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]\!].