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We show that the action of the Lie algebra HH^1(A) of outer derivations of anassociative algebra A on the Hochschild cohomology HH^*(A) of A given by theGerstenhaber bracket can be computed in terms of an arbitrary projectiveresolution of A as an A-bimodule, without having recourse to comparison mapsbetween the resolution and the bar resolution.

A little bit of extra functoriality for Ext and the computation of the Gerstenhaber bracket