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

TL;DR

This paper investigates universal deformation rings for objects in Ext-finite categories of modules over a $ ext{K}$-algebra, establishing their existence under certain conditions and computing explicit forms for specific gentle algebras.

Contribution

It proves the existence of universal deformation rings for objects with trivial endomorphism rings in Ext-finite categories and explicitly describes these rings for certain gentle algebras.

Findings

Universal deformation rings exist for objects with endomorphism ring isomorphic to $ ext{K}$.

Explicit forms of deformation rings are classified for local two-point gentle algebras.

Deformation rings are either trivial, first-order, or complete local power series rings.

Abstract

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