Let X = Γ\G/K be an arithmetic quotient of a symmetric space of non-compact type. In the case that G has Q-rank 1, we construct Γ-equivariant deformation retractions of D = G/K onto a set D0. We prove that D0 is a spine, having dimension equal to the virtual cohomological dimension of Γ. In fact, there is a (k − 1)-parameter family of such deformation retractions, where k is the number of Γ-con...

The purpose of this note is to introduce a trick which relates the (non)-vanishing top-dimensional $\ell^2$-Betti numbers actions with that sub-actions. We provide three different types applications: we prove Aut($F_n$) and Out($F_n$) (and their Torelli subgroups) do not vanish in degree equal virtual cohomological dimension, subgroups 3-manifold groups have vanishing 3 2 for instance $F_2^d \t...

Let K be a field with a valuation satisfying the following conditions: both K and the residue field k have characteristic zero; the value group is not 2-divisible; there exists a maximal subfield F in the valuation ring such that Gal(F̄ /F ) and Gal(k̄/k) have the same 2-cohomological dimension and this dimension is finite. Then Hilbert’s Tenth Problem has a negative answer for any function field...

Introduction 1 0.1. Associated primes 2 0.2. Depth 2 1. Regular local rings and their cohomological properties 4 1.1. Notions of dimension 4 1.2. Dualizing functor 5 1.3. Local rings 6 1.4. Serre’s theorem on regular local rings 6 1.5. Cohomology of D when global dimension is finite 8 2. Depth 8 2.1. Property (Sk) 8 2.2. Serre’s criterion of normality 10 2.3. Cohen-Macaulay modules 11 3. Diviso...

It is a well established fact that the solutions of many problems involving families of vector bundles should essentially depend on good estimates for the dimension of the Hilbert schemes of coherent quotients of a given bundle. Deformation theory provides basic cohomological dimension bounds, but most of the time the cohomology groups involved are hard to estimate accurately and moreover do no...