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
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
, its Hochschild cohomology
is concentrated in
degree and is equal
to the center of .
We introduce a notion of topological Azumaya algebra and show that in the case when the
ground –algebra
is an
Eilenberg–Mac Lane spectrum of a commutative ring this notion specializes to
classical Azumaya algebras. A canonical example of a topological Azumaya
–algebra is the
endomorphism –algebra
of a finite cell
–module. We show that
the spectrum of mod
topological –theory
is a nontrivial topological Azumaya algebra over the
–adic completion of
the –theory spectrum
. This leads to the determination
of , the topological
Hochschild cohomology of .
As far as we know this is the first calculation of
for a noncommutative
–algebra
.
