Let $X/k$ be a smooth projective variety over a finite field of characteristic $p$ and $\mathscr{A}/X$ be an Abelian scheme.

Is then $H^1_\mathrm{SYN}(X,\mathscr{A}[p]) = H^1_\mathrm{fppf}(X,\mathscr{A}[p])$ finite?

This is true if $X$ is a curve, see [Milne, Arithmetic Duality Theorems http://jmilne.org/math/Books/ADTnot.pdf ], p. 292, Lemma III.8.9.

Edit: I know it in the case $\mathscr{A} = A \times_k X$ is a constant Abelian scheme since then $H^1(X,\mathscr{A})[p]$ is finite and the Kummer sequence induces a short exact sequence $$0 \to \mathscr{A}(X)/p \to H^1_\mathrm{SYN}(X,\mathscr{A}[p]) \to H^1(X,\mathscr{A})[p] \to 0$$ and $\mathscr{A}(X)/p$ is finite by the Mordell-Weil theorem.

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    $\begingroup$ From a quick inspection it seems that Theorem III.5.6 implies the case where $\mathcal A[p]$ or its dual has height one. The only place where the proof of Lemma 8.9 uses that $X$ is a curve is to make a filtration by height one pieces (using Lemma B.1). Is this known to fail for higher-dimensional varieties? $\endgroup$ Dec 17 '16 at 7:49
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    $\begingroup$ @R.vanDobbendeBruyn I don't think any nontrivial filtration can exist of $\mathcal A[p]$ for $A$ the moduli space of abelian varieties - over the ordinary locus, only the canonical subgroup is preserved by monodromy, but that doesn't extend to the supersingular points. Presumably things aren't much better on compact Shimura subvarieties, say. $\endgroup$
    – Will Sawin
    Dec 18 '16 at 11:08
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    $\begingroup$ @WillSawin: ah, that makes sense. In fact, going from the ordinary to the supersingular locus makes me appreciate the existence of such a filtration over a curve much more. $\endgroup$ Dec 18 '16 at 11:56
  • $\begingroup$ So do you think the statement is wrong for $X$ higher dimensional? Or just that there is no easy generalisation of Milne's argument? $\endgroup$
    – user19475
    Dec 18 '16 at 11:58
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    $\begingroup$ So far we only discussed the latter. It is very well possible that the result is still true by a different argument. $\endgroup$ Dec 18 '16 at 11:59

Let $k$ be a finite field. Let $X$ be a normal proper variety. Let $G$ be a finite flat commutative group scheme over $X$ of order a power of $p$.

Lemma 1. If $T$ is a $G$-torsor over $X$ and $T$ is trivial over the generic point of $X$, then $T$ is trivial.

Proof. Namely, let $X' \subset T$ be the scheme theoretic closure of the rational section we get from the triviality of $T$ over the generic point. Then $X' \to X$ is a birational finite morphism, hence an isomorphism as $X$ is normal. QED

Lemma 2. Let $Z$ be a proper scheme over $X$. Then $G(Z)$ is finite.

Proof. We may pull back $G$ to $Z$. Then we see it suffices to show that given $\pi : Y \to Z$ finite flat, there are only a finite number of sections $\tau$ of $\pi$. This is clear because a section is given by a $\mathcal{O}_Z$-algebra map $\pi_*\mathcal{O}_Y \to \mathcal{O}_Z$ and we have finiteness of $H^0(Z, \mathcal{H}om(\pi_*\mathcal{O}_Y, \mathcal{O}_Z))$ as $Z$ is proper over the finite field $k$.

Lemma 3. Let $Y \to X$ be an alteration of proper varieties with $X$ normal. Then $\ker(H^1(X, G) \to H^1(Y, G))$ is finite.

Proof. If $Y \to X$ is a blow up, then the kernel is trivial by Lemma 1. There exists a blowup $X' \to X$ such that the strict transform $Y'$ of $Y$ is flat over $X'$ (google "flattening by blowing up"). Of course we may assume $X'$ is normalize (if not then just normalize $X'$). Combining these two steps we may assume $Y \to X$ is finite flat.

Assume $Y \to X$ is finite flat. Say $T \to X$ is a $G$-torsor which becomes trivial over $Y$. Choose a section $\sigma : Y \to T \times_X Y$. Using that $T \times_X (Y \times_X Y)$ is a $G$-torsor over $Y \times_X Y$ we can take the "difference" between $\sigma \circ \text{pr}_0$ and $\sigma \circ \text{pr}_1$ over $Y \times_X Y$ to get an element $$ \tau = \sigma \circ \text{pr}_0 - \sigma \circ \text{pr}_1 \in G(Y \times_X Y) $$ I leave it to the reader to see that $\tau$ determines the isomorphism class of the torsor $T$ by descent theory for the fppf covering $\{Y \to X\}$. By Lemma 2 there are only a finite number of $\tau$. QED

Proposition. $H^1(X, G)$ is finite.

Proof. By Lemma 3 we may replace $X$ by an alteration. Hence we may assume that over the function field of $X$ we have a filtration of $G$ by closed subgroup schemes such that the successive quotients have order $p$. (This step requires you to know about finite group schemes over fields; you can read about this in the book by Mumford about abelian varieties for example.) Any closed subgroup scheme of the generic fibre of $G$ extends to a finite flat closed subgroup scheme over a blowup of $X$ by the same flattening techniques as used in the proof of Lemma 3. Thus finally we may assume there is a filtration $$ (0) \subset G_1 \subset \ldots \subset G_{n - 1} \subset G_n $$ where $G_i/G_{i - 1}$ is a finite flat group scheme of order $p$. In this way (using the long exact cohomology sequence) we reduce to the case where $G$ has order $p$.

Now we use the classification of such group schemes over the normal variety $X$ (you can easily deduce what I say from the Oort-Tate paper). There are two cases.

The first case is where $G$ is a closed subgroup scheme of a line bundle $L$ over $X$. In this case there is a sequence $$ 0 \to G \to L \to L^{(p)} \to 0 $$ for some additive map as indicated. Since the cohomology of $L$ and $L^{(p)}$ are finite, we win.

The second case is where $G$ is a Galois twist of $\mu_p$ in the generic point. Here we can (after replacing $X$ by an alteration again if necessary) assume that $G|_U = \mu_{p, U}$ for some Zariski op $U$. Then we can use that $H^1(X, G) \subset H^1(U, G)$ by Lemma 1 and use the finiteness of $H^1(U, \mu_p)$. QED

  • $\begingroup$ Thank you very much! I will read this in the next weeks! $\endgroup$
    – user19475
    Dec 22 '16 at 5:55
  • $\begingroup$ Can you please give more details for: "By Lemma 3 we may replace $X$ by an alteration. Hence we may assume that over the function field of $X$ we have a filtration of $G$ by closed subgroup schemes such that the successive quotients have order $p$." and why the Tate-Oort classification gives exactly the two cases you mention? $\endgroup$
    – user19475
    Dec 23 '16 at 18:03
  • $\begingroup$ Regarding 1): The category of finite flat commutative group schemes of $p$-power order is Abelian, and over an algebraically closed field, its simple objects are $\mu_p,\alpha_p$ and $\mathbf{Z}/p$. Perhaps this helps; regarding 2): Since $X$ has characteristic $p$, the element $w_p$ of [Tate-Oort] is $= 0$, and if $a=0$, the $p$-Lie algebra of $G$ is $\mathscr{L}$ with the $p$-power morphism. $\endgroup$
    – user19475
    Dec 23 '16 at 20:13
  • $\begingroup$ If $Y \to X$ is an alteration with $Y$ normal and $H^1(Y, G)$ is finite, then Lemma 3 tells us $H^1(X, G)$ is finite. Hence we may replace $X$ by $Y$. $\endgroup$
    – darx
    Dec 23 '16 at 23:15
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    $\begingroup$ 1) If you have a closed subscheme of the generic fibre, then the closure is a closed subscheme of $G$ over $X$, but may not be flat. Then after a blowup the strict transform is flat and still a closed subscheme. 2) Just look at the equations for the group schemes in Oort-Tate paper. As you say one gets $\mu_p$, $Z/pZ$, of $\alpha_p$ in the generic point. In the last two cases looking at the invertible sheaf you mention gives a group scheme homomorphism $G \to L$ and then you just show the quotient is another line bundle $L'$ but the map $L \to L'$ is not linear, just additive. $\endgroup$
    – darx
    Dec 24 '16 at 11:22

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