Universal Deformation Rings of Finitely Generated Gorenstein-Projective Modules over Finite Dimensional Algebras

TL;DR

This paper proves that for Gorenstein-projective modules over finite dimensional algebras, the versal deformation ring is universal under certain conditions and shows that these rings are preserved under singular equivalences of Morita type, with specific examples provided.

Contribution

It extends the universality of deformation rings to Gorenstein-projective modules over arbitrary finite dimensional algebras and demonstrates their invariance under singular equivalences of Morita type.

Findings

Universal deformation rings are isomorphic to  or [ t ]/(t^2) for certain Gorenstein algebras.

Singular equivalences of Morita type preserve isomorphism classes of versal deformation rings.

Every indecomposable Gorenstein-projective module over specific monomial algebras has a universal deformation ring of the described form.

Abstract

Let $\mathbf{k}$ be a field of arbitrary characteristic, let $\Lambda$ be a finite dimensional $\mathbf{k}$-algebra, and let $V$ be a finitely generated $\Lambda$-module. F. M. Bleher and the third author previously proved that $V$ has a well-defined versal deformation ring $R(\Lambda,V)$. If the stable endomorphism ring of $V$ is isomorphic to $\mathbf{k}$, they also proved under the additional assumption that $\Lambda$ is self-injective that $R(\Lambda,V)$ is universal. In this paper, we prove instead that if $\Lambda$ is arbitrary but $V$ is Gorenstein-projective then $R(\Lambda,V)$ is also universal when the stable endomorphism ring of $V$ is isomorphic to $\mathbf{k}$. Moreover, we show that singular equivalences of Morita type (as introduced by X. W. Chen and L. G. Sun) preserve the isomorphism classes of versal deformation rings of finitely generated Gorenstein-projective modules over Gorenstein algebras. We also provide examples. In particular, if $\Lambda$ is a monomial algebra in which there is no overlap (as introduced by X. W. Chen, D. Shen and G. Zhou) we prove that every finitely generated indecomposable Gorenstein-projective $\Lambda$-module has a universal deformation ring that is isomorphic to either $\mathbf{k}$ or to $\mathbf{k}[\![t]\!]/(t^2)$.