Algebraic and Geometric Topology 4 (2004), paper no. 29, pages 623-645.

Topological Hochschild cohomology and generalized Morita equivalence

Andrew Baker, Andrey Lazarev

Abstract. We explore two constructions in homotopy category with algebraic precursors in the theory of noncommutative rings and homological algebra, namely the Hochschild cohomology of ring spectra and Morita theory. The present paper provides an extension of the algebraic theory to include the case when $M$ is not necessarily a progenerator. Our approach is complementary to recent work of Dwyer and Greenlees and of Schwede and Shipley.
A central notion of noncommutative ring theory related to Morita equivalence is that of central separable or Azumaya algebras. For such an Azumaya algebra A, its Hochschild cohomology HH^*(A,A) is concentrated in degree 0 and is equal to the center of A. We introduce a notion of topological Azumaya algebra and show that in the case when the ground S-algebra R is an Eilenberg-Mac Lane spectrum of a commutative ring this notion specializes to classical Azumaya algebras. A canonical example of a topological Azumaya R-algebra is the endomorphism R-algebra F_R(M,M) of a finite cell R-module. We show that the spectrum of mod 2 topological K-theory KU/2 is a nontrivial topological Azumaya algebra over the 2-adic completion of the K-theory spectrum widehat{KU}_2. This leads to the determination of THH(KU/2,KU/2), the topological Hochschild cohomology of KU/2. As far as we know this is the first calculation of THH(A,A) for a noncommutative S-algebra A.

Keywords. $R$-algebra, topological Hochschild cohomology, Morita theory, Azumaya algebra

AMS subject classification. Primary: 16E40, 18G60, 55P43. Secondary: 18G15, 55U99.

DOI: 10.2140/agt.2004.4.623

E-print: arXiv:math.AT/0209003

Submitted: 6 February 2004. Revised 23 June 2004. Accepted: 21 August 2004. Published: 23 August 2004.

Notes on file formats

Andrew Baker, Andrey Lazarev
Mathematics Department, Glasgow University, Glasgow G12 8QW, Scotland
Mathematics Department, Bristol University, Bristol, BS8 1TW, England.


AGT home page

EMIS/ELibM Electronic Journals

Outdated Archival Version

These pages are not updated anymore. They reflect the state of 21 Apr 2006. For the current production of this journal, please refer to