44 research outputs found

### 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