Obstruction theory for $A$-infinity bimodules

TL;DR

This paper develops an obstruction theory for extending truncated minimal $A$-infinity bimodule structures, introducing a new cohomology theory and spectral sequence analysis to understand their extensions and formality properties.

Contribution

It introduces a novel cohomology theory and spectral sequence framework for obstruction theory of $A$-infinity bimodules, extending previous methods and applying to graded operads.

Findings

Obstructions are located in specific pages of a spectral sequence.

A new cohomology theory relates to Hochschild cohomology and self-extensions.

Results establish criteria for formality and almost formality of bimodules.

Abstract

We develop an obstruction theory for the extension of truncated minimal $A$-infinity bimodule structures over truncated minimal $A$-infinity algebras. Obstructions live in far-away pages of a (truncated) fringed spectral sequence of Bousfield--Kan type. The second page of this spectral sequence is mostly given by a new cohomology theory associated to a pair consisting of a graded algebra and a graded bimodule over it. This new cohomology theory fits in a long exact sequence involving the Hochschild cohomology of the algebra and the self-extensions of the bimodule. We show that the second differential of this spectral sequence is given by the Gerstenhaber bracket with a bimodule analogue of the universal Massey product of a minimal $A$-infinity algebra. We also develop a closely-related obstruction theory for truncated minimal $A$-infinity bimodule structures over (the truncation of) a fixed minimal $A$-infinity algebra; the second page of the corresponding spectral sequence is now mostly given by the vector spaces of self-extensions of the underlying graded bimodule and the second differential is described analogously to the previous one. We also establish variants of the above for graded algebras and graded bimodules that are $d$-sparse, that is they are concentrated in degrees that are multiples of a fixed integer $d\geq1$. These obstruction theories are used to establish intrinsic formality and almost formality theorems for differential graded bimodules over differential graded algebras. Our results hold, more generally, in the context of graded operads with multiplication equipped with an associative operadic ideal, examples of which are the endomorphism operad of a graded algebra and the linear endomorphism operad of a pair consisting of a graded algebra and a graded bimodule over it.