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:

 

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