2017/03/24 by José A. Vélez-Marulanda, Velez-Marulanda, Jose A.
Mathematics · #Algebraic structures and combinatorial models #Homotopy and Cohomology in Algebraic Topology #Advanced Topics in Algebra
paper · pdf · doi:10.48550/arxiv.1703.08569
Let \k be field of arbitrary characteristic and let \Λ be a\nfinite dimensional \k-algebra. From results previously obtained by\nF.M Bleher and the author, it follows that if V^ bullet is an object of the\nbounded derived category \Db(\Λ textup-mod) of \Λ,\nthen V^ bullet has a well-defined versal deformation ring R(\Λ,\nV^ bullet), which is complete local commutative Noetherian\n\k-algebra with residue field \k, and which is universal\nprovided that textupHom_\Db(\Λ textup-mod)(V^ bullet,\nV^ bullet)=\k. Let \D_ textupsg(\Λ textup-mod)\ndenote the singularity category of \Λ and assume that V^ bullet is a\nbounded complex whose terms are all finitely generated Gorenstein projective\nleft \Λ-modules. In this article we prove that if\n textupHom_\D_ textupsg(\Λ textup-mod)(V^ bullet,\nV^ bullet)=\k, then the versal deformation ring R(\Λ,\nV^ bullet) is universal. We also prove that certain singular equivalences of\nMorita type (as introduced by X. W. Chen and L. G. Sun) preserve the\nisomorphism class of versal deformation rings of bounded complexes whose terms\nare finitely generated Gorenstein projective \Λ-modules.\n