I'm studying the Weil Restriction $\mbox{Res}_{k/\textbf{F}_p}$ where $\textbf{F}_p$ is the field with $p$ elements and $k$ is a perfect field over $\textbf{F}_p$, not necessarily finite.

In this case, it is known that the Weil restrictions for affine schemes over $k$ are representable by ind-schemes, and I want to know what happens for projective schemes.

More concretely, the question is: Is $\mbox{Res}_{k/\textbf{F}_p} \textbf{P}^1_k$ representable by an ind-scheme?

There are two things I have thought so far:

  1. I tried to construct $\mbox{Res}_{k/\textbf{F}_p} \textbf{P}^1$ by trying to "glue" two copies of $\mbox{Res}_{k/\textbf{F}_p} \textbf{A}^1$ along the induced map $f: \mbox{Res}_{k/\textbf{F}_p} \textbf{G}_m \rightarrow \mbox{Res}_{k/\textbf{F}_p} \textbf{A}^1$, but it does not work since the map $f$ does not seem to be an "open immersion" (being invertible is not an open condition in the ind-scheme $\mbox{Res}_{k/\textbf{F}_p} \textbf{A}^1$).

  2. Another way is to study the quotient of $\mbox{Res}_{k/\textbf{F}_p} \textbf{A}^2 - \{0\}$ by the action of $\mbox{Res}_{k/\textbf{F}_p} \textbf{G}_m$, but I do not know how to proceed.

Any help or suggestion would be greatly appreciated.


This is more of a long comment than an answer, but I hope you find it useful.

I will first explain one way to construct the Weil restriction of a quasi-projective variety $X$ over $L$, when $L/K$ is a finite separable extension of degree $n$. One first considers the finite product

$$\prod_{\sigma: L \to K^s} X^{\sigma},$$

where the product is over all embeddings of $L$ into a separable closure $K^s$ of $K$, and $X^{\sigma}$ denotes the conjugate of $X$ with respect to $\sigma$ (when $L/K$ is Galois the picture is simpler as one can work with Galois groups instead of embeddings). This admits a natural action of $\mbox{Gal}(K^s/K)$, and on taking the quotient by this action we obtain $\mbox{Res}_{L/K} X$.

So for example, the Weil restriction of the projective line $\mathbb{P}_L^1$ is isomorphic to $(\mathbb{P}_{K^s}^1)^n$ over $K^s$, but is not isomorphic to $(\mathbb{P}_{K}^1)^n$. In particular one sees that your approach (1) does not work even in this case. Indeed, whilst the Weil restriction respects open immersions, it does not respect open coverings. Here we have $\mbox{Res}_{L/K} \mathbb{A}_L^1 = \mathbb{A}_K^n$, however two copies of this are clearly not enough to cover $\mbox{Res}_{L/K} \mathbb{P}_L^1$.

Your approach (2) seems more likely to work, and perhaps after translating the problem you should recover the construction I just gave above.

I would personally approach the problem by trying to mimic the construction I just gave for the case of infinite field extensions, by taking limits of appropriate products and then quotienting out by suitable Galois actions. Good luck.

  • $\begingroup$ Thanks a lot for your comment. In approach (1), what I really meant is to construct an inductive limit of ind-schemes, each of which is obtained from glueing finite number of copies of ind-affine schemes. Anyway, the point is that Weil restriction does NOT respect open immersion in general when $k / \textbf{F}_p$ is not finite, so the glueing process cannot be done. And I think the construction by Galois action does not work neither when $k / \textbf{F}_p$ is not finite, since $\mbox{Gal}(k / \textbf{F}_p)$ is profinite but is NOT an inductive limit of FINITE subgroups in general. $\endgroup$
    – david
    Jun 23 '14 at 19:30

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