Contemporary Mathematics

Volume 607, 2014

http://dx.doi.org/10.1090/conm/607/12086

Fine and coarse moduli spaces in the representation theory

of finite dimensional algebras

B. Huisgen-Zimmermann

Dedicated to Ragnar-Olaf Buchweitz on the occasion of his seventieth birthday

Abstract. We discuss the concepts of fine and coarse moduli spaces in the

context of finite dimensional algebras over algebraically closed fields. In par-

ticular, our formulation of a moduli problem and its potential strong or weak

solution is adapted to classification problems arising in the representation the-

ory 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 sam-

ple 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.

1. Introduction and notation

The desire to describe/classify the objects of various algebro-geometric cate-

gories via collections of invariants is a red thread that can be traced throughout

mathematics. Prominent examples are the classification of similarity classes of

matrices in terms of normal forms, the classification of finitely generated abelian

groups in terms of annihilators of their indecomposable direct summands, and the

classification of varieties of fixed genus and dimension up to isomorphism or bi-

rational equivalence, etc., etc. – the reader will readily extend the list. In each

setting, one selects an equivalence relation on the collection of objects to be sorted;

the “invariants” one uses to describe the objects are quantities not depending on

the choice of representatives from the considered equivalence classes; and the cho-

sen data combine to finite parcels that identify these classes, preferably without

redundancy. In case the relevant parcels of invariants consist of discrete data –

as in the classification of finitely generated abelian groups up to isomorphism for

instance – there is typically no need for additional tools to organize them. By con-

trast, if the objects to be classified involve a base field K and their invariants are

structure constants residing in this field – suppose one has established a one-to-one

correspondence between the equivalence classes of objects and certain points in an

aﬃne or projective space over K – it is natural to ask whether these invariants

trace an algebraic variety over K. In the positive case, one is led to an analysis

The author was partially supported by a grant from the National Science Foundation.

c 2014 American Mathematical Society

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