Néron model

In algebraic geometry, the Néron model (or Néron minimal model, or minimal model) for an abelian variety A_K defined over the field of fractions K of a Dedekind domain R is the "push-forward" of A_K from Spec(K) to Spec(R), in other words the "best possible" group scheme A_R defined over R corresponding to A_K.

They were introduced by André Néron (1961, 1964) for abelian varieties over the quotient field of a Dedekind domain R with perfect residue fields, and Raynaud (1966) extended this construction to semiabelian varieties over all Dedekind domains.

Suppose that R is a Dedekind domain with field of fractions K, and suppose that A_K is a smooth separated scheme over K (such as an abelian variety). Then a Néron model of A_K is defined to be a smooth separated scheme A_R over R with fiber A_K that is universal in the following sense.

In particular, the canonical map ${\displaystyle A_{R}(R)\to A_{K}(K)}$ is an isomorphism. If a Néron model exists then it is unique up to unique isomorphism.

In terms of sheaves, any scheme A over Spec(K) represents a sheaf on the category of schemes smooth over Spec(K) with the smooth Grothendieck topology, and this has a pushforward by the injection map from Spec(K) to Spec(R), which is a sheaf over Spec(R). If this pushforward is representable by a scheme, then this scheme is the Néron model of A.

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