Abelian schemes over a base

Let $S$ be a scheme. An abelian scheme over $S$ of relative dimension $g$ is an $S$-group scheme $f:A\to S$ in the sense of Group schemes over a base scheme such that:

1. $f$ is smooth (Smooth morphism of schemes);
2. $f$ is proper (Proper morphisms);
3. $f$ is locally of finite presentation (Locally finite presentation morphisms);
4. every geometric fibre $A_{\bar{s}}=A{\times }_S\mathrm{Spec}\bar{\kappa }(s)$ is connected of dimension $g$ (Geometric properties of fibres, Scheme-theoretic fibre); equivalently, the relative dimension is constant equal to $g$ (Relative dimension of a smooth morphism at a point) and every geometric fibre is nonempty and connected.

Equivalently, an abelian scheme over $S$ is a smooth proper $S$-group scheme whose geometric fibres are abelian varieties of dimension $g$ in the sense of Abelian varieties over a field; smoothness and properness make the fibres smooth proper connected group schemes, and conversely a family of abelian varieties of constant dimension which is a smooth proper $S$-group scheme is an abelian scheme. The condition on geometric fibres makes $g$ locally constant on $S$ and equal to $g$ on each connected component of $S$. Since $f$ is proper, it is separated, and the unit section $e:S\to A$ is a closed immersion, being a section of a separated morphism. The group law, inverse and unit are those of the $S$-group scheme structure; they are automatically $S$-morphisms of finite presentation.

No projectivity of $A$ over $S$ is asserted: an abelian scheme over a general base need not be projective over that base, and none of the results on this page assumes it. The base $S$ is not required to be Noetherian.