On Universal Deformation Rings for Gorenstein Algebras

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