Pre-Lie systems and obstruction to A_infty-structures over a ring

TL;DR

This paper establishes an obstruction theorem for the existence of A_infty-structures over a commutative ring R on an algebra A, linking it to Hochschild cohomology without restrictive assumptions.

Contribution

It provides a general obstruction criterion for A_infty-structures over any commutative ring, removing previous limitations on the ring or complex bounds.

Findings

Obstruction theorem for A_infty-structures over rings

No assumptions needed on the ring R

Applicable to unbounded complexes

Abstract

In this note, we prove an obstruction theorem for the existence of A infinite-structures over a commutative ring R on an algebra A associative up to homotopy, in terms of the Hochschild cohomology of the associative algebra H(A). The hidden purpose of the note is to show that there are no assumptions needed on the commutative ring R nor bounded assumptions on the complex A.