CHAPTER 0

Background

In this preliminary chapter we describe the setting and present the background

material from the literature which will be used later. The main parts of this work

will start with Chapter 1. We recommend to browse through this chapter or even

start reading the work with Chapter 1 and look up items here when necessary.

0.1. Notation

We work over an arbitrary field k. If not otherwise specified, all categories will

be k-categories and all functors will be k-functors and covariant. If the isomorphism

classes of objects in a category C form a set, then we call C small. (This is often

called skeletally-small in the literature.) If X is an object in C we write X ∈ C

instead of X ∈ Ob(C).

All rings and algebras are associative with identity. If not otherwise specified,

by modules we mean right modules, and all modules are unitary. The category of

all R-modules is denoted by Mod(R). The full subcategory of finitely presented R-

modules is denoted by mod(R). Since we will only consider noetherian situations,

these are just the finitely generated modules. If R is an algebra graded by an

abelian group H we denote by

ModH

(R) the category of H-graded R-modules; the

morphisms are those of degree zero. The subcategory

modH

(R) is similarly defined

like in the ungraded situation.

0.2. One-parameter families, generic modules and tameness

In this section we briefly recall the notions of one-parameter families and tame-

ness. Although we will not explicitly use these facts later in the text, they serve as

one of the main motivations.

In the representation theory of finite dimensional algebras certain modules often

form sets with geometric structure. By the Tame and Wild Theorem of Drozd [32]

(see also [17]) the indecomposable modules over a non-wild (= tame) finite di-

mensional algebra over an algebraically closed field k essentially lie in rational one-

parameter families, that is, families indexed by (an aﬃne part of) the projective line

P1(k).

(We use the rather unusual notation k in order to stress that temporarily

the field is assumed to be algebraically closed.)

0.2.1. Let A be a finite dimensional algebra over an algebraically closed field

k. Let M be a k[T ]-A-bimodule which is free of finite rank as left k[T ]-module.

Consider the associated functor

FM = − ⊗k[T

]

M : mod(k[T ]) −→ mod(A).

11