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In this article, we present actions by central elements on Hochschildcohomology groups with arbitrary bimodule coefficients, as well as aninterpretation of these actions in terms of exact sequences. Since ourconstruction utilises the monoidal structure that the category of bimodulespossesses, we will further recognise that these actions are compatible withmonoidal functors and thus, as a consequence, are invariant under Moritaequivalences. By specialising the bimodule coefficients to the underlyingalgebra itself, our efforts in particular yield a description of thedegree-$(n,0)$-part of the Lie bracket in Hochschild cohomology, and therebyclose a gap in earlier work by S. Schwede.

Exact sequences, Hochschild cohomology, and the Lie module structure over the $M$-relative center