Download PDF by Jörgen Backelin, Jürgen Herzog, Herbert Sanders (auth.),: Algebra Some Current Trends: Proceedings of the 5th National

By Jörgen Backelin, Jürgen Herzog, Herbert Sanders (auth.), Luchezar L. Avramov, Kerope B. Tchakerian (eds.)

ISBN-10: 3540459944

ISBN-13: 9783540459941

ISBN-10: 3540503714

ISBN-13: 9783540503712

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THEOREM 3 . 9 . i. Up to equ/va/ence and cyclic permutation of the factors, there exists a unique indecomposable m a t r i x factorization f = ao . . . a,i-1, such that for all t and a~ 1 <_ i < j < n. ft. f f n is odd, then all oq are pairwise equivalent. iii. I f n is even, then oq is equivalent to ~i if and only i f i =- j (mod r). Proof. Since the C ( f , ~)-matrix factorizations correspond to Z / d Z - g r a d e d modules over C ( f , ~), the relation eyei = ~eiei for i < j implies that Ct+l(ej) o Ct(ei) = ~¢t+l(ei) o Ct(ej).

BERGMAN, The diamond lemma for ring theory, Advances in Math. 29 (1978), 178-218. [BEH] R. O. BUCHWEITZ, D. EISENBUD, J. ). [BHU] J. P. BRENNAN, J. HERZOG, B. ULRICH, Maxima/ly generated Cohen-Macaulay modules, to appear in Math. Scand.. [c] L. N. CHILDS, Linearizing of n-lc forms and generalized Clifford a/gebras, Linear and Multilinear Algebra 5 (1978), 267-278. [E] D. EISENBUD, Homologica/ a/gebra on a complete intersection with an application to group representations, Trans. AMS 260 (1980), 35-64.

El-1 is a regular sequence, it follows t h a t R / ( x l , ~ o ' . . " ~ - 1 ) ~-- C [ X 1 , X 2 , X s ] / ( X I , a o ' . . " a~-l) is C M of d i m R - 1. Hence ( x l , a 0 • . . • 0~-1) is a non-principal CM-ideal, which implies that [ ( ~ , a 0 . . - < - 1 ) ] ¢ 0. [] REMARK. For d = 2 the divisor class group C£(R) of R -. C[X1,X2, X 3 ] / ( X d + X d + X d) equals Z[M] and is isomorphic to Z / 2 Z . However, for d > 2, Z - [M] only generates t h e discrete part of ~ ( R ) , as we were told by Storch (see [St]).

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Algebra Some Current Trends: Proceedings of the 5th National School in Algebra held in Varna, Bulgaria, Sept. 24 – Oct. 4, 1986 by Jörgen Backelin, Jürgen Herzog, Herbert Sanders (auth.), Luchezar L. Avramov, Kerope B. Tchakerian (eds.)


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