2
$\begingroup$

Does anyone know a proof or reference for the following statement? Or if it's false (which seems unlikely to me), a counterexample?

Let $k$ be a field (maybe we need it to be perfect) and $A$ an abelian variety over $k$. Then the sheaf of abelian groups $\underline{\text{Ext}}^2(A,\mathbb{G}_m)$ on the fppf site of $k$-schemes is trivial.

$\endgroup$
4
$\begingroup$

For abelian schemes $\mathscr{A}/S$, $S$ being regular, $\mathscr{Ext}^i_{(\text{Sch}/S)_{\text{fppf}}}(\mathscr{A}, \mathbf{G}_m)$ is torsion for all $i\ge 2$. I recall this is in Breen's Inventiones paper "Extensions of abelian sheaves...", $\S$7. On the other hand if $p$ is a prime such that $p!$ is invertible locally on $S$, then for $1<i<2p-1$ such higher ext's actually vanish. This is discussed in Breen's paper "Un théorème d'annulation [...]", 1975, and relies on the former.

$\endgroup$
  • $\begingroup$ I think in that paper he only proves that the Ext group is torsion. $\endgroup$ – Justin Campbell Aug 10 '16 at 1:20
  • 4
    $\begingroup$ For $n > 0$, if we apply $\mathbf{R}\mathscr{H}om(\cdot, \mathbf{G}_m)$ to the $n$-torsion Kummer sequence of $\mathscr{A}$ then we get an exact sequence of sheaves $\cdots \rightarrow \mathscr{A}^{\vee} \stackrel{n}{\rightarrow} \mathscr{A}^{\vee} \rightarrow \mathscr{E}xt^1(\mathscr{A}[n], \mathbf{G}_m) \stackrel{\delta}{\rightarrow} \mathscr{E}xt^2(\mathscr{A}, \mathbf{G}_m)[n] \rightarrow 0$, so $\delta$ is an isomorphism. But the source of $\delta$ vanishes (SGA7, VIII, 3.3.1). So this $\mathscr{E}xt^2$-sheaf is torsion-free, and hence vanishes whenever it is torsion. $\endgroup$ – nfdc23 Aug 10 '16 at 1:30
  • 4
    $\begingroup$ @JustinCampbell: Sheafifying Ext$^i$ gives $\mathscr{E}xt^i$, by an erasable $\delta$-functor argument (since it holds for $i=0$). But the points of this sheaf valued in the base are generally not at all the original Ext$^i$, so is your use of the phrase "field-valued points" what you meant? $\endgroup$ – nfdc23 Aug 10 '16 at 1:33
  • $\begingroup$ I agree that one has to sheafify. My use of that phrase was confusing, I removed it. Can one deduce that the Ext^2 sheaf is torsion from Breen's theorem that the Ext^2 group is torsion? He works in the relative situation, but only over a regular base, so I'm worried about non-regular test schemes. $\endgroup$ – Justin Campbell Aug 10 '16 at 13:02

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Not the answer you're looking for? Browse other questions tagged or ask your own question.