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

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Theorem

The category AC kAC_k of affine commutative kk-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 ABAB 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:GHf:G\to H be a morphism in AC kAC_k..

  1. The following conditions are equivalent: ff is a monomorphism, O(f)O(f) is surjective (i.e. GG is a closed subgroup of HH), ff is a kernel.

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

Corollary

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

Corollary
Corollary
Corollary
Corollary
Corollary
Theorem
  1. The category AC kAC_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 kAC_k are algebraic groups. Any object of AC kAC_k is the directed limit of its algebraic quotients.

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

Revised on May 27, 2012 13:32:50 by Stephan Alexander Spahn (79.227.168.80)