Existence of Neron models for abelian varieties over a discrete valuation ring

By [F1] construct the separated minimal model $X$ of $A_K$, its strictification $U$, and finally by [F2] the descended smooth separated finite-type $R$-group scheme $G$ with generic fibre $G_K=A_K$ and $X\subseteq G$ as an $R$-dense open.

To prove the mapping property, let $Z$ be a smooth $R$-scheme and $u_K:Z_K\to A_K$ a $K$-morphism. Work first with $Z$ of finite type; arbitrary $Z$ is covered by finite-type opens and the unique extensions glue. Put $Y=Z{\times }_{R}X$. On its generic fibre define ${\theta }_K:Y_K\to G_K$ by ${\theta }_K(z,x)=u_K(z)x$, using the group law of $G_K=A_K$. For each generic point $\eta$ of an irreducible component of $Z_k$, the local ring $R^{\mathrm{\prime }}={\mathcal{O}}_{Z,\eta }$ is a DVR, with fraction field $K^{\mathrm{\prime }}$, and restriction of $u_K$ along $\mathrm{Spec}{K}^{\mathrm{\prime }}\to Z_K$ gives a point of $A_K(K^{\mathrm{\prime }})$. The translation supplier Separated minimal union and translations extends translation by this point to an $R^{\mathrm{\prime }}$-birational self-map of $X_{R^{\mathrm{\prime }}}$ which is an open immersion on its $R^{\mathrm{\prime }}$-dense domain. This domain contains the generic point of every component of the special fibre of $X_{R^{\mathrm{\prime }}}$, so ${\theta }_K$ extends at the corresponding generic points of $Y_k$. Since $R^{\mathrm{\prime }}$ is the filtered colimit of the rings of affine neighbourhoods of $\eta$, finite-presentation descent [F5] spreads a quasi-compact open neighbourhood and its morphism to $G$ to a neighbourhood in $Y$ of each such point. There are finitely many vertical generic points. Together with the generic fibre, these neighbourhoods form an $R$-dense open in $Y$; the local maps agree on overlaps because the generic fibre is schematically dense in each overlap and $G$ is separated ([F4]), so they glue to an $R$-rational map $\theta :Y⇢G$. It is defined at every height-one point of $Y$: the horizontal ones lie in $Y_K$, and each vertical height-one point is the generic point of a component of $Y_k$ just treated. Since $Y$ is smooth over the regular Noetherian DVR and $G$ is a smooth separated finite-type group scheme, [F3] extends $\theta$ uniquely to a morphism $\mathrm{\Theta }:Y\to G$.

Let $j:X\hookrightarrow G$ be the dense open embedding from [F2], and let ${\iota }_G$ and $m_G$ be inversion and multiplication on $G$. Define $h:Y=Z{\times }_{R}X\to G$ by $h(z,x)=m_G(\mathrm{\Theta }(z,x),{\iota }_G(j(x)))\mathrm{.}$ On $Y_K$ this is $u_K(z)x{x}^{-1}=u_K(z)$, independent of $x$. The projection $p:Y\to Z$ is faithfully flat, quasi-compact and locally finitely presented because $X\to \mathrm{Spec}R$ is smooth, finite type and faithfully flat. On $Y{\times }_{Z}Y=Z{\times }_{R}X{\times }_{R}X$, the two pullbacks of $h$ agree on the generic fibre, hence everywhere by [F4] and separatedness of $G$. Fppf descent [F5] therefore gives a unique $R$-morphism $u:Z\to G$ with $u\circ p=h$; its generic fibre is $u_K$. Uniqueness follows because any two extensions agree on the schematically dense generic fibre and $G$ is separated. This proves existence and uniqueness of the extension for all smooth $Z$.

Uniqueness of the model: if $N$ and $N^{\mathrm{\prime }}$ both satisfy the extension property with generic fibre $A_K$, then the identity of $A_K$ extends to $R$-morphisms $N\to N^{\mathrm{\prime }}$ and $N^{\mathrm{\prime }}\to N$. Both composites extend the identity of $A_K$, hence equal the respective identities by the uniqueness clause of the extension property (applied to the models themselves); so $N\cong N^{\mathrm{\prime }}$ uniquely, and [F4] identifies the canonical isomorphism.

The construction used only an arbitrary discrete valuation ring: the minimal model, strictification, strict-henselian completion, effective group descent and Weil extension all hold without excellence, completeness, perfect residue field or any restriction on the reduction type, so the stated class is exactly as claimed.