Lectures on Representations of Quivers by William Crawley-Boevey ...
Lectures on Representations of Quivers by William Crawley-Boevey ...
Lectures on Representations of Quivers by William Crawley-Boevey ...
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CONSEQUENCES.<br />
i<br />
(1) If X is a left A-module, then proj.dim ¤ X 1, ie Ext (X,Y)=0 ¡ 2. Y,i£<br />
PROOF. f and g are A-module maps and Ae V is isomorphic to the direct sum<br />
i<br />
<strong>of</strong> dim V copies <strong>of</strong> Ae , so is a projective left A-module. Thus the standard<br />
i<br />
resoluti<strong>on</strong> is a projective resoluti<strong>on</strong> for X.<br />
(2) A is hereditary, ie if X P with P projective, then X is projective.<br />
1 2<br />
PROOF. Ext (X,Y) Ext (P/X,Y) = 0 ¡ Y.<br />
1<br />
(3) If X,Y are f.d., then dim Hom(X,Y) - dim Ext (X,Y) = .<br />
PROOF. Apply Hom(-,Y) to the standard resoluti<strong>on</strong>:<br />
1<br />
0 ¡ Hom(X,Y) ¡ Hom(¥ Ae e X,Y) ¡ Hom(¥ Ae e X,Y) ¡ Ext (X,Y) ¡ 0.<br />
i i k i ¢ t(¢ ) k s(¢ )<br />
Now dim Hom(Ae e X,Y) = (dim e X)(dim Hom(Ae ,Y)) = (dim X) (dim Y) .<br />
i j j i j i<br />
1<br />
(4) If X is f.d., then dim End(X) - dim Ext (X,X) = q(dim X).<br />
PROOF. Put X=Y in (3).<br />
REMARK.<br />
Let i be a vertex in Q and suppose that either no arrows start at i, or no<br />
arrows terminate at i. Let Q¥ be the quiver obtained <strong>by</strong> reversing the<br />
directi<strong>on</strong> <strong>of</strong> every arrow c<strong>on</strong>nected to i. We say that Q¥ is obtained from Q<br />
be reflecting at the vertex i. The two categories Rep(Q) and Rep(Q¥ ) are<br />
closely related, <strong>by</strong> means <strong>of</strong> so-called reflecti<strong>on</strong> functors. See<br />
I.N.Bernstein, I.M.Gelfand and V.A.P<strong>on</strong>omarev, Coxeter functors and<br />
Gabriel’s Theorem, Uspekhi Mat. Nauk. 28 (1973), 19-33, English Translati<strong>on</strong><br />
Russ. Math. Surveys, 28 (1973), 17-32.<br />
8