in SGA 3, exposé X, we find the following classification result (corollaire 1.2):

Let $S$ be a connected scheme and let $\xi:\mathrm{Spec}(\Omega) \to S$ be a geometric point. Let $\pi=\pi_1(S,\xi)$ be the étale fundamental group. Then the category of isotrivial group schemes over $S$ is anti-equivalent to the category of discrete $\pi$-modules.

Here the equivalence is given by taking a isotrivial group scheme $G$ over $S$ to $\mathrm{Hom}_{\Omega\textrm{-Grp}}(G_\xi,\Bbb G_{m,\Omega})$ which carries a $\pi$-action.

A group scheme $G$ over $S$ is called isotrivial if we find a finite étale surjective morphism $S' \to S$ such that $G'=:G \times_S S'$ is diagonalizable.

My confusion is this: Suppose for simplicity that $S=\mathrm{Spec}(k)$ is the spectrum of a field. Then $\pi$ is isomorphic to the absolute Galois group of $k$ and we are supposed to have an anti-equivalence between $\pi$-modules and isotrivial $k$-group schemes given by base-changing to an algebraic closure and then taking the group of characters. But due to the definition of isotrivial group schemes, every isotrivial group scheme becomes diagonalizable over a finite field extension $L/k$, but this implies that the associated $\pi$-module has the property that not only all stabilizers are open, but also the intersection of all stabilizers, which is not true for all $\pi$-modules, e.g. if $\pi=\widehat{\Bbb Z}$ acts on $M=\bigoplus \Bbb Z/n\Bbb Z$ on each component via the projection $\widehat{\Bbb Z} \to \Bbb Z/n\Bbb Z$.

So I think that there's a mistake in the above result and the category of isotrivial group schemes over $S$ is actually anti-equivalent to the category of $\pi$-modules whose action factors through a finite quotient.

At least over a field, it's not hard to change the definitions so that we actually obtain all discrete $\pi$-modules. Just say that a $k$-group scheme is geometrically diagonalizable if the fiber over a geometric point $\mathrm{Spec}(\Omega) \to k$ is diagonalizable. Fix a separable closure $k^{sep}/k$. Then the category of geometrically diagonalizable $k$-group schemes is anti-equivalent to discrete $\pi$-modules via the functors $G \mapsto \mathrm{Hom}_{k^{sep}\textrm{-Grp}}(G_{k^{sep}},\Bbb G_{m,k^{sep}})$ and $M \to \mathrm{Spec}((k^{sep}[M])^{\pi})$ where for a $\pi$-module $M$, the action of $\pi=\mathrm{Gal}(k^{sep}/k)$ on the group Hopf algebra $k^{sep}[M]$ is induced from the actions on $k^{sep}$ and $M$. It's not hard to obtain this result directly with Galois descent (without using any of the much harder results from SGA), but I'm not sure how to adapt this to the case of a general base scheme $S$.

  • 1
    $\begingroup$ I think you are correct. Either Grothendieck needs to relax his definition of "isotrivial" or in X 1.2 he needs to consider only Galois modules on which an open subgroup of $\pi$ acts trivially (or better, consider only groups of finite type and finitely generated modules). This doesn't seem to have been fixed in the online draft of the new edition of SGA 3. $\endgroup$ – anon Feb 9 at 19:55
  • $\begingroup$ @anon Thanks! Do you happen to know a suitable relaxed definition of "isotrivial" (maybe one can replace "finite étale" by "proétale" or something like that, but I'm not knowledgable about this)? $\endgroup$ – Lukas Heger Feb 9 at 21:15

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.