Homotopy Theory of A∞ Ring Spectra and Applications to MU-Modules

Abstract

We give a definition of a derivation of an A ∞ ring spectrum and relate this notion to topological Hochschild cohomology. Strict multiplicative structure is introduced into Postnikov towers and generalized Adams towers of A ∞-ring spectra. An obstruction theory for lifting multiplicative maps is constructed. The developed techniques are then applied to show that a broad class of complex-oriented spectra admit structures of M U-algebras where M U is the complex cobordism spectrum. Various computations of topological derivations and topological Hochschild cohomology are made.

Homology and cohomology of E? ring spectra

“…When A is cofibrant, one typically writes A e for A ∧ R A op . More generally, A e denotes A ′ ∧ R A ′op , for some fixed choice of cofibrant approximation A ′ → A. Lazarev [7] identifies Topological Quillen Cohomology in terms of Topological Hochschild Cohomology, and identifies the module of infinitesimal deformations of an associative algebra A as the homotopy fiber of the multiplication map A e → A. Part 1 of the previous theorem then has the following corollary.…”

Postnikov extensions of ring spectra

“…Our attempts to give a careful proof, however, always seemed to fail. A construction of k -invariants for ring spectra had already been given in Lazarev [14], but that construction does not seems well-suited for the above classification questions. A construction of k -invariants for commutative ring spectra appeared in Basterra [1]; while this construction also did not meet all of our needs, many of the techniques of [1] are used in our Section 6.…”

“…In [14] the bimodule R!C .C / is defined to be the homotopy fiber of the above multiplication map. The hard work is then to prove that the mapping spaces C -bimod.…”

“…R!C .C /; † nC1 M / and R-Alg =C .C; C _ † nC1 M / are weakly equivalent. The proof of this fact in [14] contains gaps.…”

Differential graded categories

“…The isomorphism classes of such algebras in the homotopy category of dg Z-algebras are parametrized by the Hochschild cohomology group HH 4 Z (Z/2, Z/2). Their isomorphism classes in the homotopy category of S-algebras are parametrized by the topological Hochschild cohomology group T HH 4 S (Z/2, Z/2) as shown in [97]. The computation of the Hochschild cohomology group HH 4 Z (Z/2, Z/2) is elementary and, thanks to Franjou-Lannes-Schwartz' work [50], the topological Hochschild cohomology algebra T HH * S (Z/2, Z/2) is known.…”