# Descent on Vector Spaces and Cohomology

It is quite often of interest to study the properties of some variety ${X/\mathbb Q}$. However, it is generally much easier to study varieties over algebraically closed fields and so we need some way of translating a property of ${X_{\overline{\mathbb Q}}/\overline{\mathbb Q}}$ to ${X/\mathbb Q}$. 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 ${L/K}$ be an extension of fields and ${V}$ a vector space over ${K}$. Consider ${W = V\otimes_K L }$. The Galois Group ${G = \mathop{Gal}(L/K)}$ acts on ${W}$ through the second factor (not by linear actions but by semi-linear ones – see below). One can consider the ${K}$-vector space ${W^G}$. This is the vector space fixed by ${G}$.

Theorem 1 (Descent of Vector Spaces) The natural map ${W^G\otimes_K L \rightarrow W}$ is an isomorphism. In particular, if ${W}$ is finite dimensional, then ${\dim_K W^G = \dim_L W}$.

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 ${{GL}_n(L)}$ be the group of invertible ${n}$-dimensional matrices over ${L}$ and consider the cohomology ${H^1(G,{GL}_n(L))}$. This is not a group unless ${n=1}$ since ${{GL}_n(L)}$ is non-commutative in general. However, it is a pointed set and we have the following theorem:

Theorem 2 (Hilbert’s 90) $\displaystyle H^1(G,{GL}_n(L)) = \{0\}.$

Hilbert stated the above theorem (in a disguised form) for ${n=1}$ and ${L/K}$ 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: