nLab Demazure, lectures on p-divisible groups, II.6, the category of affine k-groups

This entry is about a section of the text

• Michel Demazure, lectures on p-divisible groups, web

Theorem

The category $AC_k$ of affine commutative $k$-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 $AB$ of abelian groups) and has finite limits and finite colimits and every monomorphism is a kernel and every epimorphism is a cokernel.

Theorem

Let $f:G\to H$ be a morphism in $AC_k$..

1. The following conditions are equivalent: $f$ is a monomorphism, $O(f)$ is surjective (i.e. $G$ is a closed subgroup of $H$), $f$ is a kernel.

2. The following conditions are equivalent: $f$ is a epimorphism, $O(f)$ is injective, $O(f):O(H)\to O(G)$ exhibits $O(G)$ as a faithful flat $O(H)$ module, $f$ is a cokernel.

Corollary

If $k\hookrightarrow k^\prime$ is a field extension skalar extension $G\mapsto G\otimes_k k^\prime$ is an exact functor.

Corollary

Corollary

Corollary

Corollary

Corollary

Theorem

1. The category $AC_k$ satisfies the axiom (AB5): it has directed limits and the directed limit of an epimorphism is an epimorphism.

2. The artinian objects? of $AC_k$ are algebraic groups. Any object of $AC_k$ is the directed limit of its algebraic quotients.

3. By Cartier duality, the dual statements hold for the category of com- mutative $k$-formal-groups.