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Universal deformation rings and derived equivalences

Shengyong Pan

TL;DR

The paper addresses the invariance of versal deformation rings under stable functors arising from derived and Morita-type equivalences for Gorenstein-projective modules over finite dimensional $k$-algebras. It develops a stable functor framework for non-negative derived functors and applies it to deformation theory, proving isomorphisms $R(A,V)\, ext{≅}\, R(B,ar{F}(V))$ when transferring through derived equivalences, and extends these results to stable equivalences of Morita type. The main contributions include generalizing Veléz-Marulanda-type results to non-Gorenstein and singular Morita-type settings, and establishing preservation of versal deformation rings under stable Morita-type equivalences. Overall, the work strengthens deformation-theoretic invariants across a broad class of equivalences, enabling transfer of deformation information between derived- or stably-equivalent algebras.

Abstract

In this paper, we show that stable functors of derived equivalences preserve the isomorphism classes of versal deformation rings of finitely generated Gorenstein-projective modules over finite dimensional $k$-algebras. Then we generalize Veléz-Marulanda's result \cite{V} in the case of singular equivalences of Morita type with levels for Gorenstein algebras. Moreover, we also prove that stable equivalences of Morita type preserve the isomorphism classes of versal deformation rings of finitely generated Gorenstein-projective modules over finite dimensional $k$-algebras.

Universal deformation rings and derived equivalences

TL;DR

The paper addresses the invariance of versal deformation rings under stable functors arising from derived and Morita-type equivalences for Gorenstein-projective modules over finite dimensional -algebras. It develops a stable functor framework for non-negative derived functors and applies it to deformation theory, proving isomorphisms when transferring through derived equivalences, and extends these results to stable equivalences of Morita type. The main contributions include generalizing Veléz-Marulanda-type results to non-Gorenstein and singular Morita-type settings, and establishing preservation of versal deformation rings under stable Morita-type equivalences. Overall, the work strengthens deformation-theoretic invariants across a broad class of equivalences, enabling transfer of deformation information between derived- or stably-equivalent algebras.

Abstract

In this paper, we show that stable functors of derived equivalences preserve the isomorphism classes of versal deformation rings of finitely generated Gorenstein-projective modules over finite dimensional -algebras. Then we generalize Veléz-Marulanda's result \cite{V} in the case of singular equivalences of Morita type with levels for Gorenstein algebras. Moreover, we also prove that stable equivalences of Morita type preserve the isomorphism classes of versal deformation rings of finitely generated Gorenstein-projective modules over finite dimensional -algebras.
Paper Structure (7 sections, 20 theorems, 46 equations)

This paper contains 7 sections, 20 theorems, 46 equations.

Table of Contents

  1. Introduction
  2. The stable functor of a non-negative functor
  3. Non-negative functors
  4. Gorenstein projective objects under the stable functor
  5. Lifts, deformations, and (uni)versal deformation rings
  6. Derived equivalences and nice two-sided tilting complexes
  7. Stable equivalences of Morita type

Key Result

Theorem 1.1

(=Theorem 4.4) Let $A$ be a finite dimensional $k$-algebra and let $V$ be a non-zero finitely generated Gorenstein-projective $A$-module.

Theorems & Definitions (30)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Lemma 2.2
  • Proposition 2.3
  • Lemma 2.4
  • Proposition 2.5
  • Theorem 2.6
  • Corollary 2.7
  • Corollary 2.8
  • ...and 20 more