Universal deformation rings of finitely generated Gorenstein-projective modules over finite dimensional algebras

摘要

Let k be a field of arbitrary characteristic, let Lambda be a finite dimensional k-algebra, and let V be a finitely generated Lambda-module. F. M. Bleher and the third author previously proved that V has a well-defined versal deformation ring R(Lambda, V). If the stable endomorphism ring of V is isomorphic to k, they also proved under the additional assumption that Lambda is self-injective that R(Lambda, V) is universal. In this paper, we prove instead that if Lambda is arbitrary but V is Gorenstein-projective then R(Lambda, 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 Lambda 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 A-module has a universal deformation ring that is isomorphic to either k or to k [t]]/(t(2)). (C) 2019 Elsevier B.V. All rights reserved.