Suppose I have a finite extension $K\subseteq L$ of perfect fields. To any smooth affine $L$-variety $V$ I can associate the Weil restriction $W=\mathrm{Res}_{L/K}(V)$, a smooth $K$-variety with an $L$-linear structure on the sheaf of differentials, compatible with the $K$-linear structure; that is, an extension of the action of $\mathcal{O}_W$ on $\Omega_{W/K}$ to $\mathcal{O}_W\otimes_K L$ such that the extension to $\overline{W}=W\otimes_K\overline{K}$ is a locally free $\mathcal{O}_{\overline W}\otimes_K L$-module. Call a smooth variety equipped with such a structure a smooth almost-$L$ variety, in analogy with almost-complex manifolds. My questions are:
- Is the Weil restriction functor from affine smooth $L$-varieties to affine smooth almost-$L$-varieties fully faithful?
- Is it essentially surjective? I doubt it is.
- If the answer to 2 is no, can we describe its essential image?
Of course, we can ask the same questions about locally analytic manifolds over complete DVRs or smooth rigid analytic spaces. I ask because the analogy of questions 1+2 came up for locally analytic manifolds over p-adic fields a few years ago and I still don't know a good answer.