MEMOIRS

CHAPTER 0BackgroundIn this preliminary chapter we describe the setting and present the backgroundmaterial from the literature which will be used later. The main parts of this workwill start with Chapter 1. We recommend to browse through this chapter or evenstart reading the work with Chapter 1 and look up items here when necessary.0.1. NotationWe work over an arbitrary field k. If not otherwise specified, all categories willbe k-categories and all functors will be k-functors and covariant. If the isomorphismclasses of objects in a category C form a set, then we call C small. (This is oftencalled skeletally-small in the literature.) If X is an object in C we write X ∈Cinstead 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 ofall 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 anabelian group H we denote by Mod H (R) the category of H-graded R-modules; themorphisms are those of degree zero. The subcategory mod H (R) is similarly definedlike in the ungraded situation.0.2. One-parameter families, generic modules and tamenessIn this section we briefly recall the notions of one-parameter families and tameness.Although we will not explicitly use these facts later in the text, they serve asone of the main motivations.In the representation theory of finite dimensional algebras certain modules oftenform 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 dimensionalalgebra over an algebraically closed field k essentially lie in rational oneparameterfamilies, that is, families indexed by (an affine part of) the projective lineP 1 (k). (We use the rather unusual notation k in order to stress that temporarilythe field is assumed to be algebraically closed.)0.2.1. Let A be a finite dimensional algebra over an algebraically closed fieldk. Let M be a k[T ]-A-bimodule which is free of finite rank as left k[T ]-module.Consider the associated functorF M = −⊗ k[T ]M :mod(k[T ]) −→ mod(A).11