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

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

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One in every of Che hallmarks of major advances in arithmetic is the invention of unforeseen relationships among diversified components. might be the main striking fresh instance of such an improve is D« Quillen! s improvement of upper algebraic K-theory within which algebra and topology are comparable in a new and primary approach.

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The e x p r e s s i o n normal E ie S i are right A-subspaces called a block. function of U(C) x~U(C). -- of M, where m = ge, geU(C)). and d(m)¢J A then d(gmJeJ A. This proposition ~(x) meM. above integer such for a block B is called its for all i. Any block has some normal form (it to replace S i by Si+Si+A). Lemma Z. Any block has only one normal any block B there are uniquely determined such that B = ZSi(B)Mi_ I (ieJ A for almost For any block B we shall construct form. In other words, right A-subspaces for Si(B)CA all i).

3. Let f = X 1d + . . 9. L e t M be the linear M C M - m o d u l e Coker ¢1, where t h e endomorphism ¢1: R m --* R m is given b y a l . T h e n diM] = 0 in G ( R ) . I f n = 3, then [M] defines an dement of order d in the d/visor c/ass group of R. Proof. Consider the filtration 0 = Ua C Ud--1 C . . C U1 C Uo = F given by the factorization f = a l • ... • aa. 9 that all 04 are pairwise equivalent. Therefore all linear MCM-modules U ~ - I / U i are isomorphic to M, which implies that diM] = O.

The following theorems then hold: THEOREM I. KD = I1@I 2 = I3@14, where II=KD(I+b); I2=KD(I-b); 13=KD(1+ab); I4=KD(]-ab). THEOREM 2. ,I 4 defined above. THEOREM 3. Let M I and M 2 be finitely generated K(a)-torsion-free KD-modules, let Mi (i=I,2) be the submodule of M i generated by the elements (1+b)x, xeM i. Then M i and M 2 are KD-isomorphic if and only if they are K(a)-isomorphic and the factor modules MI/M I and M2/M 2 are K(a)-isomorphic. TflEORJ/M 4. The Krull-Schmidt theorem does not hold for the finitely generated K(a)-torsion-free KD-modules, but each such module M can be defined uniquely up to isomorphism by the vector (M)=(r,nl,n2), where r is the K(aj-rank of M, n I is the number of submodules isomorphic to K(a)/(a+]) in the direct decomposition of the K(a)-module M/~, and n 2 is the number of submodules isomorphic to K(a)/(a-1) in the direct de- composition of M/M.

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