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).
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