Fields Of Cohomological Dimension Research Articles

The eta-inverted sphere over the rationals

We calculate the motivic stable homotopy groups of the two-complete sphere spectrum after inverting multiplication by the Hopf map eta over fields of cohomological dimension at most 2 with characteristic different from 2 (this includes the p-adic fields and the finite fields of odd characteristic) and the field of rational numbers; the ring structure is also determined.

Lie bialgebras, fields of cohomological dimension at most 2 and Hilbert's Seventeenth Problem

We investigate Lie bialgebra structures on simple Lie algebras of non-split type A. It turns out that there are several classes of such Lie bialgebra structures, and it is possible to classify some of them. The classification is obtained using Belavin–Drinfeld cohomology sets, which are introduced in the paper. Our description is particularly detailed over fields of cohomological dimension at most two, and is related to quaternion algebras and the Brauer group. We then extend the results to certain rational function fields over real closed fields via Pfister's theory of quadratic forms and his solution to Hilbert's Seventeenth Problem.

Local-global questions for tori over 𝑝-adic function fields

We study local-global questions for Galois cohomology over the function field of a curve defined over a p p -adic field (a field of cohomological dimension 3). We define Tate-Shafarevich groups of a commutative group scheme via cohomology classes locally trivial at each completion of the base field coming from a closed point of the curve. In the case of a torus we establish a perfect duality between the first Tate-Shafarevich group of the torus and the second Tate-Shafarevich group of the dual torus. Building upon the duality theorem, we show that the failure of the local-global principle for rational points on principal homogeneous spaces under tori is controlled by a certain subquotient of a third étale cohomology group. We also prove a generalization to principal homogeneous spaces of certain reductive group schemes in the case when the base curve has good reduction.

On the non-injectivity of the Vaserstein symbol in dimension three

L.N. Vaserstein proved over a two dimensional ring that the orbit space of unimodular rows of length three modulo elementary action has a Witt group structure. R.A. Rao and W. van der Kallen showed that the Vaserstein symbol need not be injective over three dimensional affine algebras over the real field, but is injective over three dimensional affine algebras over a field of cohomological dimension one whose characteristic ≠2,3. R.G. Swan, R.A. Rao and J. Fasel gave another example of a real affine algebra of dimension three for which the Vaserstein symbol is not injective. We demonstrate an uncountably infinite family of non-isomorphic affine algebras of dimension three over the real field for which the Vaserstein symbol is not injective.

Some elementary theorems about divisibility of 0-cycles on abelian varieties defined over finite fields

If X is an abelian variety over a field, and L ∈ Pic(X), we know that the degree of the 0-cycle Lg is divisible by g!. As a 0-cycle, it is not, even over a field of cohomological dimension 1, but we show that over a finite field, there is some hope.

Galois cohomology of the classical groups over fields of cohomological dimension?2

Galois cohomology of the classical groups over fields of cohomological dimension?2

On the cohomological dimension of non-standard number fields

The study made in this paper was motivated by the following question: given n 2 0, is the class of fields of cohomological dimension in elementary? It is relatively easy to see that this class is closed under elementary substructures, see Proposition 2.5; for n12 it is however unknown whether it is closed under ultraproducts, or even ultrapowers. Klingen showed in [6] that the class of fields having a class formation is elementary. As fields having a class formation have cohomological dimension 2, this suggests the following type of question: given an elementary class generated by fields of cohomological dimension 5 2, do all its elements have cohomological dimension 12? In this paper we propose to show that the elementary class S generated by all (totally imaginary) global fields and their algebraic extensions contains only fields of cohomological dimension ~2 (Theorem 3.6). This implies in particular that cd(Q*(i)) =2. We also study, for Q* a non-standard extension of Q, some of the properties of G(Q*); in contrast with the standard case, G(Q*) contains non-pro-cyclic abelian subgroups (Proposition 3.7), and cd(Q*(p))> 1 (Corollary 3.8). However, as in the standard case, we have cd(Q*“b)= 1 (Theorem 3.11). Our study allows us, given a prime number p, to find a sentence ulp such that for every field KES

A field of cohomological dimension 1 which is not $C_1$

A field of cohomological dimension 1 which is not $C_1$