## Representability of sheaf of Ext^1 of a Néron model by $\mathbb{G}_m$

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Let's work over a trait $S=\mathrm{Spec}R$, where $R$ is a dvr with fraction field $K$, residue field $k$. Given an abelian variety $A_K$ with semi-stable reduction, let $A$ over $S$ be its Néron model and $A^{\circ}$ the neutral component. We know the sheaf $\mathscr{E}xt^1(A^{\circ},\mathbb{G}_m)$ is represented by the Néron model of the dual of $A_K$ over the category of smooth scheme over $S$, see (Mazur and Messing's LNM Universal extensions and one dimensional crystalline cohomology, chapter I section 5)

My question is: Is the sheaf $\mathscr{E}xt^1(A^{\circ},\mathbb{G}_m)$ also representable over the category of schemes over $S$?

Also we know the Poincaré biextension $W_K$ of $A_K$ and $A_K^'$ by $\mathbb{G}_m$ extends to a biextension $W$ of certain open subgroups of $A$ and $A'$ (i.e. the subgroups making the component pairing vanish), my second question is if $W$ is represented by a scheme over $S$?

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## 1 Answer

First question: no. Assume, to fix ideas, that $R$ is complete with uniformizer $\pi$, $k$ is algebraically closed, and $A$ is an elliptic curve with multiplicative reduction. Denote by $\mathscr{E}$ the Ext sheaf in question. Then the restriction of $A^\circ$ to `$S_n:=\mathrm{Spec\,}(R/(\pi^{n+1}))$` is isomorphic to $\mathbb{G}_{m}$, so $\mathscr{E}(S_n)$ is zero for all $n$. If $\mathscr{E}$ were a scheme, this would imply $\mathscr{E}(R)=0$ (the functor of points of a scheme "commutes with completion" for local rings), a contradiction.

Second question: yes, because $W$ is a $\mathbb{G}_m$-torsor over the product, and torsors under affine group schemes are schemes.

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Thanks a lot! Could you please tell me the reference for 1. "the functor of points of a scheme "commutes with completion" for local rings"; 2. "torsors under affine group schemes are schemes" – Heer Nov 25 at 16:53
@ Laurent Moret-Bailly: Is $A^{'\circ}$ like an "open" subsheaf of $\mathscr{E}xt^1(A,\mathbb{G}_m)$ over the category of schemes over S? How to understand the relation between the two? – Heer Nov 25 at 17:48
Concerning completions: I don't know a reference but this is an exercise. First observe that for an affine scheme, the functor of points commutes with arbitrary inverse limits of rings (this is trivial).    For an arbitrary scheme $X$, a filtered inverse system of local rings $R_i$ (with local transition maps) and compatible elements $f_i$ of $X(R_i)$, observe that all closed points of the spectra must map to the same point $x$ of $X$, hence all $f_i$'s must factor through any affine neighborhood of $x$, so we are reduced to the affine case. – Laurent Moret-Bailly Nov 26 at 18:52
Concerning representability of torsors: this is just flat descent of affine morphisms (SGA1, VIII, th. 2.1). For the special case of torsors, I am sure this is stated in Demazure-Gabriel.   – Laurent Moret-Bailly Nov 26 at 19:01
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