Let p be a prime, F a field containing a primitive pth root of unity, and E/F a cyclic extension of degree p. Using the Bloch–Kato Conjecture we determine precise conditions for the cohomology group Hn(E) := H(GE ,Fp) to be free or trivial as an Fp[Gal(E/F )]-module, and we examine when these properties for Hn(E) are inherited by Hk(E), k > n. By analogy with cohomological dimension, we introdu...

We rephrase Gromov’s definition of Markov compacta, introduce a subclass of Markov compacta defined by one building block and study cohomological dimensions of these compacta. We show that for a Markov compactum X, dimZ(p) X = dimQ X for all but finitely many primes p where Z(p) is the localization of Z at p. We construct Markov compacta of arbitrarily large dimension having dimQ X = 1 as well ...

We introduce the notion of asymptotic cohomology based on the bounded cohomology and define cohomological asymptotic dimension asdimZ X of metric spaces. We show that it agrees with the asymptotic dimension asdimX when the later is finite. Then we use this fact to construct an example of a metric space X of bounded geometry with finite asymptotic dimension for which asdim(X × R) = asdimX. In pa...