On Universal Deformation Rings for Gorenstein Algebras

TL;DR

This paper proves that for Gorenstein algebras, certain Cohen-Macaulay modules have universal deformation rings that are complete local Noetherian algebras, and it characterizes these rings for a specific non-self-injective Gorenstein algebra.

Contribution

It establishes the existence of universal deformation rings for Cohen-Macaulay modules over Gorenstein algebras and characterizes these rings for a particular algebra with infinite global dimension.

Findings

Universal deformation rings exist for Cohen-Macaulay modules over Gorenstein algebras.

For a specific Gorenstein algebra, the deformation rings are either trivial or isomorphic to a dual number ring.

The deformation rings are stable under syzygies.

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 $\mathbf{k}[[t]]/(t^2)$.