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

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 $\mathbf{k}[\![t]\!]/(t^2)$.