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 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 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 on the following sense:

Let {W} be a {L} -vector space. We will say that a group {G} acts semi-linearly on it if \sigma(lv) = \sigma(l)\sigma(v) \text{ for all }\sigma \in G.

The typical example is when {G = \mathop{Gal}(L/K)} acts co-ordinate wise on {W = L^n} or equivalently {W = V\otimes_K L} for a {K} -vector space {V} . We will show that this is essentially the only example by proving:

Theorem 3 There is a bijection:

\displaystyle H^1(G,{GL}_n(L)) \longleftrightarrow \frac{\{\text{n-dimensional L- vector spaces with semilinear G-action}\}}{\text{isomorphisms}}

Proof: I will use {x^g} to denote {g} acting on {x} throughout:

Let us first establish the maps. Given a 1-cocyle {\eta:G \rightarrow \ mathop{GL}_n(L)} , let the corresponding vector space {W_\eta} be {L^n} with the action for {(g,w) \in G\times W} being given by {(g,w) \rightarrow \eta_g(w^g)} where {w^g} stands for the action of {g} co-ordinate wise. It is easy to verify that this is well defined:

Given {l\in L} , {\eta_g((lw)^g) = \eta_g(l^gw^g) = l^g\eta_g(w^g)} . This shows that the action is semi-linear. Then, given a one-cocycle cohomologous to {\eta} (that is, given {\tau_g = A^{-1}\eta_g A^g} ), we have the following isomorphism of vector spaces with group actions given by:

\displaystyle W_\eta \rightarrow W_\tau, w \rightarrow A^{-1}w.

That is, {A\tau_g((A^{-1}w)^g) = \eta_g(w^g)} as can be easily checked. This establishes that {\eta \rightarrow W_\eta} is well defined.

To construct the inverse, let {W} be a {L} -vector space with a semilinear {G} -action. Fix a basis {e_1,\dots, e_n} . Denote the column vector corresponding to this basis as {[e]} . Define {\eta_g} to be the unique transformation such that {\eta_g[e]^g = [e]} . To check that this is a 1-cocycle, note that:

\displaystyle \eta_g\eta_h^g[e]^{gh} = \eta_g(\eta_h[e]^h)^g = \eta_g[e]^g = [e]

and hence by uniqueness {\eta_{gh} = \eta_g\eta_h^g} .

Finally, it is easy to verify that these maps really are inverses. \Box

To see that we really have shown that Theorem 1 and Theorem 2 are equivalent, note that Theorem 1 is clearly true for {L^n = K^n\otimes_KL} . Then, if {H^1(G,{GL}_n(L)) = \{0\}} , there is a unique {L} -dimensional vector space with a semi-linear action and we can check Theorem {1} on this unique vector space.

Conversely, if Theorem {1} is true, then {H^1(G,{GL}_n(L))} 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.

 

 

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