Homological epimorphisms, recollements and Hochschild cohomology - with a conjecture by Snashall-Solberg in view

TL;DR

This paper investigates how recollements of module categories induce structure-preserving maps between Hochschild cohomology algebras, extending known sequences and proposing a variation of the Snashall-Solberg finite generation conjecture.

Contribution

It demonstrates that recollements induce Gerstenhaber structure-preserving homomorphisms between Hochschild cohomologies and generalizes a long exact sequence to all surjective homological epimorphisms, also proposing a new conjecture.

Findings

Recollements induce Gerstenhaber algebra homomorphisms in Hochschild cohomology.

Generalization of Koenig-Nagase long exact sequence to all surjective homological epimorphisms.

Formulation of a variation of the Snashall-Solberg finite generation conjecture.

Abstract

We show that recollements of module categories give rise to homomorphisms between the associated Hochschild cohomology algebras which preserve the strict Gerstenhaber structure, i.e., the cup product, the graded Lie bracket and the squaring map. We review various long exact sequences in Hochschild cohomology and apply our results in order to realise that the occurring maps preserve the strict Gerstenhaber structure as well. As a byproduct, we generalise a known long exact cohomology sequence of Koenig-Nagase to arbitrary surjective homological epimorphisms. We use our observations to motivate and formulate a variation of the finite generation conjecture by Snashall-Solberg.