The short exact sequence of normal coinvariants #
For a geometrically reduced affine group G of finite type over a field and a normal
closed subgroup N, the inclusion of coinvariants and restriction to N form the
coordinate Hopf algebra maps of a short exact sequence 1 → N → G → G/N → 1.
The subgroup may be nonreduced, and the field need not be perfect.
This combines faithfullyFlat_coinvariantsι, kernelHopfIdeal_coinvariantsι_eq,
and isShortExact_mkQuotient_kernelHopfIdeal, without using fppf sheaves.
References #
- W. C. Waterhouse, Introduction to Affine Group Schemes, §16.3.
- J. S. Milne, Algebraic Groups (2017), §5.c.
theorem
TauCeti.CommHopfAlgCat.isShortExact_coinvariantsι_mkQuotient
{k : Type u}
[Field k]
{H : CommHopfAlgCat k}
[Algebra.FiniteType k ↑H]
[Algebra.IsGeometricallyReduced k ↑H]
{I : HopfIdeal k ↑H}
(hI : I.IsNormal)
:
IsShortExact (coinvariantsι hI) (mkQuotient H I)
The inclusion of normal coinvariants and restriction to the normal subgroup form a short exact sequence of affine groups. The subgroup need not be reduced.