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

CONSEQUENCES.
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(1) If X is a left A-module, then proj.dim ¤ X 1, ie Ext (X,Y)=0 ¡ 2. Y,i£
PROOF. f and g are A-module maps and Ae V is isomorphic to the direct sum
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of dim V copies of Ae , so is a projective left A-module. Thus the standard
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resolution is a projective resolution for X.
(2) A is hereditary, ie if X P with P projective, then X is projective.
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PROOF. Ext (X,Y) Ext (P/X,Y) = 0 ¡ Y.
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(3) If X,Y are f.d., then dim Hom(X,Y) - dim Ext (X,Y) = .
PROOF. Apply Hom(-,Y) to the standard resolution:
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0 ¡ Hom(X,Y) ¡ Hom(¥ Ae e X,Y) ¡ Hom(¥ Ae e X,Y) ¡ Ext (X,Y) ¡ 0.
i i k i ¢ t(¢ ) k s(¢ )
Now dim Hom(Ae e X,Y) = (dim e X)(dim Hom(Ae ,Y)) = (dim X) (dim Y) .
i j j i j i
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(4) If X is f.d., then dim End(X) - dim Ext (X,X) = q(dim X).
PROOF. Put X=Y in (3).
REMARK.
Let i be a vertex in Q and suppose that either no arrows start at i, or no
arrows terminate at i. Let Q¥ be the quiver obtained by reversing the
direction of every arrow connected to i. We say that Q¥ is obtained from Q
be reflecting at the vertex i. The two categories Rep(Q) and Rep(Q¥ ) are
closely related, by means of so-called reflection functors. See
I.N.Bernstein, I.M.Gelfand and V.A.Ponomarev, Coxeter functors and
Gabriel’s Theorem, Uspekhi Mat. Nauk. 28 (1973), 19-33, English Translation
Russ. Math. Surveys, 28 (1973), 17-32.
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