Notation and general preliminary results
In this chapter we try to make this paper as selfcontained as possible, as we
attempt to include all the different notation, definitions and preliminary results
that we need in the succeeding chapters.
We start by recalling some notions, definitions and results from the represen-
tation theory of artin algebras. For a general reference on this we refer the reader
A quiver is an oriented graph T, where To denotes the set of vertices and Ti
denotes the set of arrows. A path in the quiver T is an ordered sequence of arrows
anan-i • • • ai or v for some v in TQ, where the arrow ai ends where the arrow a^+i
starts for i = 1,2,... , n — 1. The paths v for v in FQ are called the trivial paths.
Now, let k be any field. Let kT denote the vector space over k with the paths of T
as a fc-basis. The path algebra of T over k is kT, where the multiplication is defined
as follows. It is enough to define the multiplication of two paths p and q. We define
if q ends where p starts and p and q are both nontrivial paths,
if q — v for the vertex v where p starts in T,
if p = v for the vertex v where q ends in T,
A finite dimensional algebra A over a field k is said to be basic if A ~ &i=\Pi for
some indecomposable projective A-modules Pi with Pi gk Pj for i ^ j .
It is known that for a finite dimensional basic algebra A over an algebraically
closed field /c, there exists a finite quiver T and an ideal / in kT, such that A ~ kT/I.
Moreover, the quiver and the ideal / may be chosen such that I contains some power
of the ideal J in kT generated by the arrows in T and / is contained in J
. An ideal
/ satisfying these conditions is called an admissible ideal. The radical of kT/I is
J jI and kT / J ~
where n is the number of vertices in T.
The following is a list of definitions related to subcategories of the category
mod£ of finitely generated (left) E-modules for an artin algebra E.
0.1. Let E be an artin algebra, and let X be a full subcategory of
(a) The subcategory X is called extension closed if for all exact sequence 0 —»
A -* B -+ C — • 0 with A and C in X, the module B is in X.
(b) The subcategory X is called resolving if X is closed under kernels of epi-
morphisms and extensions and X contains the finitely generated projective
(c) The full subcategory X consists of all modules C in mod E such that there
exists an exact sequence 0 — • Xn — • X
_i —••••—» X2 — • X\ — » XQ —
C — 0 for some n and with Xi in X for all i 0.