Singular Hochschild cohomology via the singularity category

TL;DR

This paper establishes an isomorphism between singular Hochschild cohomology and the Hochschild cohomology of the dg singularity category for certain noetherian algebras, linking these concepts through derived category smoothness.

Contribution

It proves a graded algebra isomorphism connecting singular Hochschild cohomology with the Hochschild cohomology of the dg singularity category, extending recent theoretical insights.

Findings

Isomorphism between singular Hochschild cohomology and dg singularity category cohomology.

Applicable to noetherian algebras with smooth bounded dg derived categories.

Supports recent conjectures in the theory of algebraic singularities.

Abstract

We show that for a noetherian algebra $A$ whose bounded dg derived category is smooth, the singular Hochschild cohomology (=Tate--Hochschild cohomology) is isomorphic, as a graded algebra, to the Hochschild cohomology of the dg singularity category of $A$. The existence of such an isomorphism is suggested by recent work of Zhengfang Wang.