$k$ structures on $K$ vector spaces

Question

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?

Answer 1

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.

Answer 2

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 $[K:k]<\infty$.

Probably the stronger Borel/Bass version just quoted is never actually needed or referred to later on, though to verify that would require some checking. In this subject it's fairly unnatural to work with dual spaces of infinite dimensional vector spaces. However, polynomial algebras (infinite dimensional over some base field) do play a major role in the traditional foundations of algebraic geometry.

Anyway, Chapter AG is telegraphic in style, with few worked-out details and with only broad references to the literature then available such as Mumford's "red book". So the non-obvious points do need checking. It's worth recalling the context from which Borel's second edition (1991) arose. He was invited to give a short lecture course at Columbia in Spring 1968, an "interesting" time there as indicated in the footnote to the original Introduction of the W.A. Benjamin lecture notes published in 1969. In that Introduction, reproduced in the second edition, Borel points out that Bass wrote up the lecture notes and was largely responsible for assembling the ad hoc background material in AG. Later circumstances didn't favor careful refinement by Borel of the original lecture notes; he mainly expanded and revised some later parts. He had not originally planned to do a second edition, which motivated me to write my own book. But his later lectures on the structure theory led him to streamline some arguments and made it natural to add some further topics.