# What are the higher $\mathrm{Ext}^i(A,\mathbf{G}_m)$'s, where $A$ is an abelian scheme?

Let $S$ be a base scheme, let $A/S$ be an abelian scheme, and let $\mathbf{G}_m/S$ be the multiplicative group; consider $A$ and $\mathbf{G}_m$ as objects in the abelian category of sheaves of abelian groups on the fppf site of $S$, and take $\mathrm{Ext}$'s between them. We know that $\mathrm{Ext}^1(A,\mathbf{G}_m)$ is the dual abelian scheme; but what is $\mathrm{Ext}^i(A,\mathbf{G}_m)$ for $i>1$?

Here is an argument why these higher $\mathrm{Ext}$'s contain no important information: the dual abelian scheme captures already all the data of $A/S$, since applying $\mathrm{Ext}^1(\cdot, \mathbf{G}_m)$ one more time recovers $A$; therefore the higher $\mathrm{Ext}$'s cannot hold any more information. Still, we should know explicitly what they are.

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just wondering, can you give me a reference about Ext^1(A;Gm)=A^v? I never know this thing. thank you! –  natura Jan 26 '10 at 8:12
@basic: I'm afraid I know no published reference, but will give you a proof if you give me a bigger margin: please make this a separate question and I will respond there. –  Thanos D. Papaïoannou Jan 26 '10 at 13:59

It seems that here you are talking about the local exts, i.e. about the sheaves $\underline{Ext}^{i}(A,G_m)$ on $S$. The question is rather tricky actually. The problem is that before we try to answer it, we should first specify what we mean by $\underline{Ext}^{i}(A,G_m)$. We could mean exts in the category of sheaves of abelian groups in the flat topology on $S$, or we could mean exts in the category of commutative group schemes. The latter is not an abelian category so you have to do something before you can define exts. It is possible to do this though. The standard lore is to use Yoneda exts. This is carried out in detail in the LNM 15 book by Oort. Among other things Oort checks that if $S$ is the spectrum of an algebraically closed field, then ext sheaves are all zero for $i \geq 2$.

Over general base schemes the situation is more delicate. First of all there are examples of Larry Breen showing that the ext sheaves in the category of sheaves of abelian groups are strictly larger than the ext sheaves in the category of commutative group schemes. In his thesis

Breen, Lawrence Extensions of abelian sheaves and Eilenberg-MacLane algebras. Invent. Math. 9 1969/1970 15--44.

Breen also showed that over a regular noetherian base schemes the global ext groups (in either category) are torsion if $i \geq 2$. Later in

Breen, Lawrence Un théorème d'annulation pour certains $E{\rm xt}\sp{i}$ de faisceaux abéliens. Ann. Sci. École Norm. Sup. (4) 8 (1975), no. 3, 339--352.

he strengthened his result to show that the higher exts sheaves are always zero for $1 < i < 2p-1$, where $p$ is a prime which is smaller than the (positive) residue characteristic of any closed point in $S$.

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Thank you, but where precisely in Oort, LNM 15 is the results that over an algebraically closed field the higher Ext's are zero? A cursory search revealed nothing. –  Thanos D. Papaïoannou Oct 19 '09 at 21:05
It is Theorem 14.1 in Chapter II. Incidentally, the characteristic zero case was originally proven by Serre in his paper on proalgebraic groups: archive.numdam.org/article/PMIHES_1960__7__5_0.pdf –  Tony Pantev Oct 20 '09 at 12:29