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

Abstract

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 ℓ≥0 and a pair of bimodules (XΛΓ,YΓΛ) that induces a singular equivalence of Morita type with level ℓ (as introduced by Z. Wang) between Λ and Γ. Then the left Γ-module X⊗ΛV is also Gorenstein-projective with stable endomorphism ring isomorphic to k, and the universal deformation ring R(Γ,X⊗ΛV) is isomorphic to R(Λ,V).