Fields of Cohomological Dimension 1

Abstract

The fields of cohomological dimension 1 are exactly the fields with projective absolute Galois group. The analogue of the weak section conjecture holds for geometrically connected curves over such fields if and only if the field is PAC, see Theorem 238. At first glance, it seems strange to examine the assertions of the section conjecture for varieties over such fields. Among those fields are the finite fields, which were discussed at length in Chap 15. In Theorem 243, we will construct an infinite algebraic extension of ${\mathbb{F}}_{p}$ such that the profinite Kummer map for every smooth projective subvariety of an abelian variety is injective with dense image with respect to the topology from Chap. 4, but never surjective.Another source of fields with cohomological dimension 1 is given by the maximal cyclotomic extensions of algebraic number fields. A thorough understanding of the assertions of the section conjecture in this case might actually provide valuable insights, see Theorem 247, and may ultimately justify this chapter.