We prove a tight lower bound on the algebraic Betti numbers of tree and forest ideals and an upper bound on certain graded Betti numbers of squarefree monomial ideals.

We study L-Betti numbers for von Neumann algebras, as defined by D. Shlyakhtenko and A. Connes in [1], in the presence of a bi-finite correspondence and proof a proportionality formula.

The Atiyah conjecture predicts that the L-Betti numbers of a finite CW -complex with torsion-free fundamental group are integers. We establish the Atiyah conjecture, under the condition that it holds for G and that H G is a normal subgroup, for amalgamated free products G ∗H (H ⋊ F ). Here F is a free group and H ⋊ F is an arbitrary semi-direct product. This includes free products G∗F and semi-...

There are notions of L2-Betti numbers for discrete groups (CheegerGromov, Lück), for type II1-factors (recent work of Connes-Shlyakhtenko) and for countable standard equivalence relations (Gaboriau). Whereas the first two are algebraically defined using Lück’s dimension theory, Gaboriau’s definition of the latter is inspired by the work of Cheeger and Gromov. In this work we give a definition o...

In 2018 Celikbas, Laxmi, Kraśkiewicz, and Weyman exhibited an interesting family of perfect ideals codimension three, with five generators, Cohen-Macaulay type two trivial multiplication on the Tor algebra. All previously known had been found by Brown in 1987. Brown's all have non-trivial We prove that are obtained from (non-homogeneous) specialization. also both families ideals, when built usi...