Universal Deformation Rings of Finitely Generated Gorenstein-Projective Modules over Finite Dimensional Algebras (2020)
Let k be a field of arbitrary characteristic, let Λ be a finite dimensional k-algebra, and let V be a finitely generated Λ-module. F. M. Bleher and the third author previously proved that V has a well-defined versal deformation ring R(Λ, V ). If the stable endomorphism ring of V is isomorphic to k, they also proved under the additional assumption that Λ is self-injective that R(Λ, V ) is universal. In this paper, we prove instead that if Λ is arbitrary but V is Gorenstein-projective then R(Λ, V ) is also universal when the stable endomorphism ring of V is isomorphic to k. Moreover, we show that singular equivalences of Morita type (as introduced by X. W. Chen and L. G. Sun) preserve the isomorphism classes of versal deformation rings of finitely generated Gorenstein-projective modules over Gorenstein algebras. We also provide examples. In particular, if Λ is a monomial algebra in which there is no overlap (as introduced by X. W. Chen, D. Shen and G. Zhou) we prove that every finitely generated indecomposable Gorenstein-projective Λ-module has a universal deformation ring that is isomorphic to either k or to k[[t]]/(t2).