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.

We define degree two cohomological invariants for GGalois algebras over fields of characteristic not 2, and use them to give necessary conditions for the existence of a self–dual normal basis. In some cases (for instance, when the field has cohomological dimension ≤ 2) we show that these conditions are also sufficient.