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

1. A development of the Malcev´s description for torsion free abelian groups Alexander Fomin Mathematics in the contemporary world Vologda, 2013, October 8

2. Finitely presented modules over the ring U U=pZp where Zpis the ring of p-adic integers introduced by Kurt Hensel in 1900. Um UkM0

3. 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

4. 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.

5. 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

6. 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.

7. (2) (3) TFFR QD It is a duality of two categories introduced by Wickless and Fomin in 1998.

8. 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)

9. (1) (3) • S QD • It is an equivalence of two categories S and QD which presents a generalization of the Kurosh´s Theorem (1937).

10. Derry Malcev Kurosh prim t.f.f.r. q.d.,1998

11. Example 1 • (S) The sequence of zeros 0,0,…,0. • (TFFR) The group is free. • (QD) The group is divisible.

12. 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

13. 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.

14. 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.

15. 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.

16. 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.

17. Thank you For your attention