Generic representation theory of quivers with relations

TL;DR

This paper develops a theory of generic modules for finite dimensional algebras, describing their existence, uniqueness, and properties, especially for algebras of wild type and truncated path algebras.

Contribution

It establishes the existence and uniqueness of generic modules for irreducible components of representation varieties, with explicit constructive methods for certain classes of algebras.

Findings

Existence and uniqueness of generic modules for irreducible components

Constructive description of generic modules via minimal projective resolutions

Specialized results for truncated path algebras

Abstract

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.