  The Hochschild cohomology of a tensor product of algebras is isomorphic to a
graded tensor product of Hochschild cohomology algebras, as a Gerstenhaber
algebra. A similar result holds when the tensor product is twisted by a
bicharacter. We present new proofs of these isomorphisms, using Volkov's
homotopy liftings that were introduced for handling Gerstenhaber brackets
expressed on arbitrary bimodule resolutions. Our results illustrate the utility
of homotopy liftings for theoretical purposes.
