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The category of affine commutative -groups is an abelian category.
As such it has in particular kernels and cokernels.
Remark: A category is abelian if it is Ab-enriched( i.e. enriched over the category of abelian groups) and has finite limits and finite colimits and every monomorphism is a kernel and every epimorphism is a cokernel.
Let be a morphism in ..
The following conditions are equivalent: is a monomorphism, is surjective (i.e. is a closed subgroup of ), is a kernel.
The following conditions are equivalent: is a epimorphism, is injective, exhibits as a faithful flat module, is a cokernel.
If is a field extension skalar extension is an exact functor.
The category satisfies the axiom (AB5): it has directed limits and the directed limit of an epimorphism is an epimorphism.
By Cartier duality, the dual statements hold for the category of com- mutative -formal-groups.