FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2006, VOLUME 12, NUMBER 2, PAGES 17-38

Almost completely decomposable groups with primary regulator quotients and their endomorphism rings

E. A. Blagoveshchenskaya

Abstract

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Let $X$ be a block-rigid almost completely decomposable group of ring type with regulator $A$ and $p$-primary regulator quotient $X/A$ such that $pl=expX/A$ with natural $l\; >\; 1$. From the well-known fact $plEnd\; A\; \subset \; End\; X\; \subset \; End\; A$ it follows that $End\; X\; =\; End\; X\; \cap \; End\; A$ and $plEnd\; A=\; End\; X\; \cap \; plEnd\; A$. Generalizing these, we determine the chain $End\; X\; =\; \backslash mathcal\; E$_A^(l) ⊂ \mathcal E_A^(l−1) ⊂ \mathcal E_A^(l−2) ⊂ ... ⊂ \mathcal E_A⁽¹⁾ ⊂ \mathcal E_A⁽⁰⁾ = End A, satisfying $pl-k\backslash mathcal\; E$_A^(k) = End X ∩ p^l−k End A, and construct groups $X\text{'}$_k and $\backslash widetilde\{X$_k} such that $\backslash mathcal\; E$_A^(k) = Hom (X'_k, \widetilde{X_k}), where $k\; =\; 1,2,...,l$- 1.

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Last modified: June 17, 2006