# Download Algebra Some Current Trends: Proceedings of the 5th National by Jörgen Backelin, Jürgen Herzog, Herbert Sanders (auth.), PDF By Jörgen Backelin, Jürgen Herzog, Herbert Sanders (auth.), Luchezar L. Avramov, Kerope B. Tchakerian (eds.)

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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). This implies the identities in i.. 7. For n even, it only remains to show that (in the terminology of section 2) the operation of a , on the set of isomorphism classes of indecomposable Co (f, ~)-modules is transitive and that the only orbit consists of r elements.

Suslin). Corollary 2" There exists a noncommutative algebra R whose general linear group GLn(R ) has nonstandard automorphisms. Z. V. I. Zel'manov ). §4. Embeddings into matrix algebras Let F be a field. A variety M of F-algebras is called (triangularly) presentable if any algebra from M is embeddable into the algebra of (upper triangular) matrices over some commutative F-algebra. Z. Anan'in). Let F be a field of characteristic O. [Xn,Yn] some n ~ 1. ,z n] = 0 for Now let R be an algebra over a field F, and let X be an infinite set.

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.