The dimension of any module over an algebra of affiliated operators U of a finite von Neumann algebra A is defined using a trace on A. All zero-dimensional U-modules constitute the torsion class of torsion theory (T,P). We show that every finitely generated U-module splits as the direct sum of torsion and torsion-free part. Moreover, we prove that the theory (T, P) coincides with the theory of ...

We review the notion of Gieseker stability for torsion-free Higgs sheaves. This notion is a natural generalization of the classical notion of Gieseker stability for torsion-free coherent sheaves. We prove some basic properties that are similar to the classical ones for torsion-free coherent sheaves over projective algebraic manifolds. In particular, we show that Gieseker stability for torsion-f...

In this paper we seek to describe the structure of self-dual torsion-free LCA groups. We first present a proof of the structure theorem of self-dual torsion-free metric LCA groups. Then we generalize the structure theorem to a larger class of selfdual torsion-free LCA groups. We also give a characterization of torsion-free divisible LCA groups. Consequently, a complete classification of self-du...

In this paper the cotypeset of some torsion-free abelian groups of finite rank is studied. In particular, we determine the cotypeset of some rank two groups using the elements of their typesets. Introduction One of the important and known tools in the theory of torsion-free abelian groups is type and the typeset of a group. This set which is determined from the beginning of the the study the to...

An unstable torsion free sheaf on a smooth projective variety gives a GIT unstable point in certain Quot scheme. To a GIT unstable point, Kempf associates a “maximally destabilizing” 1-parameter subgroup, and this induces a filtration of the torsion free sheaf. We show that this filtration coincides with the Harder-Narasimhan filtration.

Suppose that n ≥ 2 and that S, T are sets of primes. Then the classification problem for the S-local torsion-free abelian groups of rank n is Borel reducible to the classification problem for the T -local torsion-free abelian groups of rank n if and only if S ⊆ T .

Let A and B be rational groups, i.e. torsion-free groups of rank-1 and thus subgroups of the rational numbers. This paper gives a short overview of the structure of Ext(A,B) especially considering some interesting classes of torsion-free pairs. Mathematics Subject Classification (2010). 20K15, 20K35.

Mal’cev showed in the 1950s that there is a correspondence between radicable torsion-free nilpotent groups and rational nilpotent Lie algebras. In this paper we show how to establish the connection between the radicable hull of a finitely generated torsion-free nilpotent group and its corresponding Lie algebra algorithmically. We apply it to fast multiplication of elements of polycyclically pre...

The Atiyah conjecture predicts that the L-Betti numbers of a finite CW -complex with torsion-free fundamental group are integers. We show that the Atiyah conjecture holds (with an additional technical condition) for direct and inverse limits of directed systems of groups for which it is true. As a corollary it holds for residually torsion-free solvable groups, e.g. for pure braid groups or for ...