Exact kernels of normal affine quotient projections

For a geometrically reduced affine group of finite type over a field, the projection defined by the coinvariants of a normal closed subgroup has that subgroup as its scheme-theoretic kernel. The subgroup itself need not be reduced or smooth.

Over an algebraically closed field, a Chevalley line realizes the subgroup as a line stabilizer in a representation spanned by its subgroup character spaces. Endomorphisms preserving each character space are subgroup-invariant in the Hom representation. The coinvariant kernel fixes those endomorphisms, so its universal point centralizes them and consequently stabilizes the line. Testing at the universal point retains the full defining ideal, including infinitesimal structure. The result over an arbitrary field follows by scalar extension to an algebraic closure and descent.

This combines HopfIdeal.IsNormal.exists_finite_weightSum_line_stabilizer, the character-space centralizer criterion, the fixed-vector comparison for coinvariant kernels, and CommHopfAlgCat.kernelHopfIdeal_coinvariantsι_baseChange_eq_iff.

References

• J. E. Humphreys, Linear Algebraic Groups, §11.5.
• W. C. Waterhouse, Introduction to Affine Group Schemes, §16.3.

@[simp]

theorem TauCeti.CommHopfAlgCat.kernelHopfIdeal_coinvariantsι_eq {k : Type u} [Field k] {H : CommHopfAlgCat k} [Algebra.FiniteType k ↑H] {I : HopfIdeal k ↑H} [Algebra.IsGeometricallyReduced k ↑H] (hI : I.IsNormal) : kernelHopfIdeal (coinvariantsι hI) = I

The normal affine quotient has exactly the prescribed scheme-theoretic kernel. For a geometrically reduced finite-type affine group over any field, the kernel of the coinvariant projection is the original normal subgroup, including its possibly nonreduced structure.