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

Let be a -algebra where a field of arbitrary characteristic, and let A_ be a full subcategory of -Mod, the abelian category of left -modules. Following M. Kleiner and I. Reiten, A_ is Hom-finite if the hom-space between any two objects in A_ is finite-dimensional over. We further say that A_ is Ext-finite if _ⁱ_ (X, Y) < for all objects X and Y in A_. Let V be an object in A_. In this note we prove that if _ (V) is isomorphic to, then V has a universal deformation ring R (, V), which is a local complete Noetherian commutative -algebra whose residue field is also isomorphic to. We use this result to prove that if is a local two-point infinite dimensional gentle -algebra (in the sense of V. Bekkert et al), then R (, V) is isomorphic either to, to \![t\!]/ (t²) or to \![t\!].