Varieties of uniserial representations IV. Kinship to geometric quotients

Abstract:

Let $\Lambda$ be a finite dimensional algebra over an algebraically closed field, and ${\mathbb {S}}$ a finite sequence of simple left $\Lambda$-modules. Quasiprojective subvarieties of Grassmannians, distinguished by accessible affine open covers, were introduced by the authors for use in classifying the uniserial representations of $\Lambda$ having sequence ${\mathbb {S}}$ 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 (very large) classical affine varieties $\operatorname {Mod-Uni} ({\mathbb {S}})$ parametrizing the pertinent uniserial representations, modulo the usual conjugation action of the general linear group. We show that, 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 quasi-projective, respectively affine, uniserial varieties are embedded. For that purpose, a general correspondence is established, between Grassmannian varieties of submodules of a projective module $P$ on one hand, and classical varieties of factor modules of $P$ on the other. Our findings are applied towards the third objective, concerning the existence of geometric quotients. The main results are then 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.