A1 OBSTRUCTION THEORY AND THE STRICT ASSOCIATIVITY OF E=I VIGLEIK ANGELTVEIT Abstract. We prove that for a ring spectrum K with a perfect univer- sal coeOEcient formula, the obstructions to extending the multiplication to an A1 multiplication lie in Ext*,*K*Kop(K*, K*). As a corollary, we show that if E is even and I = (x1, x2, . .).is a regular sequence in E*, then any product on E=I can be extended to an A1 multiplication. 1. Introduction In [9], Alan Robinson developed an obstruction theory for extending a homotopy associative multiplication on a ring spectrum K to an A1 multi- plication, based on Hochschild cohomology. On closer inspection, one can see that this obstruction theory requires special care if K is not homotopy com- mutative. To be precise, if K is not homotopy commutative, then K* is no longer a K*K-module, but rather a K*Kop-module. Thus HH*K*(K*K, K*) is not even deoned. The purpose of this paper is to correct that deociency. We show that by replacing K*K with K*Kop, Robinson's obstruction the- ory works out as stated. This has the added advantage that K*Kop tends to have a better structure than K*K. Also, we make the theory relative to a commutative S-algebra E. We arrive at the following theorem: Theorem 1.1. Let K be an E-ring spectrum with a perfect universal coef- ocient formula. Then the obstructions to extending the multiplication on K to an A1 multiplication lie in Extn,3-nss*(K^EKop)(K*, K*), for n 4. The obstructions to uniqueness lie in Extn,2-nss*(K^EKop)(K*, K*), for n 3. Actually, there is a Bousoeld-Kan spectral sequence converging to the space of A1 structures with E2-term Exts,tss*(K^EKop)(K*, K*) of the same type as considered in [8]. The groups Extn,2-n give the connected components of this space. Also, one might want to consider AutE (K) acting on the space of A1 structures, and regard two A1 structures as equivalent if some element of AutE (K) carries one into the other. In the last section we will see an example of this, where two A2 structures, which can be extended to A1 structures, 1 are equivalent in this way if K is regarded as an S-ring spectrum, but not as an E-ring spectrum. If K* is central in ss*(K ^E Kop), then these Ext groups are the same as the Hochschild cohomology groups HH*K*(ss*(K ^E Kop), K*), and if K is homotopy commutative, then ss*(K ^E Kop) ~=ss*(K ^E K), and we get back Robinson's obstruction theory. It is also worth noting ([5, theorem IX.1.6]) that if K is A1 , then Ext*,*ss*(K^EKop)(K*, K*) is the E2-term of a spectral sequence converging to ss*T HHE (K), the homotopy groups of topological Hochschild cohomology of K over the ground ring E. We use this to show that if E* is even and I = (x1, x2, . .).is a regular sequence in E*, then any product on E=I can be extended to an A1 multi- plication. There are many partial results in this direction in the literature; see for example [6]. As a corollary, we show that all Morava K-theories are A1 at any prime. This result will also be used in [2] to study non-commutative multiplica- tions on 2-periodic Morava K-theories. The author would like to thank Haynes Miller for reading several versions of this paper and for many useful suggestions, and also Andrey Lazarev for some useful comments. 2. Universal coefficient and K#nneth isomorphisms We work in the category of E-modules, where E is a commutative S- algebra, as in [5]. Thus spectrum means E-module, X ^ Y means X ^E Y , X*Y means ss*(X ^E Y ) and X*Y means ss*FE (Y, X). For aesthetic reasons, we will make one exception to this rule, by representing x 2 ssdX by a map Sd -! X rather than a map dE -! X, and smash products are over the sphere spectrum when smashing spheres, as in Sd1 ^ Sd2. We will assume that E is q-coobrant, and that all E-modules are cell E-modules. By a ring spectrum (E-ring spectrum) we mean a spectrum K with a multiplication OE : K ^ K -! K and a unit j : E -! K which makes K left and right unital, and associative, up to homotopy. Note that we can always promote the multiplication to a strictly unital one, in the same way as one can promote the multiplication on a homotopy unital H-space to a strictly unital one. For a ring spectrum K, we consider K*X as a K*-bimodule, where the left action of K* is the expected one and the right action involves switching K and X, i.e., for a 2 K*X and r 2 K*, a r is sent to the composite Sd1+d2 ~=Sd1 ^ Sd2 a^r-!K ^ X ^ K 1^o-!K ^ K ^ X OE^1-!K ^ X, where o is the twist map. With X = K this gives a right K*-module struc- ture on K*K which is dioeerent from the one considered by Adams ([1]). We assume that K has a perfect universal coeOEcient formula, by which we mean that the following two conditions are satisoed: 2 (A) K*K is projective as a left K*-module. This implies that K*(K(n)) ~=(K*K) n , where K(n)is the n-fold smash product of K and the tensor product is over K*. We need to be explicit about this isomorphism. We always have a map K*X K* K*Y - ! K*(X ^ Y ) sending a b to the composite Sd1 ^ Sd2 a^b-!K ^ X ^ K ^ Y 1^o^1-!K ^ K ^ X ^ Y OE^1^1-!K ^ X ^ Y. With n factors, there are as many maps (K*K) n -! K*(K(n)) as there are ways to associate a word with n letters. These are all the same because K is homotopy associative, and this is the map we assume is an isomorphism. It is perhaps also worth pointing out that K*X K*Y - ! K*(X ^ Y ) factors through K*X K* K*Y because K is homotopy associative, since the two maps (K ^ X) ^ K ^ (K ^ Y ) -! K ^ X ^ Y are homotopic. (B) There is a universal coeOEcient isomorphism K*(K(n)) ~=HomK* (K*(K(n)), K*). When K*K is projective over K*, this condition holds if K has a universal coeOEcient spectral sequence. See [1, III.13] for a condition which guarantees the existence of such a spectral sequence. The isomorphism (given by the edge homomorphism in the spectral sequence if there is one) sends f : X -! K to ss* of the composite K ^ X 1^f-!K ^ K -OE!K. This is a map of left K*-modules, but not of right K*-modules. Let R = K* and = K*Kop, so that = K*K additively and the ring structure on is given by sending ~1 ~2 to (1^1^o)(1^o^1)(4)OE^OE Sd1 ^ Sd2 ~1^~2-!K(4) - ! K -! K ^ K. We will identify K*(K(n)) with Hom ( (n+1), R), after specifying the - module structure on (n+1). We can choose between (at least) two dioeerent -module structures on (n+1). We get the orst one by thinking of (n+1) as ss*(K ^ Kop ^ K(n)), with K ^ Kop acting on the orst two factors. Let i(n+1)nddenote (n+1) with this -module structure. It is induced up from the R-module structure on n , so we get an isomorphism ~= (n+1) ind: HomR ( n , R) -! Hom ( ind , R). The other -module structure on (n+1) comes from thinking of (n+1) as ss*(K ^ K(n)^ K), with K ^ Kop acting on the orst and last factors. Let b(n+1)ardenote (n+1) with this -module structure. Let oe : K(n+2)- ! K(n+2) be the permutation which oxes the orst factor and cyclically permutes the n + 1 last factors, moving the second factor to 3 the end. It induces an isomorphism oe* : i(n+1)nd-! b(n+1)arof -modules, and thus an isomorphism ~= (n+1) (oe-1 )* : Hom ( i(n+1)nd, R) -! Hom ( bar , R). Thus we get an isomorphism (1) K*(K(n)) ~=Hom ( b(n+1)ar, R), where the isomorphism sends f : K(n)- ! K to ss* of the composite OE(OE^1) K(n+2)1^f^1-!K(3) -! K. From now on (n+1) will mean b(n+1)ar. 3. A1 obstruction theory Recall the deonition of the Stasheoe associahedra A(i), i 0, which form a (non- ) A1 operad, from [10]. We deone a unital An structure on K as a map ` A(i)+ ^ K(i)-! K 0 i n satisfying the usual conditions, for n 1. Similarly, a non-unital An struc- ture is deoned as a map ` A(i)+ ^ K(i)-! K 1 i n satisfying similar conditions. Recall that A(n) ~= Dn-2, an n - 2 disk, and that A(n) has i faces of the form A(i) x A(n - i - 1), for each 2 i n - 3. Given an An-1 structure on K, if we want to extend it to a non-unital An structure, the map A(n)+ ^ K(n)- ! K is already determined on @A(n)+ ^ K(n)~= n-3K(n). If we want the An structure to be unital, thenWthe map is also determined on A(n)+ ^ @K(n), where @K(n) is the image of ni=1K(i-1)^ E ^ K(n-i) in K(n). Let ~ = coker(R -! ), where the map R -! is the right unit K* ~=K*E -! K*K. Then K*(K=E) ~=~ and K*(K(n)=@K(n)) ~=~ n as R-bimodules. Thus elementary obstruction theory gives the following: (compare with [9, 1.5-1.7]) Lemma 3.1. For n 4, given a (unital or non-unital) An-1 structure on K, the obstruction to extending it to a non-unital An structure lies in Kn-3 (K(n)) ~=Hom3-n( (n+1), R), and the obstruction to extending it to a unital An structure (if An-1 is unital) lies in Kn-3 ((K=E)(n)) ~=Hom3-n(~ n , R). The set of non-unital An-1 structures, oxing the An-2 structure, is given by Kn-3 (K(n-1)) ~=Hom3-n( n , R), 4 and the set of unital An-1 structures, oxing the An-2 structure, is given by Kn-3 ((K=E)(n-1)) ~=Hom3-n(~ (n-1) , R). Suppose we want to calculate Ext*,*(R, R). Then we need a projective resolution of R as a -module, and by (A) we get one by taking homo- topy groups of the two-sided bar resolution B(K, K, K) -! K. (See e.g. [5, deonition XII.1.1].) Thus we get a cochain complex C* with Cn = Hom ( (n+1), R), which is isomorphic to K*(K(n)) by (1), calculating Ext (R, R). Here ffi : Cn-1 - ! Cn is given as follows: For g : n - ! R, let f :PK(n-1) -! K be the image of g under the isomorphism (1). Then ffig = ni=0(-1)iffiig, where ffiig is given by taking ss* of i^OE^1n-i 1^f^1 OE(OE^1) B(K, K, K)n = K(n+2)1 -! K(n+1) -! K(3) -! K. Note that we can give C* the structure of a cosimplicial group. The code- generacy maps are given by precomposing with the maps K(n+1)~= K(i+1)^ E ^ K(n-i)-! K(n+2) for 0 i n - 1. Let us concentrate on the non-unital theory for now, and then see what changes we need to do to for the unital An structures, which is what we really care about. We need to calculate what happens to the obstruction to extending an An-1 structure to a An structure if we change the An-1 structure while oxing the An-2 structure: Proposition 3.2. Let n 4. If we alter the An-1 structure by g : n -! R, then the obstruction cn 2 Hom ( (n+1), R) to extending the multiplica- tion to an An structure is changed by ffig. Proof.Again, let f : K(n-1)- ! K correspond to g under the isomorphism (1). The geometric argument in [9, 1.8] shows that the obstruction is changed by a sum of maps K(n)- ! K of two types, corresponding to two types of (n - 3)-dimensional faces of An ~=Dn-2. The maps (2) K(n)-1^f!K ^ K -OE!K and (3) K(n)-f^1!K ^ K -OE!K give the orst and the last term in ffig, respectively, and the maps i-1^OE^1n-i-1 f (4) K(n)1 - ! K(n-1)- ! K for i = 1, . .,.n - 1 give the rest of the terms. For example, to see that (2) gives ffi0g, consider the following homotopy commutative diagram: 1^1^f^1// 1^OE^1// K(n+2) _____K(4) _____K(3) OE^1|| OE(OE^1)|| fflffl|1^f^1 OE(OE^1fflffl|) K(n+1) ____//_K(3)____//_K 5 Applying ss* to this diagram, we see that going around clockwise gives the change to the obstruction from the (n - 3)-cell, while going counterclockwise gives ffi0g. The next proposition is proved in a similar way: Proposition 3.3. The obstruction cn is a cocycle. To onish the proof of theorem 1.1, it is enough to observe that the cochain complex ~C*with ~Cn= Hom*(~ n , R) also calculates Ext (R, R). But ~C*plays the role of the normalized cochain complex, if we think of ~ n as ss*(K ^ (K=E)(n)^ K). This makes sense because of the cosimplicial structure on C*. This onishes the proof of theorem 1.1. Remark 3.4. a) The theory of A1 maps needs to be changed in a similar way. Given a map L - ! K, where L and K are A1 and K*(L(n)) ~= HomK* ((K*L) n , K*), we are led to study Ext*,*K*Lop(K*, K*). Again this is forced upon us, because K is not a K ^ L module, but a K ^ Lop module. b) The spectral sequence set up e.g. in [8] for calculating A1 structures or A1 maps based on Andr#-Quillen cohomology has to be changed accordingly. The E2-term has to be expressed as DsK*(K*Lop, K*+t), derived functors of derivations of K*Lop into K* when the spectra are not homotopy commuta- tive. 4. The strict associativity of E=I Now suppose that E has homotopy groups only in even dimensions, and that I = (x1, x2, . .).is a regular sequence in E*, with |xi| = di. Deone E=xi by the coober sequence diE -xi! E - ! E=xi in the category of E-modules. Recall from [11, proposition 3.1] that each E=xi has at least one homotopy associative multiplication, and from [11, proposition 4.8] that choosing a multiplication on each E=xi gives a multiplication on K = E=I, the homotopy colimit of the spectra E=x1^. .^.E=xi. Not all multiplications on K come from smashing together multiplications on each E=xi; we will discuss this further in [2]. It follows trivially that (E=xi)* = E*=xi and K* = E*=I. The following result has also been proved by Lazarev in [6, lemma 2.6]: Proposition 4.1. For any homotopy associative multiplication on K we have K*Kop ~= K* (ff1, ff2, . .). with |ffi| = di+ 1. Proof.There is a multiplicative K#nneth spectral sequence (see [4]) E2 = T orE**,*(K*, Kop*) =) K*Kop. 6 By using a Koszul resolution of K* = E*=I it is easy to see that E2 = K* (ff1, ff2, . .).with ffi in bidegree (1, di). The spectral sequence collaps* *es, so all we have to do is to show that there are no multiplicative extensions. Because E21,*is concentrated in odd total degree, it follows that ff2i2 K* E* Kop*~=K* in K*Kop. Now there are several ways to show that ff2i= 0. For example, we can use the fact that K is a K ^ Kop-module and study the two maps K*Kop K*Kop K* -! K*. One sends ffi ffi 1 to ff2i, the other one sends it to 0. Note that this result does not hold for K*K, in which case ffi might very well square to something nonzero in K*. We also need to observe that K satisoes conditions (A) and (B). But K*K is projective over K* by proposition 4.1, giving (A). For (B), inductively tak- ing K* of the coober sequence deoning E=(x1, . .,.xn) from E=(x1, . .,.xn-1) gives a long exact sequence which breaks into short exact sequences and proves that K*K ~= HomK* (K*K, K*). A similar argument gives (B) for n > 1. Theorem 4.2. For E even and I regular, any homotopy associative multi- plication on K = E=I can be extended to an A1 structure. Moreover, the natural map E -! K extends uniquely to a map of A1 ring spectra for any choice of A1 structure on K, making E central in K. Proof.Recall that Ext* K*(ff1,ff2,...)(K*, K*) ~=K*[~ff1, ~ff2, . .]., with ~ffiin (cohomological) bidegree (1, di + 1). Thus the Ext groups are concentrated in even total degree, and the obstructions to existence of an A1 structure on K vanish. The second part of the theorem is obvious. With E = MU or MU(p), it follows immediately that BP , BP , P (n) and k(n) are A1 . Using Bousoeld localization, which can be done in the category of A1 ring spectra ([5, theorem VIII.2.1]), it follows that also E(n), B(n) and K(n) are A1 . (See e.g. [11, p.4] for the homotopy type of these spectra.) We include one more example: Let E = [E(n) be the K(n)-localization of E(n), which is E1 , or strictly commutative, by [3, theorem 8.2]. Let I = (p, v1, . .,.vn-1), so that K(n) = E=I. Recall, from [12] that for p odd we have ss*K(n) ^S K(n)op ~= (n)(ff0, . .,.ffn-1), and from [7] that for p = 2 we have ss*K(n) ^S K(n)op ~= (n)[ff0, . .,.ffn-1]=(ff2i- ti+1), where n pi-1 (n) = K(n)*[t1, t2, . .].=(tpi - vn ti). 7 From proposition 4.1 it follows that ss*K(n) ^E K(n)op ~= K(n)*(ff0, . .,.ffn-1), and the map ss*K(n) ^S K(n)op -! ss*K(n) ^E K(n)op sends each ti to zero, because ti exists in ss*E ^S E and is obviously sent to zero under ss*E ^S E - ! ss*E ^E E ~= E*. (The map ss*E ^S E - ! E* ultimately comes from the multiplication map ss*MU ^S MU -! MU*, and the gen- erator corresponding to ti in MU*MU maps to zero in MU* by [1, theorem II.11.3.ii].) When p = 2, the multiplication OE on K(n) is not homotopy commuta- tive, but there is an automorphism of K(n) as an S-module carrying OE to OE O o. If we identify K(n)*K(n) with HomK(n)*(K(n)*K(n), K(n)*) this automorphism is given by sending tn to vn + tn. However, this is not a map of E-modules, so OE and OE O o really are dioeerent as ring structures on K(n) regarded as an E-module. However, the obstruction theory is the same for K(n) as an S-ring spec- trum or an E-ring spectrum, because of the following result: Lemma 4.3. At any prime p, Extss*(K(n)^SK(n)op)(K(n)*, K(n)*) ~=Extss*(K(n)^EK(n)op)(K(n)*, K(n)*). Proof.We concentrate on the case p = 2, because it is slightly more compli- cated, and because for p odd the proof of [9, theorem 2.2] applies. Write ss*(K(n) ^S K(n)op) ~= 1 . . . n n+1 n+2 . .,. where i= R[ffi-1]=(ff2n+1i-1- v2i-1nff2i-1) and j = R[tj]=(t2nj- v2j-1ntj). Of course, vn only serves to keep track of the grading, so the result follows from the calculations that for any ring R and any k 2, Ext*R[t]=(tk-t)(R, R) = R concentrated in degree zero and Ext*R[ff]=(ff2k-ff2)(R, R) ~=R[~ff]. Thus, K(n) has the same space of A1 structures regarded as an S-ring spectrum or an E-ring spectrum. But the group of automorphisms of K(n) is larger in the orst case, so we expect the number of non-equivalent A1 structures to be smaller. Indeed, we just saw that even the number of non- equivalent A2 structures are dioeerent at p = 2. References [1]J. F. Adams. Stable homotopy and generalised homology. University of Chicag* *o Press, Chicago, Ill., 1974. Chicago Lectures in Mathematics. [2]Vigleik Angeltveit. Morita theory, multiplications on E=I and topological H* *ochschild cohomology. In preparation. [3]A. Baker and B. Richter. -cohomology of rings of numerical polynomials and* * E1 - structures on K-theory. Preprint. 8 [4]Andrew Baker and Andrej Lazarev. On the Adams spectral sequence for R-modul* *es. Algebr. Geom. Topol., 1:173~199 (electronic), 2001. [5]A. D. Elmendorf, I. Kriz, M. A. Mandell, and J. P. May. Rings, modules, and* * algebras in stable homotopy theory. American Mathematical Society, Providence, RI, 19* *97. With an appendix by M. Cole. [6]A. Lazarev. Towers of MU-algebras and the generalized Hopkins-Miller theore* *m. Proc. London Math. Soc. (3), 87(2):498~522, 2003. [7]Christian Nassau. On the structure of P(n)*P((n)) for p = 2. Trans. Amer. M* *ath. Soc., 354(5):1749~1757 (electronic), 2002. [8]Charles Rezk. Notes on the Hopkins-Miller theorem. In Homotopy theory via a* *lgebraic geometry and group representations (Evanston, IL, 1997), volume 220 of Conte* *mp. Math., pages 313~366. Amer. Math. Soc., Providence, RI, 1998. [9]Alan Robinson. Obstruction theory and the strict associativity of Morava K-* *theories. In Advances in homotopy theory (Cortona, 1988), volume 139 of London Math. S* *oc. Lecture Note Ser., pages 143~152. Cambridge Univ. Press, Cambridge, 1989. [10]James Dillon Stasheoe. Homotopy associativity of H-spaces. I, II. Trans. Am* *er. Math. Soc. 108 (1963), 275-292; ibid., 108:293~312, 1963. [11]N. P. Strickland. Products on MU -modules. Trans. Amer. Math. Soc., 351(7):* *2569~ 2606, 1999. [12]Nobuaki Yagita. On the Steenrod algebra of Morava K-theory. J. London Math.* * Soc. (2), 22(3):423~438, 1980. Department of Mathematics, Massachusetts Institute of Tech- nology, Cambridge, Massachusetts 02139, USA E-mail address: vigleik@math.mit.edu 9