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

§6. More homological algebra
FROM NOW ON we suppose that Q is a quiver without oriented cycles, so the
path algebra A=kQ is finite dimensional. We still consider f.d. A-modules.
We study the properties of projective, injective, non-projective, and
non-injective modules. We give a little bit of Auslander-Reiten theory.
DUALITIES.
(1) If X is a left or right A-module, then DX = Hom (X,k), Hom(X,A) and
k
1
Ext (X,A) are all A-modules on the other side.
(2) D is duality between left and right A-modules.
PROOF. Hom(X,Y) Hom(DY,DX) and DDX X.
(3) D gives a duality between injective left modules and projective right
modules.
1 1
PROOF. Ext (DX,DY) Ext (Y,X). This is zero for all Y if and only if DX is
projective, if and only if X is injective.
(4) Hom(-,A) gives a duality between projective left modules and projective
right modules.
n n n
PROOF. If P is a summand of A then Hom(P,A) is a summand of Hom(A ,A) A ,
so is projective. Now the map P ¡ Hom(Hom(P,A),A) is an iso for all P,
since it is for P=A.
(5) The Nakayama functor (-) = DHom(-,A) gives an equivalence from
projective left modules to injective left modules. The inverse functor is
-
(-) = Hom(D(-),A) Hom(DA,-).
(6) Hom(X, P) DHom(P,X) for X,P left A-modules, P projective.
PROOF. The composition
Hom(P,A) X Hom(P,A) Hom(A,X) ¡ Hom(P,X)
A A
is an isomorphism, since it is for P=A. Thus
DHom(P,X) Hom (Hom(P,A) X,k) Hom(X,Hom (Hom(P,A),k)) = Hom(X, P).
k A k
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