#### 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.

Email: a.baker@maths.gla.ac.uk, a.lazarev@bristol.ac.uk

URL:
http://www.maths.gla.ac.uk/~ajb/,
http://www2.maths.bris.ac.uk/~maxal/

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