This theorem seems to come tantalizingly close to supplying counter-examples to the Eilenberg–Ganea conjecture, which asserts that any group of cohomological dimension 2 admits a (not necessarily finite) 2-dimensional Eilenberg–Mac Lane space. Indeed, I don’t know of any example of a finitely-presented group with an infinite 2-dimensional K(Γ, 1) but not a finite one, but such things surely exi...

Using a result of Harer, we prove certain upper bounds for the homotopical/cohomological dimension of the moduli spaces of Riemann surfaces of compact type, of Riemann surfaces with rational tails and of Riemann surfaces with at most k rational components. These bounds would follow from conjectures of Looijenga and Roth-

We prove that if a positively-graded ring R is Gorenstein and the associated torsion functor has finite cohomological dimension, then the corresponding noncommutative projective scheme Tails(R) is a Gorenstein category in the sense of [10]. Moreover, under this condition, a (right) recollement relating Gorensteininjective sheaves in Tails(R) and (graded) Gorenstein-injective R-modules is given.

We discuss secondary (and higher) characteristic classes for algebraic vector bundles with trivial top Chern class. We then show that if X is a smooth affine scheme of dimension d over a field k of finite 2cohomological dimension (with char(k) 6= 2) and E is a rank d vector bundle over X , vanishing of the Chow-Witt theoretic Euler class of E is equivalent to vanishing of its top Chern class an...

We give a survey about some recent work on tight closure and Hilbert-Kunz theory from the viewpoint of vector bundles. This work is based in understanding tight closure in terms of forcing algebras and the cohomological dimension of torsors of syzygy bundles. These geometric methods allowed to answer some fundamental questions of tight closure, in particular the equality between tight closure a...

We give an alternative proof of the Hasse principle for principal homogeneous spaces de ned over elds of virtual cohomological dimension at most one which is based on a special decomposition of elements in Chevalley groups. 1991 Mathematics Subject Classi cation: Primary 20G10; Secondary 11E72, 12D15.