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Topological Andreґ–Quillen Cohomology and EN ARTICLE IN PRESS Advances in Mathematics 177 (2003) 227–279 http://www.elsevier.com/locate/aim Topological Andre´ –Quillen Cohomology and EN Andre´ –Quillen Cohomology Michael A. Mandell1 Department of Mathematics, University of Chicago, 5734 S University Avenue, Chicago, IL 60637, USA Received 10 January 2001; accepted 29 April 2002 Communicated by Mark Hovey Abstract We show that the Andre´ –Quillen cohomology of an EN simplicial algebra with arbitrary coefficients and the topological Andre´ –Quillen cohomology of an EN ring spectrum with Eilenberg–Mac Lane coefficients may be calculated as the Andre´ –Quillen cohomology of an associated EN differential graded algebra. r 2003 Elsevier Science (USA). All rights reserved. MSC: 55P43 (primary); 18G55; 18D50 (secondary). Keywords: Andre´ -Quillen cohomology; EN ring spectrum 0. Introduction Andre´ [1] and Quillen [17] introduced a cohomology theory for commutative rings now called Andre´ –Quillen cohomology. For a commutative ring k; a commutative k- algebra A; and an A-module M; the Andre´ –Quillen cohomology of A relative to k with coefficients in M can be defined as a sort of derived functor of derivations. This functor satisfies the appropriate axioms for a cohomology theory and is closely related to ‘‘square zero’’ commutative k-algebra extensions of A: Andre´ –Quillen cohomology computations play an important role in the construction of the ring spectrum EO2 by Hopkins and Miller. The existence of EO2 and the closely related ring spectrum eo2 have already had a big impact on our E-mail address: [email protected]. 1 Supported in part by NSF Postdoctoral Research Fellowship DMS 9804421. 0001-8708/03/$ - see front matter r 2003 Elsevier Science (USA). All rights reserved. PII: S 0 0 0 1 - 8708(02)00017-8 ARTICLE IN PRESS 228 M.A. Mandell / Advances in Mathematics 177 (2003) 227–279 understanding of the stable category and of the homotopy groups of spheres. Recent work of Goerss and Hopkins has shown that EO2 is actually a ‘‘commutative S- algebra’’ in the sense of [4], the analogue in the category of spectra of a commutative (differential graded) algebra. Their proof develops an obstruction theory for the existence of commutative S-algebra structures. The obstruction groups are algebraically defined in terms of the homology of the underlying ring spectrum, and are a generalization of Andre´ –Quillen cohomology to EN simplicial algebras in a category of comodules. In practice, these groups often reduce to the Andre´ –Quillen cohomology of an EN simplicial algebra. The category of commutative S-algebras has enough of the formal properties of the category of commutative algebras that many of the different definitions of Andre´ –Quillen cohomology also make sense in this category. Basterra [2] compares these definitions, and shows that they are equivalent. We therefore obtain a ‘‘Topological Andre´ –Quillen’’ cohomology theory for the category of commutative S-algebras. This version of Andre´ –Quillen cohomology is also closely related to square zero extensions, and it leads to a different obstruction theory for the existence of commutative S-algebra structures, and in addition, an obstruction theory for the existence of commutative S-algebra maps. This is because the Postnikov tower of a connective commutative S-algebra provides a canonical example of a sequence of square zero extensions. As explained in [2] (following [9]), the k-invariants of a commutative S-algebra lift to unique elements of topological Andre´ –Quillen cohomology in the following sense. Given an S-module (or spectrum) A; a commutative S-algebra structure on the nth stage of the Postnikov tower A½n can be lifted to the ðn þ 1Þth stage A½n þ 1 if and only if the k-invariant can be lifted to the nþ2 topological Andre´ –Quillen cohomology D ðA½n; Hpnþ1AÞ; and the set of homotopy classes of lifts of the commutative S-algebra structure is in one-to-one correspondence with the set of lifts of the k-invariant. This is the obstruction theory for the construction of commutative S-algebra structures. From here, obstruction theory for the construction of commutative S-algebra maps works just as the analogous theory works for spaces or spectra: a map of commutative S-algebras B-A½n lifts to A½n þ 1 if and only if the k-invariant pulls back to zero in nþ2 D ðB; Hpnþ1AÞ and the set of homotopy classes of lifts has a free transitive action nþ1 of D ðB; Hpnþ1AÞ: In this way, the topological Andre´ –Quillen cohomology of commutative S-algebras provides a natural and complete obstruction theory for connective commutative S-algebras. These generalizations of Andre´ –Quillen cohomology to very different looking categories have similar, interesting applications. A third theory is the Andre´ –Quillen cohomology of EN differential graded algebras. This last theory is less rigid than the theory for EN simplicial algebras and less technically demanding than the theory for commutative S-algebras. Moreover, a number of general tools exist for calculating the Andre´ –Quillen cohomology of EN differential graded algebras. In this paper we show that the Andre´ –Quillen cohomology of an EN simplicial algebra and the topological Andre´ –Quillen cohomology of a commutative S-algebra may be calculated as the Andre´ –Quillen cohomology of an associated EN differential graded algebra. In any of the three categories, Andre´ –Quillen ARTICLE IN PRESS M.A. Mandell / Advances in Mathematics 177 (2003) 227–279 229 cohomology is defined for an algebra A with coefficients in an A-module M and is natural in both A and M as reviewed below. It is a cohomology theory in the algebra A and a homology theory in the module M: There is a ‘‘transitivity’’ long exact sequence associated to a map of algebras A-B (cf. [17, 5.1]), and a ‘‘coefficient’’ long exact sequence associated to a map of modules M-N: In the obstruction theory context above, the coefficients are of a simple form: In the simplicial context, the simplicial structure on M is constant and M is merely a p0A-module. In the topological context, M ¼ Hp for some p0A-module p; i.e. p * M is concentrated in degree zero, and the A-module structure is determined by the canonical map of commutative S-algebras from A to the Eilenberg–Mac Lane spectrum Hp0A: We call coefficients of this form ‘‘constant’’ in the simplicial context and ‘‘Eilenberg–Mac Lane’’ in the topological context. In either context, we form the corresponding coefficients for EN differential graded algebras using the p0A-module p0M (with zero differential). Our main results are then the following theorems. (See Corollaries 1.8 and 7.9 for generalizations to other coefficients.) Theorem. Let k be a (graded) commutative ring. There is a functor N from EN simplicial k-algebras to EN differential graded k-algebras that induces an isomorphism of Andre´–Quillen cohomology in constant coefficients, which preserves the transitivity and coefficient long exact sequences. Theorem. Let R be a connective cofibrant commutative S-algebra, and let k ¼ p0R: There is a functor X from commutative R-algebras to EN differential graded k- algebras that induces an isomorphism of Andre´–Quillen cohomology in Eilenberg– Mac Lane coefficients, which preserves the transitivity and coefficient long exact sequences. In the simplicial context, the functor N is the normalization functor on the underlying simplicial k-module and we have a canonical isomorphism of D commutative k-algebras H * NA p * A: We give an outline of the argument and generalizations in Section 1. In the topological context, the functor X is the composite of a cellular approximation functor and a cellular chain functor, and D we have a canonical isomorphism of commutative k-algebras H * XA H * A; where H * A ¼ p * ðA4RHkÞ is the ‘‘ordinary’’ homology theory of [4, Section IV.2]. The condition that R be connective means that its homotopy groups are concentrated in non-negative degrees; the condition that R be cofibrant is a technical one. Both conditions are satisfied by the sphere spectrum S: We give an outline of the argument and generalizations in Section 7. These comparison theorems allow us to apply the tools developed in the differential graded context to the other two contexts. In a future paper, Maria Basterra and the author will describe these tools and use them to calculate topological Andre´ –Quillen cohomology and EN simplicial algebra Andre´ –Quillen cohomology in some interesting examples. ARTICLE IN PRESS 230 M.A. Mandell / Advances in Mathematics 177 (2003) 227–279 1. EN simplicial algebras and EN differential graded algebras In this section, we state the main results for the comparison of EN simplicial algebras and EN differential graded algebras. We work relative to a base commutative ring k; in categories of k-modules exclusively, and the tensor product should be understood to be the tensor product over k: In the context of [6], k is the coefficient ring of a cohomology theory and so we must allow k to be graded. In this case, k-modules are graded and differential graded k-modules are bigraded. The internal degree plays no role in the theory below and for the most part can be ignored; in arguments below ‘‘degree’’ always refers to the differential or simplicial degree. Before stating in detail our comparison theorems, we begin with a review of the definition of Andre´ –Quillen cohomology. The same basic ingredients go into the definition for EN algebras as go into the definition for commutative algebras. Recall that for a commutative k-algebra A and a simplicial A-module M; we can define the ‘‘square-zero extension’’ ZM ¼ A"M to be the simplicial commutative A-algebra obtained by taking the product of any two elements in M to be zero.
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