Weak corestriction principle for non-abelian Galois cohomology

Abstract

We introduce the notion of (Weak) Corestriction Principle and prove some relations between the validity of this principle for various connecting maps in non-abelian Galois cohomology over fields of characteristic 0. We also prove the validity of Weak Corestriction Principle for images of coboundary maps H(k, G) → H(k, T ), where T is a finite commutative k-group of multiplicative type, G is adjoint, semisimple and contains only almost simple factors of certain inner types. Introduction. Let G be a commutative algebraic group defined over a field k of characteristic 0. Let H(k, G) denote the usual Galois cohomology H(Gal(k/k), G(k)). It is well-known that there exists corestriction homomorphism Cores := Coresk′/k : Hi(k′, G) → H(k, G) for any i > 1 and any finite extension k′ of k, which gives rise to a map of functors (G 7→ Hi(k′, G)) → (G 7→ H(k, G)). In particular, if 1 → A j → B p → C → 1 is an exact sequence of commutative algebraic k-groups, {α1, α2, ...} (resp. {α′ 1, α′ 2, ...}) denotes the sequence of homomorphisms appearing in the long exact sequence of cohomology deduced from (∗) as cohomology of Gal(k/k)-modules (resp. as Gal(k/k′)-modules), then we have Cores ◦ α′ m = αm ◦ Cores for all m > 1. However, if in (∗) one of the groups is not commutative, then it turns out that there is no corestriction map between these two long exact sequences in general. (In [R1], C. Riehm has found some sufficient conditions for the existence of corestriction map.) It leads us to the following definition. Let A,B be algebraic groups defined over k. Assume that we are given a map of functors f : (L 7→ H(L,A)) → (L 7→ H(L,B)), where L denotes a field extension of k, i.e., a collection of maps of cohomology sets fL : H(L, A) → H(L,B), where fL is functorial in L. Received March 21, 2003, revised June 30, 2003; published on July 18, 2003. 2000 Mathematics Subject Classification: Primary 11G72; Secondary 18G50, 20G10.