Iterated extensions in module categories

TL;DR

This paper explores the structure of certain length categories of modules over an algebra, using iterated extensions to connect them with noncommutative deformations and providing new constructive characterizations.

Contribution

It introduces new methods to analyze Mod(F) via iterated extensions and offers a constructive proof of uniserial length category characterization.

Findings

Relates length categories to noncommutative deformations

Provides a new constructive proof for uniserial categories

Explicitly describes categories of holonomic D-modules

Abstract

Let k be an algebraically closed field, let R be an associative k-algebra, and let F = {M_a: a in I} be a family of orthogonal points in R-Mod such that End_R(M_a) = k for all a in I. Then Mod(F), the minimal full sub-category of R-Mod which contains F and is closed under extensions, is a full exact Abelian subcategory of R-Mod and a length category in the sense of Gabriel. In this paper, we use iterated extensions to relate the length category Mod(F) to noncommutative deformations of modules, and use some new methods to study Mod(F) via iterated extensions. In particular, we give a new proof of the characterization of uniserial length categories, which is constructive. As an application, we give an explicit description of some categories of holonomic and regular holonomic D-modules on curves which are uniserial length categories.