Topological Hochschild cohomology and generalized Morita equivalence

Baker, A. and Lazarev, A. (2004) Topological Hochschild cohomology and generalized Morita equivalence. Algebraic and Geometric Topology, 4, pp. 623-645. (doi: 10.2140/agt.2004.4.623)

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 FR(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 KÛ2. 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.

• Item Type: Articles
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Baker, Dr Andrew
• Authors: Baker, A., and Lazarev, A.
• Subjects: Q Science > QA Mathematics
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: Algebraic and Geometric Topology
• ISSN: 1472-2747
• ISSN (Online): 1472-2739