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

Abstract

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