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 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 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 on the following sense:
Let be a -vector space. We will say that a group acts semi-linearly on it if
The typical example is when acts co-ordinate wise on or equivalently for a -vector space . We will show that this is essentially the only example by proving:
Theorem 3 There is a bijection:
Proof: I will use to denote acting on throughout:
Let us first establish the maps. Given a 1-cocyle , let the corresponding vector space be with the action for being given by where stands for the action of co-ordinate wise. It is easy to verify that this is well defined:
Given , . This shows that the action is semi-linear. Then, given a one-cocycle cohomologous to (that is, given ), we have the following isomorphism of vector spaces with group actions given by:
That is, as can be easily checked. This establishes that is well defined.
To construct the inverse, let be a -vector space with a semilinear -action. Fix a basis . Denote the column vector corresponding to this basis as . Define to be the unique transformation such that . To check that this is a 1-cocycle, note that:
and hence by uniqueness .
Finally, it is easy to verify that these maps really are inverses.
To see that we really have shown that Theorem 1 and Theorem 2 are equivalent, note that Theorem 1 is clearly true for . Then, if , there is a unique -dimensional vector space with a semi-linear action and we can check Theorem on this unique vector space.
Conversely, if Theorem is true, then is a one-element set.
Proofs for Theorem 1 and Theorem 2 can be found in many places. Serre’s Galois Cohomology is a good place to read about Group Cohomology generally and Theorem 2 in particular.
UPDATE: I later discovered that the content of this post is Exercise 1.9 in Poonen’s “Rational Points on Varieties”. He also proves Theorem 1 as Lemma 1.3.10.