53 research outputs found
Fine and coarse moduli spaces in the representation theory of finite dimensional algebras
We discuss the concepts of fine and coarse moduli spaces in the context of
finite dimensional algebras over algebraically closed fields. In particular,
our formulation of a moduli problem and its potential strong or weak solution
is adapted to classification problems arising in the representation theory of
such algebras. We then outline and illustrate a dichotomy of strategies for
concrete applications of these ideas. One method is based on the classical
affine variety of representations of fixed dimension, the other on a projective
variety parametrizing the same isomorphism classes of modules. We state sample
results and give numerous examples to exhibit pros and cons of the two lines of
approach. The juxtaposition highlights differences in techniques and attainable
goals
Strongly tilting truncated path algebras
For any truncated path algebra , we give a structural description of
the modules in the categories and
, consisting of the finitely generated
(resp. arbitrary) -modules of finite projective dimension. We deduce
that these categories are contravariantly finite in and
, respectively, and determine the corresponding minimal
-approximation of an arbitrary -module from a
projective presentation. In particular, we explicitly construct - based on the
Gabriel quiver and the Loewy length of - the basic strong tilting
module (in the sense of Auslander and Reiten) which is coupled
with in the contravariantly finite
case. A main topic is the study of the homological properties of the
corresponding tilted algebra , such as its finitistic dimensions and the
structure of its modules of finite projective dimension. In particular, we
characterize, in terms of a straightforward condition on , the situation
where the tilting module is strong over
as well. In this --symmetric situation, we obtain
sharp results on the submodule lattices of the objects in , among them a certain heredity
property; it entails that any module in is an extension of a projective
module by a module all of whose simple composition factors belong to
Co- Versus Contravariant Finiteness of Categories of Representations
This article supplements recent work of the authors. (1) A criterion for
failure of covariant finiteness of a full subcategory of
is given, where is a finite dimensional algebra. The criterion is
applied to the category of all finitely
generated -modules of finite projective dimension, yielding a negative
answer to the question whether is always
covariantly finite in . Part (2) concerns contravariant
finiteness of . An example is given where
this condition fails, the failure being, however, curable via a sequence of
one-point extensions. In particular, this example demonstrates that curing
failure of contravariant finiteness of
usually involves a tradeoff with respect to other desirable qualities of the
algebra
Understanding finite dimensional representations generically
We survey the development and status quo of a subject best described as
"generic representation theory of finite dimensional algebras", which started
taking shape in the early 1980s. Let be a finite dimensional algebra
over an algebraically closed field. Roughly, the theory aims at (a) pinning
down the irreducible components of the standard parametrizing varieties for the
-modules with a fixed dimension vector, and (b) assembling generic
information on the modules in each individual component, that is, assembling
data shared by all modules in a dense open subset of that component.
We present an overview of results spanning the spectrum from hereditary
algebras through the tame non-hereditary case to wild non-hereditary algebras.Comment: To appear in "Geometric and topological aspects of group
representations" (J. Carlson, S. Iyengar, and J. Pevtsova, Eds.
Repetitive resolutions over classical orders and finite dimensional algebras
Repetitiveness in projective and injective resolutions and its influence on
homological dimensions are studied. Some variations on the theme of
repetitiveness are introduced, and it is shown that the corresponding
invariants lead to very good -- and quite accessible -- upper bounds on various
finitistic dimensions in terms of individual modules. These invariants are the
`repetition index' and the `syzygy type' of a module over an artinian ring
. The repetition index measures the degree of repetitiveness among
non-projective direct summands of the syzygies of , while the syzygy type of
measures the number of indecomposable modules among direct summands of the
syzygies of . It is proved that if is a right -module which
contains an isomorphic copy of , then the left big
finitistic dimension of is bounded above by the repetition index of
, which in turn is bounded above by the syzygy type of .
The finite dimensional -algebras , where
is a classical order over a discrete valuation ring with
uniformizing parameter and residue class field , are investigated. It
is proved that, if , then the global
repetition index of is and all finitely generated
-modules have finite syzygy type. Various examples illustrating the
results are presented
Irreducible components of module varieties: projective equations and rationality
We expand the existing arsenal of methods for exploring the irreducible
components of the varieties which parametrize the
representations with dimension vector of a finite dimensional algebra
. To do so, we move back and forth between and a projective
variety, , parametrizing the same set of isomorphism classes
of modules. In particular, we show the irreducible components to be accessible
in a highly compressed format within the projective setting. Our results
include necessary and sufficient conditions for unirationality, smoothness, and
normality, followed by applications. Moreover, they provide equational access
to the irreducible components of and techniques for deriving
qualitative information regarding both the affine and projective scenarios.Comment: 27 pages; 9 diagrams in xypic. Section 5 slightly shortened. To
appear in Contemp. Mat
Top-stable degenerations of finite dimensional representations II
Let be a finite dimensional algebra over an algebraically closed
field. We exhibit slices of the representation theory of that are
always classifiable in stringent geometric terms. Namely, we prove that, for
any semisimple object , the class of those
-modules with fixed dimension vector (say ) and top which
do not permit any proper top-stable degenerations possesses a fine moduli
space. This moduli space, , is a projective
variety. Despite classifiability up to isomorphism, the targeted collections of
modules are representation-theoretically rich: indeed, any projective variety
arises as for suitable choices of ,
, and . In tandem, we give a structural characterization of the
finite dimensional representations that have no proper top-stable
degenerations
Moduli spaces of graded representations of finite dimensional algebras
Let be a basic finite dimensional algebra over an algebraically
closed field, presented as a path algebra modulo relations; further, assume
that is graded by lengths of paths. The paper addresses the
classifiability, via moduli spaces, of classes of graded -modules with
fixed dimension and fixed top . It is shown that such moduli spaces
exist far more frequently than they do for ungraded modules. In the local case
(i.e., when is simple), the graded -dimensional -modules with
top always possess a fine moduli space which classifies these modules up to
graded-isomorphism; moreover, this moduli space is a projective variety with a
distinguished affine cover that can be constructed from quiver and relations of
. When is not simple, existence of a coarse moduli space for the
graded -dimensional -modules with top forces these modules to
be direct sums of local modules; under the latter condition, a finite
collection of isomorphism invariants of the modules in question yields a
partition into subclasses, each of which has a fine moduli space (again
projective) parametrizing the corresponding graded-isomorphism classes
Truncated path algebras are homologically transparent
It is shown that path algebras modulo relations of the form ,
where is a quiver, a coefficient field, and the ideal
generated by all paths of a given length, can be readily analyzed
homologically, while displaying a wealth of phenomena. In particular, the
syzygies of their modules, and hence their finitistic dimensions, allow for
smooth descriptions in terms of and the Loewy length of . The same
is true for the distributions of projective dimensions attained on the
irreducible components of the standard parametrizing varieties for the modules
of fixed - dimension
Generic representation theory of quivers with relations
The irreducible components of varieties parametrizing the finite dimensional
representations of a finite dimensional algebra are explored, with
regard to both their geometry and the structure of the modules they encode.
Provided that the base field is algebraically closed and has infinite
transcendence degree over its prime field, we establish the existence and
uniqueness (not up to isomorphism, but in a strong sense to be specified) of a
generic module for any irreducible component , that is, of a module
which displays all categorically defined generic properties of the modules
parametrized by ; the crucial point of the existence statement - a
priori almost obvious - lies in the description of such a module in a format
accessible to representation-theoretic techniques. Our approach to generic
modules over path algebras modulo relations, by way of minimal projective
resolutions, is largely constructive. It is explicit for large classes of
algebras of wild type. We follow with an investigation of the properties of
such generic modules in terms of quiver and relations. The sharpest specific
results on all fronts are obtained for truncated path algebras, that is, for
path algebras of quivers modulo ideals generated by all paths of a fixed
length; this class of algebras extends the comparatively thoroughly studied
hereditary case, displaying many novel features
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