibration, then the inclusion Fd-1SP 1 (K, R) ,! FdSP 1 (K, R) will be a cofibration, and there is an isomorphism of S-modules (4.1) FdSP 1 (K, R)=Fd-1SP 1 (K, R) ' K(d)^ d (R=S)^d. 14 KUHN 5.The reduced model It is sometimes useful to replace Alg by a slightly different category. Let Alg0be the category of nonunital, commutative, associative S-algebras (the category denoted NS in [B ]). Basterra observes that the functor S_ : Alg0! Alg, that wedges a unit S onto a nonunital algebra, has as right adjoint the augmentation ideal functor J : Alg ! Alg 0, defined by letting J(R) be the fiber of R -ffl!S. She then notes that, with a natural topological model category on Alg0, these adjoint functors form a Quillen pair, and thus induce adjoint equivalences on the associated homotopy categories. Example 5.1. If Z is a based space, J(D(Z+ )) = D(Z). The multiplication on D(Z) is induced by the reduced diagonal : Z ! Z ^ Z. Our SP 1(K, ) construction has a `reduced' analogue in Alg0. Definition 5.2. Let E be the category with objects n, for n 1, and with morphisms from n to m equal to all epimorphisms from {1, . .,.n} to {1, . .,.m}. As observed in [Ar ] (see also [AK ]), a based space K defines a functor K^ : Eop ! T with K^ (n) = K^n . Also, J 2 Alg0 defines J^ : E ! S in the obvious way. Definition 5.3. Given K 2 T and J 2 Alg0, let SP 1(K, J) = K^ ^E J^ . The analogues of all the properties of SP1 (K, R) proved in the last section hold in our setting, with virtually identical proofs. In particular, SP1 (K, J) is again an object in Alg0, and it agrees with the categorical tensor product K J. From the above comments, one can formally deduce the following isomor- phism in Alg. Proposition 5.4. SP 1 (K, J) _ S = SP1 (K, J _ S). Though we won't show this here, this proposition can also be given a direct proof, and there are analogues in other contexts. Readers may wish to compare this result with observations in [P ]. As usual, SP 1 (K, J) is filtered: if Ed denote the full subcategory of E with objects n for n d, then we let FdSP 1 (K, J) = K^ ^Ed J^ . If K is a C.W. complex, then the inclusion Fd-1SP 1 (K, J) ,! FdSP 1 (K, J) will be a cofibration, and there is an isomorphism of S-modules (5.1) FdSP 1 (K, J)=Fd-1SP 1 (K, J) ' K(d)^ d J^d. We note that the isomorphism of the last proposition is filtration preserv- ing. 6. Reinterpretation of Arone's tower for 1 MapT (K, X) In this section, we let K be a finite C.W. complex. In [Ar ], G. Arone described a model for the Goodwillie tower of the functor sending a based space Z to the S-module 1 MapT (K, Z). Here we show A MODEL FOR A SPACE TENSORED WITH A RING SPECTRUM 15 that, if Z is a finite complex, his tower arises as the S-dual of the filtered object K D(Z) of the last section. We recall Arone's construction and some of its properties [Ar ]. For more detail, see also [AK ]. Definitions 6.1. Let Z be a based space. (i) Let P K(Z) = Map ES(K^ , Z^ ), the spectrum of natural transformations from 1 K^ to 1 Z^ . (ii) Let PdK(Z) = Map EdS(K^ , Z^ ). (iii) Let (K, Z) : 1 MapT (K, Z) ! P K(Z) be the natural transformation that sends f : K ! Z to the natural transformation with nth component equal to 1 f^n : 1 K^n ! 1 Z^n . The spectrum P K(Z) is the inverse limit of the tower of fibrations . .!.PdK+1(Z) ! PdK(Z) ! PdK-1(Z) ! . .,. and the dth fiber is isomorphic to Map Sd(K(d), Z^d). Because K(d) is finite, and the d action on this is free away from the basepoint, this fiber is naturally homotopy equivalent to the homotopy orbit spectrum (D(K(d)) ^ Z^d)h d. From this last description one sees that the tower has the form of a Good- willie tower, and also that the connectivity of the fibers goes up if the con- nectivity of Z is greater than the dimension of K. Arone then proves that this is the Goodwillie tower of 1 MapT (K, Z) by proving Theorem 6.2. [Ar ] If the connectivity of Z is greater than the dimension of K, then (K, Z) is a homotopy equivalence. Now we connect these constructions to K D(Z). Definitions 6.3. Let Z be a based space. (i) Let ~ (K, Z) : K D(Z) ! D(MapT (K, Z)) be the map in Alg0 adjoint to the composite K eval--!MapT(MapT (K, Z), Z) D-!Map Alg0(D(Z), D(MapT (K, Z)). (ii) Let (K, Z) : 1 MapT (K, Z) ! D(K D(Z)) be the S-module map adjoint to ~ (K, Z). (iii) Let ff(K, Z) : P K(Z) ! D(K D(Z)) be the map of cofiltered S- modules defined as follows. Let i : Z^ ! D(D(Z)^) be the natural trans- formation adjoint to D(Z)^ ! D(Z^ ). Now let ff(K, Z) be the composite induced by i: Map ES(K^ , Z^ ) i-!Map ES(K^ , D(D(Z)^)) = MapS (K^ ^E D(Z)^, S). A check of the definitions verifies the next lemma. 16 KUHN Lemma 6.4. There is a commutative diagram 1 MapT (K,RZ) (K,Z)ooooo RRRR(K,Z)RR ooo RRRR wwoooo ff(K,Z) RR(( P K(Z) __________________________//D(K D(Z)) Lemma 6.5. ff(K, Z) is a homotopy equivalence if Z is a finite complex. Proof.If Z is finite, then i : Z^n ! D(D(Z)^n) is a an equivalence for all n. Now the lemma follows by observing that K^ is a cofibrant Eop-space, or more simply, note that the fibers of the towers will be equivalences, as K(d)is a free d-complex for all d. (This has been noted before; see e.g. [AK , McC ].) Summarizing, we conclude Theorem 6.6. If both K and Z are finite complexes, and the dimension of K is less than the connectivity of Z, then (K, Z) : 1 MapT (K, Z) ! D(K D(Z)) is a weak equivalence, and thus the algebra map ~(K, Z) : K D(Z) ! D(MapT (K, Z)) can be identified as the map from a spectrum to its double dual. We end this section by noting how the homological version of this discus- sion would go. Let F be a field, HF the associated commutative S-algebra, and Alg F the category of commutative, nonunital HF algebras. Let K F J 2 AlgF denote the tensor product of a based space K and an J 2 AlgF. As before, one learns that K F J = Kx ^E J^ , where smash products are taken over HF. Now let DF(Z) = Map (Z, HF), the HF-module whose homotopy groups are the cohomology groups of Z with F-coefficients. In this case, the natural map i : HF ^ Z^n ! DF(DF(Z)^n) is an equivalence for any space Z with H*(Z; F) of finite type. Reasoning as before, from Arone's theorem one deduces Theorem 6.7. If K is a finite complex of dimension less than the connec- tivity of Z, and H*(Z; F) is of finite type, then the natural map in AlgF, : K F DF(Z) ! DF(MapT (K, Z)), is an equivalence. Remark 6.8. It seems likely that this theorem can be deduced from older convergence results for the Anderson spectral sequence [An ], and then one can run our arguments backwords, and deduce Arone's theorem. The novelty would then be to identify the filtration as Arone did. A MODEL FOR A SPACE TENSORED WITH A RING SPECTRUM 17 7.Topological Hochschild homology and Topological Andre-Quillen homology Let T HH(R; M) denote the Topological Hochschild homology spectrum associated to a S-algebra R and an R-bimodule M (see e.g. [EKMM , Chap.9]). If R is commutative and augmented, then ffl : R ! S makes S into an R-bimodule. We have Proposition 7.1. S1 R = T HH(R; S). Proof.This is a variant of a theorem of J. McClure, R. Schwanzl, and R. Vogt [MSV ]. They show that if R is a commutative S-algebra, then T HH(R; R) is the tensor product of R with S1 with the tensor product in the category of commutative S-algebras. In the appendix, we note that if R is also augmented, then this would agree with S1+ R 2 Alg. Thus T HH(R; R) = S1+ R. Applying R to the pushout square in T S0 ______//* | | | | fflffl| fflffl| S1+_____//S1, yields a pushout square in Alg R ___________//S | | | | fflffl| fflffl| S1+ R _____//S1 R, and we conclude that S1 R = (S1+ R) ^R S = T HH(R, R) ^R S = T HH(R; S). Given K 2 T and R 2 Alg, there is a natural map K ^ R ! K R, and thus K ^ (L R) ! K (L R) = (K ^ L) R. This map is easily seen to be filtration preserving. Specializing to the case when K = S1, and L = Sn yield filtration pre- serving maps (Sn R) ! Sn+1 R, or, equivalently, Sn R ! (Sn+1 R). Definition 7.2. Let T AQ(R) = hocolimn!1 nSn R. 18 KUHN M. Mandell has shown the author that this definition agrees with other definitions of Topological Andr'e Quillen Homology in the literature, e.g. [B ]. In particular T AQ(R) is homotopy equivalent to the cofiber of J(R) ^ J(R) ! J(R)7. As the next example makes clear, T AQ(R) can be viewed as an `infinite delooping' of R. Example 7.3. If X is a connective S-module, T AQ( 1 ( 1 X)+ ) ' X. To see this, just recall that Sn yields the (n - 1)-connected n-fold delooping of an infinite loopspace. Note that T AQ(R) is filtered with FdT AQ(R)=Fd+1T AQ(R) ' hocolimn!1 -n Sn(d)^ d (R=S)^d. As in [AM ], let Kd be the unreduced suspension of the classifying space of the poset of nontrivial partitions of a set with d elements. Lemma 7.4. [AD ] There is a d-equivariant map hocolimn!1 -n Sn(d)! Kd that is a nonequivariant equivalence. The original short proof of this, due to Arone and Mahowald, appears in [K3 , Appendix]. Corollary 7.5. There is a homotopy equivalence FdT AQ(R)=Fd+1T AQ(R) ' ( Kd ^ (R=S)^d)h d. 8. Spectral sequences and examples Applying homology or cohomology with F-coefficients to our filtered mod- els for Sn R and T AQ(R) yields highly structured convergent spectral sequences with E1 terms equal to known functors of H*(R; F). To see why this is true, we note that there is an explicit equivariant duality map [AK ] F (Rn, d)+ ^ Sn(d)! Snd, where F (Rn, d) is the usual configuration space of d distinct points in Rn. Thus the homology calculations of [CLM ] apply. To be more precise, let {E*,*r(Sn R; F)} and {E*,*r(T AQ(R); F)} re- spectively denote the spectral sequences for computing H*(Sn RF) and H*(T AQ(R); F). Let H~*(R; F) denote the reduced cohomology of R, i.e. H*(J(R); F). Theorem 8.1. For R 2 Alg with H*(R; F) of finite type, there are natural isomorphisms as follows. ____________ 7Strictly speaking, R should be replaced by by a fibrant object in Algand th* *en J(R) replaced by a cofibrant object in Alg0. A MODEL FOR A SPACE TENSORED WITH A RING SPECTRUM 19 (1) If F has characteristic 0, then E*,*1(Sn R; F) = S*( 1-nL( -1H~*(R; F))), and E*,*1(T AQ(R); F) = L( -1H~*(R; F))). (2) If F has characteristic p, then E*,*1(Sn R; F) = S*(Rn( 1-nLr( -1H~*(R; F)))), and E*,*1(T AQ(R); F) = R( Lr( -1H~*(R; F)))). In this theorem, dV denotes the d-fold shift of a graded vector space V , L is the free Lie algebra functor, Lr is the free restricted Lie algebra functo* *r, S* is the free commutative algebra functor8, and R and Rn are appropriate free Dyer-Lashof operation functors. The author plans to write more about this elsewhere. We end with three examples. All are nontrivial, and most of what I say follows immediately from work done in [K3 ]. More detail about the last example will appear in [K4 ]. Example 8.2. If R = 1 S1+, then T AQ(R) ' HZ. At least when local- ized at 2, the filtration of T AQ(R) will correspond to the symmetric product of spheres filtration of HZ9. In particular, the dth associated graded spec- trum is contractible unless d is a power of 2, k k+1 F2k = SP 2(S), and F2k=F2k-1= L(k). Here L(k) is as in [MP ]. The boundary maps of the filtration yields the complex of spectra . .!. L(2) ! L(1) ! L(0) ! HZ occurring in the Whitehead conjecture [K1 ]. The sequence is exact in ho- motopy, but zero in mod 2 homology: indeed, the spectral sequence for computing H*( HZ; F2) = A=ASq1 collapses at E1. Example 8.3. If R = 1 Z=2+ , then T AQ(R) ' HZ=2. The dthassociated graded spectrum is contractible unless d is a power of 2, k 0 k F2k = SP 2(S ), and F2k=F2k-1= M(k). Here we recall [MP ] that SP 2k(S0) is defined to be the cofiber of the `diag- onal' : SP 2k-1(S0) ! SP 2k(S0), and M(k) = L(k) _ L(K - 1). As in the previous example, the boundary maps of the filtration yields the complex of spectra . .!.M(2) ! M(1) ! M(0) ! HZ=2 ____________ 8One has sign conventions of the usual sort. 9This presumably holds integrally: [AD , Thm.1.14] says that the filtration * *quotients are correct. 20 KUHN occurring in the mod 2 Whitehead conjecture [K2 ]. The sequence is exact in homotopy, but the spectral sequence for computing H*(HZ=2; F2) = A collapses at E1. Example 8.4. If R = D(S1+), then T AQ(R) ' -1HQ. There are various ways to see this; in [K4 ], we will prove that S2 R ' HQ _ S0. Localized at 2, the dth associated graded spectrum is contractible unless d is a power of 2, and k 0 F2k=F2k-1= -1SP 2(S ). Thus H*(F2k=F2k-1; F2) = -1A=Lk+1, where Lk is the span of all admis- sible sequences in the Steenrod algebra of length at least k. The boundary maps of the filtration yields a complex of spectra . .!. -2SP 4(S0) ! -1SP 2(S0) ! S0 that is exact in cohomology: each map sends the bottom class of the cyclic A-module to Sq1 applied to the bottom class of the next module. Appendix A. Augmented versus nonaugmented ring spectra Let Algu be the category of unital commutative S-algebras, but not nec- essarily augmented. Thus we have forgetful maps Alg ! Algu ! Alg0. Alg uis enriched over Tu, the category of unbased topological spaces, so one can look for a convenient model for K R with K 2 Tu and R 2 Algu. In this appendix we describe such a model, and compare this to the construction in x4. A.1. K R for unital commutative algebras. Let Su be the category of S-modules under S, so an object is an S-module map j : S ! X. __ Definition A.1. Given K 2 Tu and X 2 Su, let K^ X 2 Su be the pushout: K+ ^ S _____//S ^ S = S | | | | fflffl| fflffl|_ K+ ^ X _______//K^ X. It is easy to check Lemma A.2. There is an adjunction __ Map Su(K^ X, Y ) = Map Tu(K, Map Su(X, Y )). Let Set be the category of finite sets. Given K 2 Tu, there is an apparent functor Kx : Setop! Tu. Now note that, as it was in Alg, ^ is the coproduct in Algu. Then R 2 Algu defines R^ : Set ! Su. Lemma A.3. For all K, L 2 Tu, Map SetT(Kx , Lx ) = Map Tu(K, L). Lemma A.4. For all R, Q 2 Algu, Map SetSu(R^, Q^) = Map Algu(R, Q). A MODEL FOR A SPACE TENSORED WITH A RING SPECTRUM 21 Definition A.5. Given K 2 Tu and R 2 Algu, let __ ^ SP1 (K, R) = Kx ^SetR . As in x4, the lemmas combine to prove the analogue of Proposition 4.8. Proposition A.6. For all K 2 Tu and R 2 Algu, SP 1(K, R) is again in Algu and is naturally isomorphic to the categorical tensor product K R. SP 1(K, R) is filtered in the usual way, and one gets an isomorphism of S-modules (A.1) FdSP 1 (K, R)=Fd-1SP 1 (K, R) ' (K+ )(d)^ d (R=S)^d. We note that (K+ )(d)is just S if d = 0 and Kxd =(fat diagonal) if d > 0. A.2. The unbased versus the based construction. In this subsection, we denote the tensor in Alg by and the tensor in Algu by u. Given R 2 Algu , the product R x S will be in Alg , with augmentation given by projection, and unit S -! S x S ! R x S. This construction is right adjoint to the forgetful functor: Lemma A.7. Given Q 2 Alg and R 2 Algu, there is an adjunction Map Algu(Q, R) = Map Alg(Q, R x S). Note that, if Q 2 Alg and R 2 Algu, then Map Algu(Q, R) is based with basepoint Q ! S ! R. Proposition A.8. Given K 2 T , Q 2 Alg , and R 2 Alg u, there is an adjunction isomorphism Map Algu(K Q, R) = Map T(K, Map Algu(Q, R)). Proof.We have natural isomorphisms Map Algu(K Q, R)= Map Alg(K Q, R x S) = Map T(K, Map Alg(Q, R x S)) = Map T(K, Map Algu(Q, R)). Corollary A.9. If Q 2 Alg and L 2 Tu, then L+ Q = L u Q. Proof.Let K = L+ in the proposition, and note that Map T(L+ , Map Algu(Q, R))= Map Tu(L, Map Algu(Q, R)) = Map Algu(L u Q, R). 22 KUHN References [AK]S.T. Ahearn and N. J. Kuhn, Product and other fine structure in polynomial * *reso- lutions of mapping spaces, preprint, 2001. [An]D.W. Anderson, A generalization of the Eilenberg-Moore spectral seq* *uence, Bull.A.M.S. 78(1972), 784-788. [Ar]G. Arone, A generalization of Snaith-type filtration , Trans.A.M.S. 351(199* *9), 1123- 1250. [AD]G. Z. Arone and W. G. Dwyer, Partition complexes, Tits buildings and symmet* *ric products, Proc. London Math. Soc. 82 (2001), 229-256. [AM] G. Arone and M. Mahowald, The Goodwillie tower of the identity functor and* * the unstable periodic homotopy of spheres , Invent.Math. 135(1999), 743-788. [B]M. Basterra, Andr'e-Qullen cohomology of commutative S-algebras, J. Pure App* *lied Alg. 144(1999), 111-143. [BF]A. K. Bousfield and E. M. Friedlander, Homotopy theory of -spaces, spectra* *, and bisimplicial sets, in Geometric Applications of Homotopy Theory II (Evanston* *, 1977), Springer L.N.Math 658(1978), 80-130. [CLM] F. R. Cohen, T. Lada, and J. P. May, The Homology of Iterated Loop Spaces, Springer L. N. Math. 533, 1976. [EKMM] A.D.Elmendorf, I.Kriz, M.A.Mandell, J.P.May, Rings, modules, and algebr* *as in stable homotopy theory, A.M.S. Math. Surveys and Monographs 47, 1997. [FF]E.E.Floyd and W.J.Floyd, Actions of the classical small categories of topol* *ogy, un- published 200 page manuscript dating from 1990. Available from Bill Floyd's * *webpage at http://www.math.vt.edu/people/. [K1]N. J. Kuhn, A Kahn-Priddy sequence and a conjecture of G. W. Whitehead, Math. Proc. Camb. Phil. Soc. 92(1982), 467-483. [K2]N. J. Kuhn, Spacelike resolutions of spectra, Proceedings of the Homotopy T* *heory Conference (Northwestern, 1982), A. M. S. Cont. Math. Series 19(1983), 153-1* *65. [K3]N. J. Kuhn, New relationships among loopspaces, symmetric products, and Eil* *enberg MacLane spaces, in Cohomological Methods in Homotopy Theory (Barcelona, 1998* *), Birkhauser Verlag Progress in Math 196(2001), 185-216. [K4]N. J. Kuhn, On the S-dual of a circle, in preparation. [L]M. Lydakis, Smash products and -spaces, Math. Proc. Camb. Phil. Soc. 126(19* *99), 311-328. [MacL]S. MacLane, Categories for the working mathematician, Springer Graduate T* *exts in Math. 5, 1971. [M]J. P. May, E1 ring spaces and E1 ring spectra, Springer L. N. Math. 577, 197* *7. [McC]R. McCarthy, On n-excisive functors of module categories, to appe* *ar in Math.Proc.Camb.Phil.Soc. [McC]M.C.McCord, Classifying spaces and infinite symmetric products, T. A. M. S* *. 146 (1969), 273-298. [MSV] J. E. McClure, R. Schwänzl, and R. Vogt, THH(R) ' R S1 for E1 ring spectr* *a, J.Pure Appl.Algebra. 121 (1997), 137-159. [MP] S.A.Mitchell and S.B.Priddy, Stable splittings derived from the Steinberg * *module, Topology 22(1983), 285-298. [P]T. Pirashvili, Dod-Kan type therem for -groups, Math. Ann. 318(2000), 277-2* *98. [S]S. Schwede, Stable homotopy and -spaces, Math. Proc. Camb. Phil. Soc. 126(1* *999), 329-356. [Se]G.Segal, Categories and cohomology theories, Topology 13(1974), 293-312. Department of Mathematics, University of Virginia, Charlottesville, VA 22903 E-mail address: njk4x@virginia.edu