# finitness of syntomic/fppf cohomology with coefficients in a finite flat group scheme

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.

• 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? Dec 17 '16 at 7:49
• @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. Dec 18 '16 at 11:08
• @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. Dec 18 '16 at 11:56
• So do you think the statement is wrong for $X$ higher dimensional? Or just that there is no easy generalisation of Milne's argument?
– user19475
Dec 18 '16 at 11:58
• So far we only discussed the latter. It is very well possible that the result is still true by a different argument. 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

• Thank you very much! I will read this in the next weeks!
– user19475
Dec 22 '16 at 5:55
• 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?
– user19475
Dec 23 '16 at 18:03
• 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.
– user19475
Dec 23 '16 at 20:13
• 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 $Y$.
– darx
Dec 23 '16 at 23:15
• 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.
– darx
Dec 24 '16 at 11:22