A group scheme over a base scheme is an -scheme with multiplication , an identity section , and inversion satisfying the group axioms as equalities of morphisms.
A homomorphism of group schemes is a morphism over the base that commutes with multiplication: . It consequently preserves the identity and inversion.
For a commutative group scheme , the multiplication-by- morphism adds a point to itself times. On an abelian variety, it is surjective for every nonzero integer .
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In algebraic geometry and number theory, a **group scheme** is a scheme that has the structure of a group, in the sense that it supports the operations of multiplication and inversion in a way that is compatible with the geometric structure.