Basterra, Maria; Mandell, Michael A. Homology and cohomology of \(E_\infty\) ring spectra. (English) Zbl 1071.55006 Math. Z. 249, No. 4, 903-944 (2005). During the 1990’s, stable homotopy theory was revolutionized by the appearance of new categories of spectra with symmetric monoidal smash products before passage to derived homotopy categories. The original approach due to A. D. Elmendorf, I. Kriz, M. A. Mandell and J. P. May [Rings, modules, and algebras in stable homotopy theory, Mathematical Surveys and Monographs. 47. AMS (1997; Zbl 0894.55001)] led to categories of \(S\)-modules and \(S\)-algebras which support model category structures, while later constructions such as symmetric and orthogonal spectra gave alternative approaches.This paper focuses on homology and cohomology functors for the category of commutative \(R\)-algebras over a commutative \(S\)-algebra \(R\), where \(S\) denotes the sphere spectrum. Such functors have properties akin to those possessed by the usual kind of functors on spaces or spectra in that they satisfy appropriate versions of the Eilenberg-Steenrod axioms. Much of the paper is devoted to setting up analogues of the notion of spectra in the world of commutative \(R\)-algebras, leading to versions of Brown representability. Motivating examples are provided by Topological André-Quillen theory [see M. Basterra, J. Pure Appl. Algebra. 144(2), 111–143 (1999; Zbl 0937.55006); M. Basterra and B. Richter, Lond. Math. Soc. Lect. Note Ser. 315, 115–131 (2004; Zbl 1079.13008)] which is defined on a category of \(R\)-algebras over a fixed \(R\)-algebra \(B\) and Theorem 1 characterizes such cohomology theories.Theorem 5 and its corollaries give interesting explicit identifications of the so-called cotangent complex of a Thom spectrum \(M\) associated with an infinite loop map into \(BF\). In particular, it is shown that \[ \mathbf{L}\mathbf{\Omega}_SMU\simeq MU\wedge bu, \] where \(bu\) is the \(1\)-connected cover of the connective \(KU\)-spectrum. Reviewer: Andrew Baker (Glasgow) Cited in 30 Documents MSC: 55P43 Spectra with additional structure (\(E_\infty\), \(A_\infty\), ring spectra, etc.) 55P48 Loop space machines and operads in algebraic topology 55U35 Abstract and axiomatic homotopy theory in algebraic topology Keywords:(co)homology theory; \(S\)-algebra; \(E_\infty\) ring spectrum Citations:Zbl 0894.55001; Zbl 0937.55006; Zbl 1079.13008 PDFBibTeX XMLCite \textit{M. Basterra} and \textit{M. A. Mandell}, Math. Z. 249, No. 4, 903--944 (2005; Zbl 1071.55006) Full Text: DOI arXiv References: [1] Basterra, M.: André-Quillen cohomology of commutative S-algebras. J. Pure Appl. Algebra. 144(2), 111-143 (1999) · Zbl 0937.55006 [2] Brown, E.H.: Abstract homotopy theory. Trans. Amer. Math. 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