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

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.