It is quite often of interest to study the properties of some variety . However, it is generally much easier to study varieties over algebraically closed fields and so we need some way of translating a property of to . This idea is known as descent and in this post, I would like to say a little bit about the simplest example of descent – over vector spaces.
Let be an extension of fields and a vector space over . Consider . The Galois Group acts on through the second factor (not by linear actions but by semi-linear ones – see below). One can consider the -vector space . This is the vector space fixed by .
Theorem 1 (Descent of Vector Spaces) The natural map is an isomorphism. In particular, if is finite dimensional, then .
It is not hard to prove this theorem directly (Theorem 2.14 in these notes of K. Conrad) but I would like to relate it to another theorem. This is also well known and is a generalization of the famous Hilbert’s Theorem 90. Let be the group of invertible -dimensional matrices over and consider the cohomology . This is not a group unless since is non-commutative in general. However, it is a pointed set and we have the following theorem:
Theorem 2 (Hilbert’s 90)
Hilbert stated the above theorem (in a disguised form) for and a finite cyclic extension. Noether generalized the theorem to arbitrary extensions. I do not know who is responsible for the generalization to general linear groups but I saw this theorem first in Serre’s “Galois Cohomology”.
In this post, I will show that the above theorems are equivalent in the following sense: