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

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CONSEQUENCES. i (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 i of dim V copies of Ae , so is a projective left A-module. Thus the standard i resolution is a projective resolution for X. (2) A is hereditary, ie if X P with P projective, then X is projective. 1 2 PROOF. Ext (X,Y) Ext (P/X,Y) = 0 ¡ Y. 1 (3) If X,Y are f.d., then dim Hom(X,Y) - dim Ext (X,Y) = . PROOF. Apply Hom(-,Y) to the standard resolution: 1 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 1 (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. 8
§2. Bricks In this section we consider finite dimensional left A-modules with A an hereditary k-algebra. In particular the results hold when A is a path algebra. We recall the Happel-Ringel Lemma and another lemma due to Ringel. INDECOMPOSABLE MODULES. Recall Fitting’s Lemma, that X is indecomposable End(X) is a local ring, ie End(X) = k1 +rad End(X), since the field k is algebraically closed. X Any module can be written as a direct sum of indecomposable modules, and by the Krull-Schmidt Theorem the isomorphism types of the summands and their multiplicities are uniquely determined. We say that X is a brick if End(X)=k. Thus a brick is indecomposable. 1 LEMMA 1. Suppose X,Y are indecomposable. If Ext (Y,X)=0 then any non-zero map ¤ :X ¡ Y is mono or epi. PROOF. We have exact sequences ¢ :0 ¡ Im(¤ ) ¡ Y ¡ Cok(¤ ) ¡ 0 and :0 ¡ Ker(¤ ) ¡ X ¡ Im(¤ ) ¡ 0. 1 From Ext (Cok(¤ ), ) we get so ¢ 1 f 1 ... ¡ Ext (Cok(¤ ),X) ¡ Ext (Cok(¤ ),Im(¤ )) ¡ 0. = f(¡ ) for some ¡ . Thus there is commutative diagram ¡ ¡ ¤ ¡ Cok(¤ ¡ ¡ :0 X Z ) 0 ¢ :0 ¡ Im(¤ ) ¡ Y ¡ Cok(¤ ) ¡ 0 Now the sequence ¨ ¤£¢ ¥ ¦ ¦ ¥£¢ § ¥ ¤© 0 ¡ X ¡ Z¥ Im(¤ ) ¡ Y ¡ 0 ¥ (¤ - § ) 1 is exact, so splits since Ext (Y,X)=0. 9
Page 1 and 2: Lectures on Repres
Page 3 and 4: §1. Path algebras Once and for all
Page 5 and 6: PROPERTIES OF PATH ALGEBRAS. These
Page 7: THE STANDARD RESOLUTION. ¡ ¥ ¡
Page 11 and 12: §3. The variety of representations
Page 13 and 14: 1 LEMMA 1. dim Rep(¤ ) - dim O = d
Page 15 and 16: §4. Dynkin and Euclidean diagrams
Page 17 and 18: PROOF. (2) By inspection the given
Page 19 and 20: §5. Finite representation type In
Page 21 and 22: §6. More homological algebra FROM
Page 23 and 24: LEMMA 3. and - give inverse bijecti
Page 25 and 26: §7. Euclidean case. Preprojectives
Page 27 and 28: LEMMA 4. If ¤ is a positive real r
Page 29 and 30: PROPERTIES. Let X be regular simple
Page 31 and 32: DEFINITION. Given a -orbit of regul
Page 33 and 34: LEMMA 2. If X is regular, X £ 0 th
Page 35 and 36: REMARKS. ~ (1) For the Kronecker qu
Page 37 and 38: (4) Auslander-Reiten theory for wil

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