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Let A be a k-algebra and G be a group acting on A. We show that G also acts on the Hochschild cohomology algebra HH⊙(A) and that there is a monomorphism of rings HH⊙(A)G→HH⊙(A[G]). That allows us to show the existence of a monomorphism from HH⊙(Ã)G into HH⊙(A), where Ã is a Galois covering with group G.

Hochschild Cohomology of skew group rings and invariants