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Topological Hochschild cohomology and generalized Morita equivalence

Andrew Baker and Andrey Lazarev

Algebraic & Geometric Topology 4 (2004) 623–645

arXiv: math.AT/0209003


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 KU2 is a nontrivial topological Azumaya algebra over the 2–adic completion of the K–theory spectrum KÛ2. This leads to the determination of THH(KU2,KU2), the topological Hochschild cohomology of KU2. As far as we know this is the first calculation of THH(A,A) for a noncommutative S–algebra A.

$R$–algebra, topological Hochschild cohomology, Morita theory, Azumaya algebra
Mathematical Subject Classification 2000
Primary: 16E40, 18G60, 55P43
Secondary: 18G15, 55U99
Forward citations
Received: 6 February 2004
Accepted: 21 August 2004
Published: 23 August 2004
Andrew Baker
Mathematics Department
Glasgow University
Glasgow G12 8QW
Andrey Lazarev
Mathematics Department
Bristol University
Bristol BS8 1TW