On universal deformation rings and stable equivalences of Gorenstein-projective modules

Abstract

Let $\mathbf{k}$ be a field and let $\Lambda$ and $\Gamma$ finite dimensional $\mathbf{k}$-algebras. Assume that ${}_{\Gamma }{X}_{\Lambda }$ and ${}_{\Lambda }{Y}_{\Gamma }$ are bimodules that define a singular equivalence of Morita type with level (in the sense of Z. Wang) between $\Lambda$ and $\Gamma$ and which also induce an equivalence between the stable categories of finitely generated Gorenstein-projective modules $\Lambda$-$\underline{\text{Gproj}}$ and $\Gamma$-$\underline{\text{Gproj}}$. We prove that if $V$ is an indecomposable object in $\Lambda$-$\underline{\text{Gproj}}$ with ${\underline{\mathrm{End}}}_{\Lambda }(V)\cong \mathbf{k}$, then $X{\otimes }_{\Lambda }V$ is an object in $\Gamma$-$\underline{\text{Gproj}}$ such that ${\underline{\mathrm{End}}}_{\Gamma }(X{\otimes }_{\Lambda }V)\cong \mathbf{k}$ and the universal deformation rings (in the sense of F.M. Bleher and the second author) $R(\Lambda ,V)$ and $R(\Gamma ,X{\otimes }_{\Lambda }V)$ are isomorphic. This result generalizes the one obtained by the second author assuming that $\Lambda$ and $\Gamma$ are Gorenstein $\mathbf{k}$-algebras.