Varieties of uniserial representations IV. Kinship to geometric quotients

TL;DR

This paper studies algebraic varieties related to uniserial representations of finite dimensional algebras, establishing their role as approximations to geometric quotients and providing criteria for the existence of such quotients, with applications in representation theory.

Contribution

It proves that certain varieties are good approximations to geometric quotients of uniserial representations and characterizes when these quotients exist, enhancing understanding of finite uniserial type algebras.

Findings

Varieties serve as approximations to classical geometric quotients.

Criteria established for the existence of geometric quotients.

Applications include a geometric characterization of finite uniserial type algebras.

Abstract

Let $\Lambda$ be a finite dimensional algebra over an algebraically closed field, and ${\Bbb S}$ a finite sequence of simple left $\Lambda$-modules. In [6, 9], quasiprojective algebraic varieties with accessible affine open covers were introduced, for use in classifying the uniserial representations of $\Lambda$ having sequence ${\Bbb S}$ of consecutive composition factors. Our principal objectives here are threefold: One is to prove these varieties to be `good approximations' -- in a sense to be made precise -- to geometric quotients of the classical varieties $\operatorname{Mod-Uni}({\Bbb S})$ parametrizing the pertinent uniserial representations, modulo the usual conjugation action of the general linear group. To some extent, this fills the information gap left open by the frequent non-existence of such quotients. A second goal is that of facilitating the transfer of information among the `host' varieties into which the considered uniserial varieties can be embedded. These tools are then applied towards the third objective, concerning the existence of geometric quotients: We prove that $\operatorname{Mod-Uni}({\Bbb S})$ has a geometric quotient by the $GL$-action precisely when the uniserial variety has a geometric quotient modulo a certain natural algebraic group action, in which case the two quotients coincide. Our main results are exploited in a representation-theoretic context: Among other consequences, they yield a geometric characterization of the algebras of finite uniserial type which supplements existing descriptions, but is cleaner and more readily checkable.