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

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.

47 pages. Final version, to appear in "Advances in Mathematics". Rearranged and fixed Section 9, made minor changes and insertions throughout in accordance with the suggestions of the referees

References in corpus (7)

• Support varieties for selfinjective algebras
• Support varieties and the Hochschild cohomology ring modulo nilpotence
• Spectral analysis of finite dimensional algebras and singularities
• Monoidal categories and the Gerstenhaber bracket in Hochschild cohomology
• Extension Fullness of the Categories of Gelfand-Zeitlin and Whittaker Modules
• Some homological properties of category O. III
• Exact sequences, Hochschild cohomology, and the Lie module structure over the $M$-relative center

Cited by in corpus (1)

• $A_{\infty }$-coderivations and the Gerstenhaber bracket on Hochschild cohomology