of the Dissertation Affine Stratifications and equivariant vector bundles on the moduli of principally polarized abelian varieties by Anant Atyam Doctor of Philosophy in Mathematics Stony Brook University 2014 We explicitly construct a locally closed affine stratification of the coarse moduli space of principally polarized complex abelian four folds A4 and using [32], produce an upper bound for...

Fields of cohomological dimension one versus C 1-fields J.-L. Colliot-Théì ene Summary. Ax gave examples of fields of cohomological dimension 1 which are not C 1-fields. Kato and Kuzumaki asked whether a weak form of the C 1-property holds for all fields of cohomological dimension 1 (existence of solutions in extensions of coprime degree rather than existence of a solution in the ground field)....

Let G be a pro-p group of finite cohomological dimension and type FP∞ and T is a finite p-group of automorphisms of G. We prove that the group of fixed points of T in G is again a pro-p group of type FP∞ (in particular it is finitely presented). Moreover we prove that a pro-p group G of type FP∞ and finite virtual cohomological dimension has finitely many conjugacy classes of finite subgroups.

In this paper, we determine the strict cohomological dimension of higher dimensional local fields.

Given a standard graded polynomial ring over commutative Noetherian $A$, we prove that the cohomological dimension and height of ideals defining any its Veronese subrings are equal. This result is due to Ogus when $A$ field characteristic zero, follows from Peskine Szpiro positive characteristic; our applies, for example, integers.