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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 =  p Z p where Z p is the ring of p-adic integers introduced by Kurt Hensel in 1900.

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Alexander Fomin

Mathematics in the contemporary world

Vologda, 2013, October 8

Finitely presented modules over the ring abelian groupsU

U=pZp

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

Um UkM0

Category “Sequences” abelian groups

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 abelian groups

• 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 abelian groups

• 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 abelian groups

• 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.

(2) (3) abelian groups

TFFR QD

It is a duality of two categories introduced by Wickless and Fomin in 1998.

Malcev´s description (1938) abelian groups

• (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) abelian groups

• S QD

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

Derry abelian groups

Malcev

Kurosh

prim

t.f.f.r.

q.d.,1998

Example 1 abelian groups

• (S) The sequence of zeros 0,0,…,0.

• (TFFR) The group is free.

• (QD) The group is divisible.

Example 2 abelian groups

• (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 abelian groups

• (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 abelian groups

• (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 abelian groups

• (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 abelian groups

• (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 abelian groups