Classifying representations by way of Grassmannians

Let $\Lambda$ be a finite dimensional algebra over an algebraically closed field. Criteria are given which characterize existence of a fine or coarse moduli space classifying, up to isomorphism, the representations of $\Lambda$ with fixed dimension $d$ and fixed squarefree top $T$. Next to providing a complete theoretical picture, some of these equivalent conditions are readily checkable from quiver and relations. In case of existence of a moduli space -- unexpectedly frequent in light of the stringency of fine classification -- this space is always projective and, in fact, arises as a closed subvariety ${\mathfrak{Grass}}^T_d$ of a classical Grassmannian. Even when the full moduli problem fails to be solvable, the variety ${\mathfrak{Grass}}^T_d$ is seen to have distinctive properties recommending it as a substitute for a moduli space. As an application, a characterization of the algebras having only finitely many representations with fixed simple top is obtained; in this case of `finite local representation type at a given simple $T$', the radical layering $\bigl( J^lM/ J^{l+1}M \bigr)_{l \ge 0}$ is shown to be a classifying invariant for the modules with top $T$. This relies on the following general fact obtained as a byproduct: Proper degenerations of a local module $M$ never have the same radical layering as $M$

The phantom menace in representation theory

Our principal goal in this overview is to explain and motivate the concept of a phantom in the representation theory of a finite dimensional algebra $\Lambda$. In particular, we exhibit the key role of phantoms towards understanding how a full subcategory $\cal A$ of the category $\Lambda\text{-mod}$ of all finitely generated left $\Lambda$-modules is embedded into $\Lambda\text{-mod}$, in terms of maps leaving or entering $\cal A$. Contents: 1. Introduction and prerequisites; 2. Contravariant finiteness and first examples; 3. Homological importance of contravariant finiteness and a model application of the theory; 4. Phantoms. Definitions, existence, and basic properties; 5. An application: Phantoms over string algebras

Top-stable degenerations of finite dimensional representations I

Given a finite dimensional representation $M$ of a finite dimensional algebra, two hierarchies of degenerations of $M$ are analyzed in the context of their natural orders: the poset of those degenerations of $M$ which share the top $M/JM$ with $M$ - here $J$ denotes the radical of the algebra - and the sub-poset of those which share the full radical layering $\bigl(J^lM/J^{l+1}M\bigr)_{l \ge 0}$ with $M$. In particular, the article addresses existence of proper top-stable or layer-stable degenerations - more generally, it addresses the sizes of the corresponding posets including bounds on the lengths of saturated chains - as well as structure and classification

Varieties of uniserial representations IV. Kinship to geometric quotients

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

The geometry of uniserial representations of algebras II. Alternate viewpoints and uniqueness

We provide two alternate settings for a family of varieties modeling the uniserial representations with fixed sequence of composition factors over a finite dimensional algebra. The first is a quasi-projective subvariety of a Grassmannian containing the members of the mentioned family as a principal affine open cover; among other benefits, one derives invariance from this intrinsic description. The second viewpoint re-interprets the `uniserial varieties' as locally closed subvarieties of the traditional module varieties; in particular, it exhibits closedness of the fibres of the canonical maps from the uniserial varieties to the uniserial representations