Fine and coarse moduli spaces in the representation theory of finite dimensional algebras

TL;DR

This paper explores the use of fine and coarse moduli spaces for classifying modules over finite dimensional algebras, comparing affine and projective approaches with practical examples.

Contribution

It formulates a moduli problem tailored to representation theory and compares two strategies for constructing moduli spaces, highlighting their advantages and limitations.

Findings

Two main approaches: affine and projective varieties for moduli spaces.

Sample results illustrating the effectiveness of each approach.

Discussion of techniques and goals in moduli space construction.

Abstract

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.