On Weak Universal Deformation Rings for Objects of EXT-FINITE Categories of Modules

Let $\text{A}$ be a $\text{k}$-algebra where $\text{k}$ a field of arbitrary characteristic, and let ${\mathcal{A}}_{\text{k}}$ be a full subcategory of $\text{A}$-Mod, the abelian category of left $\text{A}$-modules.Following M. Kleiner and I. Reiten, ${\mathcal{A}}_{\text{k}}$ is {\it Hom-finite} if the hom-space between any two objects in ${\mathcal{A}}_{\text{k}}$ is finite-dimensional over $\text{k}$. We further say that ${\mathcal{A}}_{\text{k}}$ is {\it Ext-finite} if ${\dim }_{\text{k}}{\text{Ext}}_{\text{A}}^i(X,Y)<\mathrm{\infty }$ for all objects $X$ and $Y$ in ${\mathcal{A}}_{\text{k}}$. Let $V$ be an object in ${\mathcal{A}}_{\text{k}}$. In this note we prove that if ${\text{End}}_{\text{A}}(V)$ is isomorphic to $\text{k}$, then $V$ has a universal deformation ring $R(\text{A},V)$, which is a local complete Noetherian commutative $\text{k}$-algebra whose residue field is also isomorphic to $\text{k}$. We use this result to prove that if $\text{A}$ is a local two-point infinite dimensional gentle $\text{k}$-algebra (in the sense of V. Bekkert et al), then $R(\text{A},V)$ is isomorphic either to $\text{k}$, to $\text{k}[\text{\hspace{0.17em}⁣}[t]\text{\hspace{0.17em}⁣}]\mathrm{/}(t^2)$ or to $\text{k}[\text{\hspace{0.17em}⁣}[t]\text{\hspace{0.17em}⁣}]$.