Neron models, the Neron mapping property and weak Neron models

Let $S$ be a Dedekind scheme (Dedekind domains) with function field $K$, and let $X_K$ be a smooth separated $K$-scheme of finite type. For an $S$-scheme $X$, its generic fibre is $X_K=X{\times }_S\mathrm{Spec}K$, the fibre of $X\to S$ over the generic point of $S$ (Scheme-theoretic fibre); it is a $K$-scheme. An $S$-model of $X_K$ is an $S$-scheme $X$ together with a specified isomorphism $X_K\cong X{\times }_S\mathrm{Spec}K$ of $K$-schemes.

A Neron model of $X_K$ over $S$ is an $S$-model $X\to S$ which is smooth (Smooth morphism of schemes), separated (Separated morphism of schemes), of finite type (Locally finite type and finite type morphisms), and which satisfies the Neron mapping property: for every smooth $S$-scheme $Y$ and every $K$-morphism $u_K:Y_K\to X_K$ there is a unique $S$-morphism $u:Y\to X$ extending $u_K$, that is, with $u{\times }_S\mathrm{Spec}K=u_K$ under the specified identifications.

Equivalently, $X$ represents the functor $Y\mapsto {\mathrm{Hom}}_K(Y_K,X_K)$ from smooth $S$-schemes to sets, so by the Yoneda lemma a Neron model of $X_K$ is determined up to a unique isomorphism: applying the property to $Y=X$ and to its identity $K$-morphism shows that any two Neron models of $X_K$ admit a unique $S$-isomorphism over $X_K$.

Local nature. For a closed point $s\in S$ the local ring ${\mathcal{O}}_{S,s}$ is a discrete valuation ring with fraction field $K$, and $X{\times }_S\mathrm{Spec}{\mathcal{O}}_{S,s}$ is a Neron model of $X_K$ over the local Dedekind scheme $\mathrm{Spec}{\mathcal{O}}_{S,s}$ whenever $X$ is a Neron model of $X_K$ over $S$; conversely, an $S$-model of finite type over $S$ is a Neron model if each of its localizations at closed points is. The finite-type hypothesis is essential for this converse. Thus the notion is local on $S$.

Weak Neron models. A scheme $X$ over the Dedekind scheme $S$ satisfies the extension property for etale points at a closed point $s\in S$ if for each etale local ${\mathcal{O}}_{S,s}$-algebra $R^{\mathrm{\prime }}$ (a local ring with a local homomorphism ${\mathcal{O}}_{S,s}\to R^{\mathrm{\prime }}$ that is etale, Étale morphism of schemes), with fraction field $K^{\mathrm{\prime }}$, the canonical map $X(R^{\mathrm{\prime }})\to X_K(K^{\mathrm{\prime }})$ is surjective. A weak Neron model of $X_K$ is a smooth separated finite-type $S$-model $X$ of $X_K$ satisfying the extension property for etale points at every closed point of $S$. When $X$ is separated over $S$ the displayed map is injective by the valuative criterion of separatedness, so the extension property is then a bijection.

The Neron mapping property applied to $Y=X$ shows that a Neron model of $X_K$ is unique up to a unique isomorphism inducing the specified identity on $X_K$ and, applied to etale $S$-schemes $Y$, shows that a Neron model is in particular a weak Neron model. This definition asserts no existence statement: it describes what it means for a model to be a Neron model, and every existence claim on this page is a theorem with its own hypotheses. The weak Neron property is not asserted here to characterize Neron models.