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.

We define a complete system of invariants en,Q, n ≥ 0 for quaternionic skew-hermitian forms, which are twisted versions of the invariants en for quadratic forms. We also show that quaternionic skew-hermitian forms defined over a field of 2-cohomological dimension at most 3 are classified by rank, discriminant, Clifford invariant and Rost invariant.

Paul Gunnells has developed an algorithm for computing actions of Hecke operators on arithmetic cohomology below the cohomological dimension. One version of his algorithm uses a conjecture concerning LLL-reduced matrices. We prove this conjecture for dimensions 2 through 5 and disprove it for all higher dimensions.

Assuming the Bloch-Kato Conjecture, we determine precise conditions under which Hilbert 90 is valid for Milnor k-theory and Galois cohomology. In particular, Hilbert 90 holds for degree n when the cohomological dimension of the Galois group of the maximal p-extension of F is at most n.

Let K/Q be a number field with euclidean ring of integers O. Let Γ be a finite-index torsion-free subgroup of the symplectic group Sp2n(O), and let N be the cohomological dimension of Γ. We exhibit a finite, geometrically-defined spanning set of H (Γ;Z) by generalizing the modular symbol algorithm of Ash and Rudolph for SLn(O).

In these notes we give a criterion for a central simple algebra A to be split in terms of the essential dimension of the algebraic group SL1(A). This criterion also provides an example of an algebraic group G with ed(G) = n which does not possess any non-trivial cohomological invariant. 2000 Mathematics Subject Classification: 11E72, 16K20.