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We develop a theory of
-module Thom spectra for a commutative symmetric ring spectrum
and we analyze their multiplicative properties. As an interesting source of examples, we show that
-algebra Thom spectra associated to the special unitary groups can be described in terms of quotient constructions on
. We apply the general theory to obtain a description of the
-based topological Hochschild homology associated to an
-algebra Thom spectrum.
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