André-Quillen cohomology of commutative \(S\)-algebras
The notion of commutative \(S\)-algebra, as introduced in [\textit{A. D. Elmendorf, I. Kriz, M. A. Mandell}, and \textit{J. P. May}, Rings, modules, and algebras in stable homotopy theory, Math. Surv. Monogr. 47 (1997; Zbl 0894.55001)], is a generalization to stable homotopy theory of the algebraic notion of commutative ring. In this paper, a cohomology theory of commutative S-algebras is studied, analogous to that of \textit{D. Quillen} [Proc. Sympos. Pure Math. 17, 65-87 (1970; Zbl 0234.18010)]. If \(B\) is a commutative \(A\)-algebra, and \(M\) is a \(B\)-module, this new `` André-Quillen'' cohomology \(AQ^*(B/A;M)\) is defined as Ext\(_B^*(\Omega_{B/A},M)\), where \(\Omega_{B/A}\) is an `` abelianization'' of \(B\). A spectral sequence suitable for computing the \(AQ\)-cohomology relative to \(S\) of commutative \(S\)-algebras with \(S\)-algebra maps to \(HF_p\) is defined and studied. Postnikov towers of commutative \(S\)-algebras are defined; their \(k\)-invariants lie in André-Quillen cohomology.
scientific article; zbMATH DE number 1398839