In the book "Degeneration of abelian varieties" by Faltings, Chai it reads (cf. p.81,82) as if the following holds:


take an abelian scheme $G$ over a base scheme $S$ with dual scheme $\hat{G}$ (for my applications you may always assume $S$ to be the spectrum of a field) and $P$ the Poincaré bundle on $G\times \hat{G}$.


They consider then the pullback of $P$ to the first infinitesimal neighborhood of $G\times {0}$ in $G\times \hat{G}$.


They say that so one gets a line bundle over $G\times Spec(\mathcal O\_S \oplus \epsilon \Omega\_{\hat{G}})$, which is a first order infinitesimal deformation of the trivial sheaf on $G$.


My questions:


(1) Most important: What do they mean with $Spec(\mathcal O\_S \oplus \epsilon \Omega\_{\hat{G}})$? It's not clear to me what this notation should mean and why the first inf. neighborhood of $0$ in $\hat{G}$ looks like this.


(2) Why is this a deformation?



Question about Faltings, Chai: Degeneration of abelian varieties
Let $G$ be any group scheme over $S$. Let $I\subset\mathcal{O}\_G$ be the augmentation ideal defining the identity section of $G$ over $S$. Then, by definition, the first infinitesimal neighborhood of the identity in $G$ is the closed sub-scheme of $G$ defined by the ideal $I^2$. Moreover, $I/I^2$ is a coherent sheaf on $S$: it is nothing but the sheaf of invariant differentials on $G$, namely $\Omega\_{G}$. In the book 'Neron Models', for example, you will find in Proposition 4.2.1 that there is a canonical isomorphism from $e^\*\Omega\_{G/S}$ to $\Omega\_G$; here, $\Omega\_{G/S}$ is the *full* sheaf of differentials on $G$ and $e:S\to G$ is the identity section. 


Now, $\Omega\_{G/S}$ is the ideal of the diagonal embedding of $G$ in its first order neighborhood $G^{(1)}\subset G\times\_S G$. Pull everything back by the map $g\mapsto (e,g)$ from $G$ to $G\times G$. Then the diagonal embedding pulls back to the identity section and its first order neighborhood pulls back to the first order neighborhood of $e$ in $G$. So we see that $e^\*\Omega\_{G/S}=I/I^2$.


So $\mathcal{O}\_{G}/I^2$ is non-canonically isomorphic to $\mathcal{O}\_S\oplus\epsilon\Omega\_G$, where $\epsilon^2=0$. This is because we have a sequence of maps
$$\mathcal{O}\_S\to \mathcal{O}\_G/I^2\to \mathcal{O}\_G/I=\mathcal{O}\_S$$
of sheaves of rings supported on the identity section. In particular, the short exact sequence
$$0\to \Omega\_G\to\mathcal{O}\_G/I^2\to\mathcal{O}\_S\to 0$$
admits a section. Which means that we can write $\mathcal{O}\_G/I^2$ as a direct sum $\mathcal{O}\_{S}\oplus\Omega\_G$, with the multiplication given by
$$(f,\omega)\cdot(g,\omega')=(fg,f\omega'+g\omega).$$
Writing it as $\mathcal{O}\_S\oplus\epsilon\Omega\_G$ is just a compressed way to indicate this.


That this pull-back is a deformation of the trivial sheaf follows from the fact that the restriction of the Poincare bundle to $G\times 0$ is canonically trivialized.


