2017/05/11 by Viktor Bekkert, Bekkert, Viktor, Hernán Giraldo +3
Mathematics · #Algebraic structures and combinatorial models #Commutative Algebra and Its Applications #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.1705.05230
openalex publication_date 2017/05/11 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28
Let \k be a field of arbitrary characteristic, let \Λ be a\nfinite dimensional \k-algebra, and let V be a finitely generated\n\Λ-module. F. M. Bleher and the third author previously proved that V\nhas a well-defined versal deformation ring R(\Λ,V). If the stable\nendomorphism ring of V is isomorphic to \k, they also proved under\nthe additional assumption that \Λ is self-injective that R(\Λ,V)\nis universal. In this paper, we prove instead that if \Λ is arbitrary\nbut V is Gorenstein-projective then R(\Λ,V) is also universal when the\nstable endomorphism ring of V is isomorphic to \k. Moreover, we\nshow that singular equivalences of Morita type (as introduced by X. W. Chen and\nL. G. Sun) preserve the isomorphism classes of versal deformation rings of\nfinitely generated Gorenstein-projective modules over Gorenstein algebras. We\nalso provide examples. In particular, if \Λ is a monomial algebra in\nwhich there is no overlap (as introduced by X. W. Chen, D. Shen and G. Zhou) we\nprove that every finitely generated indecomposable Gorenstein-projective\n\Λ-module has a universal deformation ring that is isomorphic to either\n\k or to \k[ ![t] !]/(t2).\n