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A short note on deformations of (strongly) Gorenstein-projective modules over the dual numbers

Jose A. Velez-Marulanda, Hector Suarez

Abstract

Let $\mathbf{k}$ be a field of arbitrary characteristic, and let $Λ$ be a finite dimensional $\mathbf{k}$-algebra. In this short note we prove that if $V$ is a finitely generated strongly Gorenstein-projective left $Λ$-module whose stable endomorphism ring $\underline{\mathrm{End}}_Λ(V)$ is isomorphic to $\mathbf{k}$, then $V$ has an universal deformation ring $R(Λ,V)$ isomorphic to the ring of dual numbers $\mathbf{k}[ε]$ with $ε^2=0$. As a consequence, we obtain the following result. Assume that $Q$ is a finite connected acyclic quiver, let $\mathbf{k} Q$ be the corresponding path algebra and let $Λ= \mathbf{k} Q[ε] = \mathbf{k} Q\otimes_{\mathbf{k}} \mathbf{k}[ε]$. If $V$ is a finitely generated Gorenstein-projective left $Λ$-module with $\underline{\mathrm{End}}_Λ(V)=\mathbf{k}$, then $V$ has an universal deformation ring $R(Λ,V)$ isomorphic to $\mathbf{k}[ε]

A short note on deformations of (strongly) Gorenstein-projective modules over the dual numbers

Abstract

Let be a field of arbitrary characteristic, and let be a finite dimensional -algebra. In this short note we prove that if is a finitely generated strongly Gorenstein-projective left -module whose stable endomorphism ring is isomorphic to , then has an universal deformation ring isomorphic to the ring of dual numbers with . As a consequence, we obtain the following result. Assume that is a finite connected acyclic quiver, let be the corresponding path algebra and let . If is a finitely generated Gorenstein-projective left -module with , then has an universal deformation ring isomorphic to $\mathbf{k}[ε]
Paper Structure (6 sections, 4 theorems, 9 equations)

This paper contains 6 sections, 4 theorems, 9 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Lifts, deformations, and (uni)versal deformation rings
  4. (Strongly) Gorenstein projective modules over the dual numbers
  5. Proof of main results
  6. Data availability statement

Key Result

Theorem 1.1

Let $\Lambda = \Bbbk Q[\epsilon]= \Bbbk Q \otimes_\Bbbk \Bbbk[\epsilon]$ where $\Bbbk Q$ is the path algebra of a finite, connected and acyclic quiver $Q$, and $\Bbbk[\epsilon]$ is the ring of dual numbers with $\epsilon^2=0$. If $V$ is a Gorenstein-projective left $\Lambda$-module with $\underline{

Theorems & Definitions (5)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1
  • Proposition 2.2
  • proof : Proof of Theorem \ref{['thm2']}