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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 ^KU2. 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.
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