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 $S$?



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


