Galois descent, cohomology, and conjugacy

Abstract

We give a concise exposition on the application of non-abelian Galois cohomology to descent problems in algebra, as developed by A. Borel, J.-P. Serre, and others in the late fifties and early sixties. Although its origins lie in algebraic number theory, this abstract framework allows one to formalize and address a very general question: If two algebraic objects defined over a field k are found to be isomorphic over a field extension Ω/k, are they also isomorphic over k?In this thesis, we focus on explicit descent problems over a field of characteristic zero in which the algebraic objects involved can be described as points in Zariski closed subsets of affine space, and whose automorphism groups are subgroups of the algebraic group GLn. In our cases of interest, the action of the automorphism group arises by the conjugation action of GLn on various spaces of k-linear maps. Our presentation follows closely the 2010 monograph G. Berhuy. Our contribution is that we fill in numerous details in the proofs found there, in particular those involving techniques from algebraic geometry. We clarify the relationship between the classical Hilbert’s Theorem 90 for cyclic extensions and the more general non-abelian Hilbert’s Theorem 90, which is one of the fundamental basic tools used in Galois cohomology. Finally, we give a complete proof that the descent problem for a finite dimensional associative k-algebra A is controlled by the Galois cohomology set H1(GΩ, Aut(A)(Ω)).