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A development of the Malcev´s description for torsion free abelian groups

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A development of the Malcev´s description for torsion free abelian groups

Alexander Fomin

Mathematics in the contemporary world

Vologda, 2013, October 8

Finitely presented modules over the ring U

U=pZp

where Zpis the ring of p-adic integers introduced by Kurt Hensel in 1900.

Um UkM0

Category “Sequences”

An object of the category S is a finite sequence m1,…,mn of elements of a finitely presented U-module M.

Morphisms {a1,…,an} {b1,…,bk} are pairs (φ,T), where

φ : <a1,…,an>U <b1,…,bk>U is a U-module homomorphism and T is a matrix with integer entries such that

(φ a1,…, φ an)= (b1,…,bk)T

Category TFFR

- Objects are torsion free abelian groups of finite rank with marked bases.
- Morphisms are homomorphisms such that the corresponding matrix is of integer entries.

Category QD

- Objects are quotient divisible groups with marked bases introduced by Beaumont-Pierce in 1961 and generalized by Fomin-Wickless in 1998.
- Morphisms are homomorphisms such that the corresponding matrix is of integer entries

The main Theorem

- Each of three following objects determines uniquely two other objects:
- 1. A sequence of the category S,
- 2. A torsion free group of the category TFFR,
- 3. A quotient divisible group of the categoty QD.

Malcev´s description (1938)

- (1) (2)
- (m1,…,mn) A
- It is a duality of two categories S and TFFR which is a development of the Malcev´s description (1938)

(1) (3)

- S QD
- It is an equivalence of two categories S and QD which presents a generalization of the Kurosh´s Theorem (1937).

Example 1

- (S) The sequence of zeros 0,0,…,0.
- (TFFR) The group is free.
- (QD) The group is divisible.

Example 2

- (S) The sequence m1,…,mn is a free basis of a free U-module M=m1U+…+ mnU
- (TFFR) The group is divisible.
- (QD) The group is free

Example 3

- (S) The sequence consists of p-adic integers and it is linearly independent over Z.
- (TFFR) The group is strongly indecomposable and it has the following property: every subgroup of infinite index is free.
- (QD) The group is a pure subgroup of Zp.

Example 4

- (S) the sequence is linearly independent over Z.
- (TFFR) The group is coreduced (it doesn´t contain nonzero free direct summands).
- (QD) The group is reduced.

Example 5

- (S)The sequence is linearly independent overU.
- (TFFR)The group is completely decomposable into a direct sum of rank-1 torsion free groups.
- (QD) The group is completely decomposable into a direct sum of rank-1 quotient divisible mixed groups.

Example 6

- (S)The sequence is almost linearly independent overU.
- (TFFR)The group is almost completely decomposable into a direct sum of rank-1 torsion free groups.
- (QD) The group is completely decomposable into a direct sum of rank-1 quotient divisible mixed groups.

Thank you

For your attention

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