Every Single Problem – Page 2

I was reading Popov and Vinberg’s book on invariant theory when they asserted a basic lemma (without proof) that I didn’t recognize. It was this (p. 155):

Lemma 2.4: Suppose $K$ is an extension of the field $k$ , and $G$ is a group of $k$ -automorphisms of $K$ . Suppose $V$ is a (not necessarily finite-dimensional) vector space over $k$ , and $W$ is a subspace of $K\otimes_k V$ that is stable under the natural action of $G$ (induced by the action on the first factor). Then $W$ is generated by invariant vectors.

I asked for a proof on Math.SE. Alex Youcis pointed me to these notes of J. Milne (see section c). They’re phrased in slightly different language, but they do the job. The heart of the matter is the following lemma:

Lemma 16.6 (this is Milne’s numbering, though I’m not being completely faithful to his statement or notation): Suppose $K/k$ is a field extension such that $G = \text{Aut}_k\,(K)$ satisfies $K^G=k$ . (Note this is a stronger set of assumptions about $K,k,G$ than the above.) Let $V$ be a $k$ -vector space. If $W\subset K\otimes_kV$ is a nontrivial $K$ -subspace that is stable under the action of $G$ , then it contains a nontrivial $G$ -invariant vector. (NB: the $G$ -invariant vectors are precisely those that lie in the image of the natural $k$ -linear map $V\rightarrow K\otimes_kV$ given by $v\mapsto 1\otimes v$ .)

The proof of this lemma is as follows. Choose a basis $\{e_i\}_{i\in I}$ for $V$ , where $I$ is some indexing set. Then any element of $K\otimes_k V$ is a finite sum of the form $\sum x_i\otimes e_i$ , with $x_i\in K\setminus\{0\}$ for each $i$ in the (finite) sum. Since $W$ is nontrivial, we can choose a $w\in W$ such that the number of terms in this representation $w=\sum x_i\otimes e_i$ is minimal among nonzero elements of $W$ , and since $W$ is a $K$ -vector space, we can scale this $w$ by an element of $K$ so that one of the coefficients in this sum is $1$ ; thus we can without loss of generality assume $w = 1\otimes e_1 + \sum x_i\otimes e_i$ (where we have labeled $e_1$ as the basis vector corresponding to the coefficient $1$ ) with the number of nonzero terms minimal. Since $W$ is $G$ -stable, for any $g\in G$ we then have $gw = 1\otimes e_1 + \sum gx_i\otimes e_i \in W$ , and therefore, their difference $\sum (gx_i - x_i)\otimes e_i$ is in $W$ as well. Note this sum has fewer nonzero terms than the sum representing $w$ , and we conclude by $w$ ‘s minimality that this new sum is zero, and thus that $gx_i - x_i = 0$ , since the $e_i$ ‘s are linearly independent over $k$ and thus the $1\otimes e_i$ ‘s are linearly independent over $K$ .

We can conclude that $gx_i=x_i$ for each $x_i$ and for each $g\in G$ . It is immediate that $w = 1\otimes e_1+\sum x_i\otimes e_i$ is $G$ -invariant, proving the existence of a nontrivial $G$ -invariant vector. Since $k = K^G$ , the $x_i$ ‘s all lie in $k$ , and this, together with similar logic as above (linear independence of $1\otimes e_i$ ‘s over $K$ ), also lets us recognize that $G$ -invariant vectors are exactly those of the form $\sum x_i\otimes e_i$ with all $x_i$ ‘s in $k$ , i.e. those that are images of some $v\in V$ under the natural $k$ -linear map $V\rightarrow K\otimes_k V$ , as per the parenthetical statement in the lemma. QED.

I recognized the proof of Lemma 16.6 as very similar to the proof I learned many years ago of Artin’s lemma, one of the building blocks in Artin’s develoment of the fundamental theorem of Galois theory. Artin’s lemma is the statement that if $K$ is a field, $G$ is a finite group of automorphisms of $K$ , and $k = K^G$ , then $[K:k]\leq |G|$ . I spent some time yesterday morning thinking through this relationship, and I realized that the proof of Artin’s lemma can be reformulated in an elegant way to use Lemma 16.6. I wanted to record that here.

Proof of Artin’s lemma using Lemma 16.6:

Consider any finite collection of elements $x_1,\dots,x_n$ of elements of $K$ that are linearly independent over $k$ . Our goal is to show that $n\leq |G|$ . We will do this by constructing a certain $K$ -linear map

$\varphi: K^n\rightarrow K^{|G|}$

and showing it is injective. The map $\varphi$ is defined by $K$ -linear extension of the following map on the standard basis $e_1,\dots,e_n$ for $K^n$ :

$\varphi(e_i) = (gx_i)_{g\in G}$ .

The right side here is an element of $K^{|G|}$ with entries indexed by elements of $G$ .

We are going to apply Lemma 16.6 with $V=k^n$ . In this situation, $K^n$ is $K\otimes_kV$ . We will take $\text{ker}\,\varphi$ as $W$ . (To invoke the lemma we will have to show it is $G$ -stable.) The assumption that the $x_i$ ‘s are linearly independent over $k$ is then going to imply that there is no $G$ -invariant vector in $W$ , from which Lemma 16.6 will imply that $W$ itself is zero, i.e. that $\varphi$ is injective, as desired. Here are the details:

First, we claim $W =\text{ker}\,\varphi$ is $G$ -stable. We see this as follows. Let

$w = \sum_{i=1}^n y_ie_i$

(where each $y_i$ lies in $K$ ) be an arbitrary element of $W$ , i.e. assume that

$\varphi(w) = \sum_{i=1}^n y_i(gx_i)_{g\in G} = 0$ .

Looking coordinate-by-coordinate, this equation states that $\sum_{i=1}^n y_igx_i = 0$ for each $g\in G$ . Then, if $\gamma\in G$ is arbitrary, we have

$\sum_{i=1}^n (\gamma y_i)(\gamma g x_i) = 0$

as well. As $g$ traverses $G$ , $\gamma g$ does as well, so these equations (for each $g\in G$ ) can be arranged into the vector equation

$\sum_{i=1}^n (\gamma y_i)(g x_i)_{g\in G}=0$ .

The left side is precisely $\varphi(\gamma w)$ . Thus for any $w\in W= \text{ker}\;\varphi$ , $\gamma w\in W=\text{ker}\;\varphi$ as well, i.e. $W$ is $G$ -stable.

Thus we can invoke Lemma 16.6: if $W$ is nontrivial, then it contains a nontrivial $G$ -invariant element $w^\star$ , which in view of the parenthetical note at the end of the lemma, has the form $w^\star = \sum_{i=1}^n y_ie_i$ with each $y_i\in k$ and some $y_i$ nonzero. But then $\varphi(w^\star)=0$ is the statement that

$\sum_{i=1}^n y_i(gx_i)_{g\in G}=0$ .

Extracting the coordinate where $g=\text{id.}$ from this vector equation, we get

$\sum_{i=1}^n y_ix_i = 0$ .

But since each $y_i\in k$ , this is a nontrivial linear relation between the $x_i$ ‘s over $k$ , which is not possible since we presumed them to be linearly independent. We conclude $W=\text{ker}\;\varphi$ is trivial, i.e. $\varphi$ is injective, and it follows that $n\leq |G|$ . This completes the proof of Artin’s lemma.