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 $\Lambda$, we give a structural description of the modules in the categories ${\cal P}^{<\infty}(\Lambda\text{-mod})$ and ${\cal P}^{<\infty}(\Lambda\text{-Mod})$, consisting of the finitely generated (resp. arbitrary) $\Lambda$-modules of finite projective dimension. We deduce that these categories are contravariantly finite in $\Lambda\text{-mod}$ and $\Lambda\text{-Mod}$, respectively, and determine the corresponding minimal ${\cal P}^{<\infty}$-approximation of an arbitrary $\Lambda$-module from a projective presentation. In particular, we explicitly construct - based on the Gabriel quiver $Q$ and the Loewy length of $\Lambda$ - the basic strong tilting module $_\Lambda T$ (in the sense of Auslander and Reiten) which is coupled with ${\cal P}^{<\infty}(\Lambda\text{-mod})$ in the contravariantly finite case. A main topic is the study of the homological properties of the corresponding tilted algebra $\tilde{\Lambda} = \text{End}_\Lambda(T)^{\text{op}}$, 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 $Q$, the situation where the tilting module $T_{\tilde{\Lambda}}$ is strong over $\tilde{\Lambda}$ as well. In this $\Lambda$-$\tilde{\Lambda}$-symmetric situation, we obtain sharp results on the submodule lattices of the objects in ${\cal P}^{<\infty}(\text{Mod-}\tilde{\Lambda})$, among them a certain heredity property; it entails that any module in ${\cal P}^{<\infty}(\text{Mod-}\tilde{\Lambda})$ is an extension of a projective module by a module all of whose simple composition factors belong to ${\cal P}^{<\infty}(\text{mod-}\tilde{\Lambda})$

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 $\Lambda\text{-mod}$ is given, where $\Lambda$ is a finite dimensional algebra. The criterion is applied to the category ${\cal P}^{\infty}(\Lambda\rm{-mod})$ of all finitely generated $\Lambda$-modules of finite projective dimension, yielding a negative answer to the question whether ${\cal P}^{\infty}(\Lambda\rm{-mod})$ is always covariantly finite in $\Lambda\text{-mod}$. Part (2) concerns contravariant finiteness of ${\cal P}^{\infty}(\Lambda\rm{-mod})$. 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 ${\cal P}^{\infty}(\Lambda\rm{-mod})$ 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 $\Lambda$ 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 $\Lambda$-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 $M$ over an artinian ring $\Lambda$. The repetition index measures the degree of repetitiveness among non-projective direct summands of the syzygies of $M$, while the syzygy type of $M$ measures the number of indecomposable modules among direct summands of the syzygies of $M$. It is proved that if $T$ is a right $\Lambda$-module which contains an isomorphic copy of $\Lambda/J(\Lambda)$, then the left big finitistic dimension of $\Lambda$ is bounded above by the repetition index of $T$, which in turn is bounded above by the syzygy type of $T$. The finite dimensional $K$-algebras $\Lambda = {\cal O}/\pi{\cal O}$, where $\cal O$ is a classical order over a discrete valuation ring $D$ with uniformizing parameter $\pi$ and residue class field $K$, are investigated. It is proved that, if $\text{gl.dim.}\, {\cal O} =d<\infty$, then the global repetition index of $\Lambda$ is $d-1$ and all finitely generated $\Lambda$-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 $Rep(A,\bold d)$ which parametrize the representations with dimension vector $\bold d$ of a finite dimensional algebra $A$. To do so, we move back and forth between $Rep(A,\bold d)$ and a projective variety, $GRASS(A,\bold d)$, 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 $GRASS(A,\bold d)$ 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 $\Lambda$ be a finite dimensional algebra over an algebraically closed field. We exhibit slices of the representation theory of $\Lambda$ that are always classifiable in stringent geometric terms. Namely, we prove that, for any semisimple object $T \in \Lambda\text{-mod}$, the class of those $\Lambda$-modules with fixed dimension vector (say $\bf d$) and top $T$ which do not permit any proper top-stable degenerations possesses a fine moduli space. This moduli space, $\mathfrak{ModuliMax}^T_{\bf d}$, is a projective variety. Despite classifiability up to isomorphism, the targeted collections of modules are representation-theoretically rich: indeed, any projective variety arises as $\mathfrak{ModuliMax}^T_{\bf d}$ for suitable choices of $\Lambda$, $\bf d$, and $T$. 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 $\Lambda$ be a basic finite dimensional algebra over an algebraically closed field, presented as a path algebra modulo relations; further, assume that $\Lambda$ is graded by lengths of paths. The paper addresses the classifiability, via moduli spaces, of classes of graded $\Lambda$-modules with fixed dimension $d$ and fixed top $T$. It is shown that such moduli spaces exist far more frequently than they do for ungraded modules. In the local case (i.e., when $T$ is simple), the graded $d$-dimensional $\Lambda$-modules with top $T$ 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 $\Lambda$. When $T$ is not simple, existence of a coarse moduli space for the graded $d$-dimensional $\Lambda$-modules with top $T$ 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 $\Lambda = KQ/I$, where $Q$ is a quiver, $K$ a coefficient field, and $I \subseteq KQ$ 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 $Q$ and the Loewy length of $\Lambda$. 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 $K$- dimension

Generic representation theory of quivers with relations

The irreducible components of varieties parametrizing the finite dimensional representations of a finite dimensional algebra $\Lambda$ 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 $\mathcal C$, that is, of a module which displays all categorically defined generic properties of the modules parametrized by $\mathcal C$; 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