Singular equivalences of Morita type with level, Gorenstein algebras, and universal deformation rings (2023)
Let k be a field of arbitrary characteristic, let Λ be a finite dimensional k-algebra, and let V be an indecomposable finitely generated non-projective Gorenstein-projective left Λ-module whose stable endomorphism ring is isomorphic to k. In this article, we prove that the universal deformation rings R(Λ, V ) and R(Λ, ΩΛV ) are isomorphic, where ΩΛV denotes the first syzygy of V as a left Λ-module. We also prove the following result. Assume that Λ is also Gorenstein and that Γ is another Gorenstein k-algebra such that there exists ` ≥ 0 and a pair of bimodules (ΓXΛ, ΛYΓ) that induces a singular equivalence of Morita type with level ` (as introduced by Z. Wang) between Λ and Γ. Then the left Γ-module X ⊗Λ V is also Gorenstein-projective with stable endomorphism ring isomorphic to k, and the universal deformation ring R(Γ, X ⊗Λ V ) is isomorphic to R(Λ, V ).