[Paper Review] Singular deformation theory and the invariance of Gerstenhaber algebra structure on the singular Hochschild cohomology

Zhengfang Wang|arXiv (Cornell University)|Jan 7, 2016

Algebraic structures and combinatorial models3 citations

TL;DR

This paper extends Keller's derived invariance result to the singular setting by developing singular deformation theory, proving that the Gerstenhaber algebra structure on singular Hochschild cohomology is invariant under singular equivalences of Morita type with level. The approach adapts infinitesimal deformation techniques to the singular context, establishing a derived invariant structure for non-smooth algebras.

ABSTRACT

Keller proved in 1999 that the Gerstenhaber algebra structure of the Hochschild cohomology of an algebra is an invariant of the derived category. In this paper, we adapt his approach and develop the singular infinitesimal deformation theory. As a consequence, we show that the Gerstenhaber algebra structure on the singular Hochschild cohomology of an algebra is preserved under singular equivalences of Morita type with level.

Motivation & Objective

• To extend Keller’s derived invariance result for Hochschild cohomology to the singular setting.
• To develop a deformation-theoretic framework for singular Hochschild cohomology.
• To establish that the Gerstenhaber algebra structure on singular Hochschild cohomology is preserved under singular equivalences of Morita type with level.
• To generalize the invariance of algebraic structures from smooth to singular algebras via deformation-theoretic methods.

Proposed method

• Adapting Keller’s deformation-theoretic approach to the singular context using infinitesimal deformations.
• Constructing a singular version of the Hochschild cochain complex that captures singular cohomological data.
• Employing the notion of singular equivalences of Morita type with level to relate algebras with non-isomorphic singular Hochschild cohomologies.
• Using the derived category framework to analyze the behavior of Gerstenhaber brackets under singular equivalences.
• Applying obstruction-theoretic techniques to verify the preservation of algebraic structures in the singular setting.
• Establishing a correspondence between deformation functors and the Gerstenhaber algebra structure on singular Hochschild cohomology.

Experimental results

Research questions

• RQ1 Does the Gerstenhaber algebra structure on singular Hochschild cohomology remain invariant under singular equivalences of Morita type with level?
• RQ2 Can Keller’s deformation-theoretic method for derived invariance be extended to singular algebras?
• RQ3 How do infinitesimal deformations behave in the singular Hochschild cohomology setting?
• RQ4 What conditions ensure that the Gerstenhaber bracket structure is preserved under singular equivalences?
• RQ5 To what extent does the singular Hochschild cohomology retain algebraic structure under derived equivalences?

Key findings

• The Gerstenhaber algebra structure on singular Hochschild cohomology is invariant under singular equivalences of Morita type with level.
• Singular deformation theory provides a viable framework for studying algebraic structures in the singular setting.
• The approach successfully generalizes Keller’s derived invariance result to non-smooth algebras.
• The bracket operation on singular Hochschild cohomology is preserved through the deformation-theoretic correspondence.
• The method establishes a link between singular equivalences and the preservation of higher algebraic structures.
• The results confirm that the singular Hochschild cohomology retains a rich algebraic structure under appropriate derived equivalences.