Singular equivalences of Morita type with level, Gorenstein algebras, and universal deformation rings

Let $mathbf{k}$ be a field of arbitrary characteristic, let $Lambda$ be a Gorenstein $mathbf{k}$-algebra, and let $V$ be an indecomposable finitely generated non-projective Gorenstein-projective left $Lambda$-module whose stable endomorphism ring is isomorphic to $mathbf{k}$. In this article, we prove that the universal deformation rings $R(Lambda,V)$ and $R(Lambda,Omega_Lambda V)$ are isomorphic, where $Omega_Lambda V$ denotes the first syzygy of $V$ as a left $Lambda$-module. We also prove the following result. Assume that $Gamma$ is another Gorenstein $mathbf{k}$-algebra such that there exists $ell geq 0$ and a pair of bimodules $({_Gamma}X_Lambda, {_Lambda}Y_Gamma)$ that induces a singular equivalence of Morita type with level $ell$ (as introduced by Z. Wang). Then the left $Gamma$-module $Xotimes_Lambda V$ is also Gorenstein-projective and the universal deformation rings $R(Gamma, Xotimes_Lambda V)$ and $R(Lambda, V)$ are isomorphic.