On Universal Deformation Rings for Gorenstein Algebras

Abstract

Let $\mathbf{k}$ be an algebraically closed field, and let $\Lambda$ be a finite dimensional $\mathbf{k}$-algebra. We prove that if $\Lambda$ is a Gorenstein algebra, then every finitely generated Cohen-Macaulay $\Lambda$-module $V$ whose stable endomorphism ring is isomorphic to $\mathbf{k}$ has a universal deformation ring $R(\Lambda ,V)$, which is a complete local commutative Noetherian $\mathbf{k}$-algebra with residue field $\mathbf{k}$, and which is also stable under taking syzygies. We investigate a particular non-self-injective Gorenstein algebra ${\Lambda }_0$, which is of infinite global dimension and which has exactly three isomorphism classes of finitely generated indecomposable Cohen-Macaulay ${\Lambda }_0$-modules $V$ whose stable endomorphism ring is isomorphic to $\mathbf{k}$. We prove that in this situation, $R({\Lambda }_0,V)$ is isomorphic either to $\mathbf{k}$ or to $\mathbf{k}[[t]]/(t^2)$.

Major error in one of the proofs

References in corpus (6)

• Singularity categories of gentle algebras
• Universal deformation rings of modules over Frobenius algebras
• Universal deformation rings and generalized quaternion defect groups
• Universal deformation rings and dihedral blocks with two simple modules
• Universal deformation rings for the symmetric group S_5 and one of its double covers
• Universal deformation rings and tame blocks

Cited by in corpus (2)

• Universal Deformation Rings for Complexes over Finite-Dimensional Algebras
• On Singular Equivalences of Morita Type and Universal Deformation Rings for Gorenstein Algebras