Uniqueness, weak Neron property, etale base change and local nature of Neron models

Let $X^{\mathrm{\prime }}$ be another Neron model of $X_K$. Applying the mapping property of $X$ to the identity $K$-morphism $X_K^{\mathrm{\prime }}\to X_K$ gives $u:X^{\mathrm{\prime }}\to X$ over $S$, and applying the mapping property of $X^{\mathrm{\prime }}$ to the identity $X_K\to X_K^{\mathrm{\prime }}$ gives $v:X\to X^{\mathrm{\prime }}$; the composites $uv$ and $vu$ extend the identity $K$-morphisms and both sides are $S$-morphisms, so by the uniqueness clause they are identities. This proves (a).

Let $s\in S$ be closed and $R^{\mathrm{\prime }}$ an etale local ${\mathcal{O}}_{S,s}$-algebra with fraction field $K^{\mathrm{\prime }}$. It is a filtered limit of pointed etale neighbourhoods $V_i\to U_i$ with $U_i\subseteq S$ open and containing $s$. The given $K^{\mathrm{\prime }}$-point of $X_K$ descends to the generic fibre of one such neighbourhood after passing to a later stage, by [F4]. Each $V_i\to U_i\to S$ is smooth, so the Neron property extends that stage point to $V_i\to X$; base change along the limit gives the required $R^{\mathrm{\prime }}$-point of $X$. Thus the extension property for etale points holds at every closed point and $X$ is a weak Neron model, proving (b).

For (c), let $S^{\mathrm{\prime }}\to S$ be etale with function field $K^{\mathrm{\prime }}$ and let $Y^{\mathrm{\prime }}$ be a smooth $S^{\mathrm{\prime }}$-scheme with a $K^{\mathrm{\prime }}$-morphism $u_{K^{\mathrm{\prime }}}:Y_{K^{\mathrm{\prime }}}^{\mathrm{\prime }}\to X_{K^{\mathrm{\prime }}}$. The composite $Y^{\mathrm{\prime }}\to S^{\mathrm{\prime }}\to S$ is smooth by [F2], and $u_{K^{\mathrm{\prime }}}$ composed with the projection $X_{K^{\mathrm{\prime }}}=X_K{\times }_K{K}^{\mathrm{\prime }}\to X_K$ is a $K$-morphism $Y_K^{\mathrm{\prime }}\to X_K$; by the mapping property it extends uniquely to an $S$-morphism $Y^{\mathrm{\prime }}\to X$. Combining this with the structure morphism $Y^{\mathrm{\prime }}\to S^{\mathrm{\prime }}$ gives an $S^{\mathrm{\prime }}$-morphism $Y^{\mathrm{\prime }}\to X{\times }_S{S}^{\mathrm{\prime }}=X_{S^{\mathrm{\prime }}}$ extending $u_{K^{\mathrm{\prime }}}$, and uniqueness follows from the uniqueness in the $S$-mapping property. Since $X_{S^{\mathrm{\prime }}}$ is smooth separated of finite type over $S^{\mathrm{\prime }}$ by [F2] and has generic fibre $X_{K^{\mathrm{\prime }}}$, it is a Neron model of $X_{K^{\mathrm{\prime }}}$.

For (d), first suppose $X$ is a Neron model over $S$. Cover a smooth $\mathrm{Spec}{\mathcal{O}}_{S,s}$-scheme by affine standard-smooth charts. Their finite-presentation presentations and invertible Jacobian minors descend along the filtered limit of open neighbourhoods of $s$ to smooth charts over some neighbourhood, by [F4]; the generic-fibre morphism to $X_K$ descends at a later stage by the same finite-presentation lemma. Apply the Neron property over $S$ to each descended smooth neighbourhood and pass to the limit. These local extensions agree on overlaps by uniqueness, so they glue to an extension over $\mathrm{Spec}{\mathcal{O}}_{S,s}$. Uniqueness follows from separatedness and density of the generic fibre.

Conversely, suppose each localization $X_{{\mathcal{O}}_{S,s}}$ is a Neron model. First take a smooth finite-type $S$-scheme $Y$ and a $K$-morphism $u_K:Y_K\to X_K$. For each closed $s$, the local property gives $Y_{{\mathcal{O}}_{S,s}}\to X_{{\mathcal{O}}_{S,s}}$. Since $Y$ and $X$ are of finite presentation over the noetherian base, [F4] descends this map to $Y_U\to X_U$ for some open neighbourhood $U$ of $s$, with generic restriction $u_K$ after a further shrinking if needed. These neighbourhoods cover the closed points of $S$, and together with the generic point cover $S$. The descended maps agree on overlaps because they agree on $Y_K$, which is schematically dense in the flat scheme $Y$, and $X$ is separated. They therefore glue to an extension $Y\to X$. For an arbitrary smooth $Y\mathrm{/}S$, cover it by finite-type open subschemes, apply this argument on each, and glue by uniqueness. This proves the converse and the locality assertion.

For (e), assume $X_K$ is a $K$-group scheme with multiplication $m_K$, inverse $i_K$ and unit $e_K$. Apply the mapping property to the smooth $S$-schemes $X{\times }_{S}X$, $X$ and $S$ and the $K$-morphisms $m_K$, $i_K$ and $e_K$: this yields $S$-morphisms $m:X{\times }_{S}X\to X$, $i:X\to X$ and $e:S\to X$ extending them uniquely. Each group identity, for example associativity, is an identity between $S$-morphisms from the reduced smooth $S$-scheme $X{\times }_{S}X{\times }_{S}X$ to the separated $S$-scheme $X$; it holds after restriction to the generic fibre, which is schematically dense by [F3], so it holds everywhere. Thus $X$ is an $S$-group scheme.