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

Abstract

Let $\mathbf{k}$ be a field of arbitrary characteristic, let $\Lambda$ be a finite dimensional $\mathbf{k}$-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 $\Lambda$ is Gorenstein and that $\Gamma$ is another Gorenstein $\mathbf{k}$-algebra such that there exists $\ell \ge 0$ 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) between $\Lambda$ and $\Gamma$. Then the left $\Gamma$-module $X{\otimes }_{\Lambda }V$ is also Gorenstein-projective with stable endomorphism ring isomorphic to $\mathbf{k}$ and the universal deformation ring $R(\Gamma ,X{\otimes }_{\Lambda }V)$ is isomorphic to $R(\Lambda ,V)$.

arXiv commentarXiv admin note: text overlap with arXiv:1705.05230