On Singular Equivalences of Morita Type and Universal Deformation Rings for Gorenstein Algebras

TL;DR

This paper investigates how universal deformation rings of modules over Gorenstein algebras are preserved under singular equivalences of Morita type, extending understanding of deformation theory in algebra.

Contribution

It proves that the isomorphism class of versal deformation rings remains invariant under singular equivalences of Morita type for Gorenstein algebras.

Findings

Versal deformation rings are well-defined for modules over finite-dimensional algebras.

Under certain conditions, these rings are universal.

The main result shows invariance of deformation rings under singular equivalences of Morita type.

Abstract

Let $\Lambda$ be a finite-dimensional algebra over a fixed algebraically closed field $\mathbf{k}$ of arbitrary characteristic, and let $V$ be a finitely generated $\Lambda$-module. It follows from results previously obtained by F.M. Bleher and the third author that $V$ has a well-defined versal deformation ring $R(\Lambda, V)$, which is a complete local commutative Noetherian $\mathbf{k}$-algebra with residue field $\mathbf{k}$. The third author also proved that if $\Lambda$ is a Gorenstein $\mathbf{k}$-algebra and $V$ is a Cohen-Macaulay $\Lambda$-module whose stable endomorphism ring is isomorphic to $\mathbf{k}$, then $R(\Lambda, V)$ is universal. In this article we prove that the isomorphism class of a versal deformation ring is preserved under singular equivalence of Morita type between Gorenstein $\mathbf{k}$-algebras.