Topological Hochschild cohomology and generalized Morita equivalence

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 $K U ∕ 2$ is a nontrivial topological Azumaya algebra over the $2$ –adic completion of the $K$ –theory spectrum ${\hat{K U}}_{2}$ . This leads to the determination of $THH (K U ∕ 2, K U ∕ 2)$ , the topological Hochschild cohomology of $K U ∕ 2$ . As far as we know this is the first calculation of $THH (A, A)$ for a noncommutative $S$ –algebra $A$ .

PDF Access Denied

Warning: We have not been able to recognize your IP address 47.88.87.18 as that of a subscriber to this journal. Online access to the content of recent issues is by subscription, or purchase of single articles.

Please contact your institution's librarian suggesting a subscription, for example by using our journal-recommendation form.

Or, visit our subscription page for instructions on purchasing a subscription. You may also contact us at contact@msp.org or by using our contact form.

Or, you may purchase this single article for USD 29.95:

Keywords

$R$–algebra, topological Hochschild cohomology, Morita theory, Azumaya algebra

Mathematical Subject Classification 2000

Primary: 16E40, 18G60, 55P43

Secondary: 18G15, 55U99

References

Forward citations

Publication

Received: 6 February 2004

Accepted: 21 August 2004

Published: 23 August 2004

Authors

Andrew Baker
Mathematics Department Glasgow University Glasgow G12 8QW Scotland

Andrey Lazarev
Mathematics Department Bristol University Bristol BS8 1TW England

Volume 4, issue 1 (2004)

Andrew Baker and Andrey Lazarev

Abstract