Abelian varieties over a field

Let $k$ be a field. A group variety over $k$ means a smooth separated finite-type $k$-scheme $G$ equipped with $k$-morphisms $m:G{\times }_{k}G\to G$, $i:G\to G$, and $e:\mathrm{Spec}k\to G$ satisfying associativity, the two identity laws, and the two inverse laws as identities of scheme morphisms. A homomorphism respects these maps. A closed subgroup scheme is a closed subscheme on which these maps restrict; it is normal if conjugation factors through it.

An abelian variety over $k$ is a proper geometrically connected group variety over $k$. This definition does not include projectivity as an assumption. Commutativity follows from A proper geometrically connected group variety is commutative ↗. The identity gives a $k$-rational point. Using AC for the referenced smoothness/regularity suppliers, smoothness and geometric connectedness imply geometric integrality: after any algebraically closed field extension the local rings are regular, so distinct irreducible components cannot meet; the finitely many irreducible components are therefore open and closed, and connectedness leaves exactly one.

A pseudo-abelian variety is a smooth connected finite-type $k$-group scheme with no nontrivial smooth connected affine normal subgroup scheme. In this terminology connectedness is ordinary connectedness, not a replacement for properness. Over imperfect fields a pseudo-abelian variety need not be proper.

Milne, Algebraic Groups, Definition 8.3, pp. 149–150, and Chapter 8 definitions of complete connected group varieties; Stacks, Section 39.9 [0BF9], Definition 39.9.1 [03RO]. The smoothness requirement excludes finite nonreduced group schemes.