# $k$ structures on $K$ vector spaces

The following statement is made in Borel's Linear Algebraic groups, section 11 on $k$ structures.

Let $V$ and $W$ be $K$ vector spaces with $k$ structures. If $f:V\to W$ is $K$ linear, then $f$ is said to be defined over $k$ if $f(V_k)\subset W_k$ and these are elements of $Hom_K(V,W)_k\subset Hom_K(V,W)$. This is a $k$ structure if $W$ is finite dimensional.

The author seems to be making no assumption on the dimension of $V$ (which is the source of my problem). If we allow $V$ to be infinite dimensional then it seems to me that it is incorrect that $Hom_K(V,W)_k$ is a $k$ structure on $Hom_K(V,W)$ as is given by the following example. Let $V_k=\oplus_{i\geq0} ke_i$ and take $K$ to be an extension which is not a finite $k$ vector space, and define $f\in Hom_K(V,K)$ by choosing $f(e_i)$ which are linearly independent over $k$. We will never be able to write such an element as $\sum f_i\otimes \alpha_i$ with $f_i\in Hom_K(V,K)_k$.

So my question is : Do we need to assume both $V,W$ to be finite dimensional $K$ vector spaces for $Hom_K(V,W)$ to have a $k$ structure?

Also, what are the other references for $k$ structures and rationality properties?

• It would help to formulate your main question explicitly, since what you've written down in the third paragraph looks confusing at first sight. The final question is easier to answer. Borel used ad hoc methods in the 1960s Columbia lectures to avoid explicit use of schemes. (The section here is AG.11.) Similarly, Springer's later textbook has a somewhat ad hoc treatment of rationality questions starting with Chapter 11. The recent Cambridge book Pseudo-reductive Groups by Conrad-Gabber-Prasad may be the best modern reference for rationality properties. – Jim Humphreys Apr 21 '11 at 14:40
• Edited the question. Thanks for the reference. – Rex Apr 21 '11 at 16:39
• I added a couple of tags and also replaced my earlier unhelpful answer. – Jim Humphreys Apr 23 '11 at 13:38

The statement appears to be wrong. What you need is for $V$ to be finitely generated. It is a general theorem of commutative algebra that, if $R\rightarrow S$ is a flat map of commutative rings, and $M$ and $N$ are $R$-modules with $M$ finitely presented over $R$, then the natural map $$Hom_R(M,N)\rightarrow Hom_S(S\otimes_RM,S\otimes_RN)$$ induces an isomorphism $$Hom_R(M,N)\otimes_RS\rightarrow Hom_S(S\otimes_RM,S\otimes_RN).$$ See Eisenbud, Proposition 2.10.
If we take $R=k$ and $S=K$, you get the result about $k$-structures as long as $V$ is finitely generated.
To replace my earlier offhand dismissal of the basic question here, it may be useful to add some comments on where things actually go wrong in Section 11.1 of Borel's Chapter AG. As Rex suggests in the question, there is an overstated claim: ... this is even a $k$-structure provided that $W$ is finite dimensional. In particular, when $W=K$, we have a $k$-structure on the dual $V^*$ of $V$. The problem here is that $W$ needs to be finite dimensional over $k$ (not just over $K$), or in other words $K/k$ needs to be a finite field extension. Only then for example can you concretely describe an arbitrary linear functional $f:V \rightarrow K$ in terms of a collection of linear functionals $V_k \rightarrow k$. (Here the dimension of $V$ is irrelevant, however.) So the answer to the original question is that $V$ can be arbitrary while $W$ should be finite dimensional over $K$ along with $[K:k]<\infty$.
• @Mariano: That's obviously an understatement, but I wanted to emphasize that it isn't enough (for rationality questions) to limit everything to finite dimensional vector spaces $V$. At the same time, the full linear dual of an infinite dimensional space doesn't seem to come up in this context. Though it needs to be checked that Borel's later treatment doesn't rely on the overstatement in AG.11. – Jim Humphreys Apr 25 '11 at 20:46