Lectures on Representations of Quivers by William Crawley-Boevey ...

§7. Euclidean case. Preprojectives and preinjectives
FROM NOW ON we set A=kQ where Q is a quiver without oriented cycles and
with underlying graph Euclidean. We denote by § the minimal positive
imaginary root for . In this section we describe the three classes of
preprojective, regular and preinjective modules.
m£
m£
X£ i¡ ¦
DEFINITIONS. If X is indecomposable, then
(1) X is preprojective
i
X=0 for i>>0 X=
-m
P(j) some 0, j.
(2) X is preinjective
-i
X=0 for i>>0 X=
m
I(j) some 0, j.
(3) X is regular
i
0 for all .
We say a decomposable module X is preprojective, preinjective or regular if
each indecomposable summand is.
The defect of a module X is < § ,dim X> = -.
N
LEMMA 1. There is N>0 such that c dim X = dim X for regular X.
PROOF. Recall that c¤ =¤ if and only if ¤ is radical, and that q(c¤ )=q(¤ ).
Thus c induces a permutation of the finite set ¢ {0}/¦
§
. Thus there is some
N
N>0 with c the identity on
N
¢ § ¢ {0}/¦ ¡
§ /¦ ¦ n
identity on .
. Since it follows that c
i
is the
N iN
Let c dim X - dim X = r § . An induction shows that c dim X = dim X + ir §
for all i¡ ¦ . If r>0, so X must be
preprojective. If r>0 this is not positive for i