Singular equivalences of Morita type with level, Gorenstein algebras, and universal deformation rings

TL;DR

This paper investigates the invariance of universal deformation rings of Gorenstein-projective modules under certain singular equivalences of Morita type with level, revealing their stability across related algebras.

Contribution

It proves that universal deformation rings are preserved under singular equivalences of Morita type with level for Gorenstein algebras and modules.

Findings

Universal deformation rings of a module and its syzygy are isomorphic.

Deformation rings are invariant under singular equivalences of Morita type with level.

Gorenstein-projective modules remain Gorenstein-projective under these equivalences.

Abstract

Let $\mathbf{k}$ be a field of arbitrary characteristic, let $\Lambda$ be a finite dimensional $\mathbf{k}$-algebra, and let $V$ be an indecomposable finitely generated non-projective Gorenstein-projective left $\Lambda$-module whose stable endomorphism ring is isomorphic to $\mathbf{k}$. In this article, we prove that the universal deformation rings $R(\Lambda,V)$ and $R(\Lambda,\Omega_\Lambda V)$ are isomorphic, where $\Omega_\Lambda V$ denotes the first syzygy of $V$ as a left $\Lambda$-module. We also prove the following result. Assume that $\Lambda$ is Gorenstein and that $\Gamma$ is another Gorenstein $\mathbf{k}$-algebra such that there exists $\ell \geq 0$ and a pair of bimodules $({_\Gamma}X_\Lambda, {_\Lambda}Y_\Gamma)$ that induces a singular equivalence of Morita type with level $\ell$ (as introduced by Z. Wang) between $\Lambda$ and $\Gamma$. Then the left $\Gamma$-module $X\otimes_\Lambda V$ is also Gorenstein-projective with stable endomorphism ring isomorphic to $\mathbf{k}$ and the universal deformation ring $R(\Gamma, X\otimes_\Lambda V)$ is isomorphic to $R(\Lambda, V)$.