Pre-Lie systems and obstructions to $$A_\infty $$ A ∞ -structures over a ring

Abstract

In this note we establish an obstruction theory for the existence of $$A_\infty $$ -algebra structures on a differential $$\mathbb Z $$ -graded $$R$$ -module $$A$$ equipped with a homotopy associative multiplication. This approach follows classical obstruction theory: the corresponding obstructions live in the Hochschild cohomology of the associative algebra $$H(A)$$ . Our theory makes use of the pre-Lie algebra structure on the Hochschild cochain complex of $$H(A)$$ . As a consequence, no additional assumptions on the commutative ring $$R$$ are needed.