Affine homogeneous quotients by linearly reductive subgroups #
Let G = Spec H be an affine group and N the closed subgroup cut out by a Hopf ideal I.
The functions on G invariant under right translation by N form the subalgebra
B = H^{co H/I}, and restricting points of G to B is a morphism G ⟶ Spec B that is
constant on left cosets. It therefore descends to a morphism from the fppf homogeneous quotient
G/N to the fppf sheaf of points of Spec B.
This file proves that the descended morphism is an isomorphism when I is generated by the
coinvariants vanishing at the identity and H is faithfully flat and of finite presentation over
B. The first condition makes the fibres of G ⟶ Spec B exactly the left cosets of N; the
second lets every point of Spec B lift to G after an fppf extension of its value algebra.
Both conditions hold when G is a geometrically reduced affine group of finite type over a field
and N is linearly reductive. Then G/N is represented by the affine scheme Spec B, where
B is a finitely generated algebra by the Hilbert--Nagata theorem. The subgroup N need not be
normal, smooth, or connected, and the field is arbitrary. In characteristic zero reductive groups
are linearly reductive, so the quotient of any affine group of finite type by a reductive closed
subgroup is affine: this is the affine direction of Matsushima's criterion.
Main declarations #
TauCeti.HopfIdeal.toCoinvariantsPoints: the restriction of points ofGto the coinvariants.TauCeti.HopfIdeal.fppfHomogeneousQuotientToCoinvariantsPoints: the descended mapG/N ⟶ Spec Bof fppf sheaves.TauCeti.HopfIdeal.isIso_fppfHomogeneousQuotientToCoinvariantsPoints: it is an isomorphism whenIis generated byB⁺andHis faithfully flat and finitely presented overB.TauCeti.HopfIdeal.fppfHomogeneousQuotientIsoCoinvariantsPoints: the homogeneous quotient by a linearly reductive subgroup is represented by the spectrum of the invariants.TauCeti.HopfIdeal.coinvariantsPointsHomEquiv: when it representsG/N, the spaceSpec Bhas the universal property of the quotient.TauCeti.HopfIdeal.isIso_fppfHomogeneousQuotientToCoinvariantsPoints_of_reductive: in characteristic zero, the same holds for every reductive closed subgroup.TauCeti.HopfIdeal.isIso_fppfHomogeneousQuotientToCoinvariantsPoints_ker_of_reductive: the same for a reductive group presented by a closed immersion.
References #
- J. S. Milne, Algebraic Groups (2017), §5.c (quotients as fppf sheaves) and Chapter 12 (linearly reductive groups).
- M. Takeuchi, A correspondence between Hopf ideals and sub-Hopf algebras, Manuscripta Math. 7 (1972), 251--270.
- Y. Matsushima, Espaces homogènes de Stein des groupes de Lie complexes, Nagoya Math. J. 16 (1960), 205--218.
- W. C. Waterhouse, Introduction to Affine Group Schemes, §§14--16.
The descent argument is CommHopfAlgCat.isIso_fppfHomogeneousQuotientHomEquiv_symm, shared with
the representation of homogeneous quotients by projective orbits in
TauCeti.Algebra.AlgebraicGroup.Representation.ProjectiveOrbit.Quotient.
Restriction of points of G = Spec H to the coinvariants H^{co H/I}: the morphism from the
points of G to the fppf sheaf of points of Spec H^{co H/I} induced by the inclusion of the
coinvariants.
Instances For
The restriction of a point g to the coinvariants is g ∘ val.
Restriction to the coinvariants is invariant under right multiplication by the subgroup.
The morphism G/N ⟶ Spec H^{co H/I} of fppf sheaves induced by restricting points of G to
the coinvariants.
Instances For
The descended map restricts to toCoinvariantsPoints along the projection G ⟶ G/N.
The descended map restricts to toCoinvariantsPoints along the projection G ⟶ G/N.
If H is faithfully flat and of finite presentation over the coinvariants, every point of
Spec H^{co H/I} lifts to G fppf locally.
Homogeneous quotients represented by coinvariants. Suppose that I is generated by the
coinvariants vanishing at the identity, and that H is faithfully flat and of finite
presentation over its coinvariants B = H^{co H/I}. Then restriction to B identifies the fppf
homogeneous quotient G/N with the fppf sheaf of points of Spec B.
Universal property of Spec H^{co H/I}. When restriction to the coinvariants identifies
the homogeneous quotient G/N with Spec B, B = H^{co H/I}, morphisms from Spec B to an fppf
sheaf F are exactly the natural maps from G to F invariant under right multiplication by
N. The correspondence is composition with the quotient map G ⟶ Spec B.
Instances For
The universal property sends a morphism out of Spec H^{co H/I} to its composite with the
quotient map.
Descending a subgroup-invariant natural map through Spec H^{co H/I} and restricting along
the quotient map recovers that map.
Morphisms out of Spec H^{co H/I} are determined by their composites with the quotient
map, when that space represents the homogeneous quotient.
For a linearly reductive closed subgroup of a geometrically reduced affine group of finite type over a field, restriction to the invariant functions identifies the fppf homogeneous quotient with the sheaf of points of the spectrum of the invariants.
The homogeneous quotient by a linearly reductive subgroup is affine. Let G be a
geometrically reduced affine group of finite type over a field and N a linearly reductive
closed subgroup. Then the fppf homogeneous quotient G/N is represented by Spec O(G)^N, the
spectrum of the functions on G invariant under right translation by N.
Instances For
The forward map of the representing isomorphism is the descended restriction map.
Matsushima's criterion, affine direction, in characteristic zero. For a reductive closed
subgroup N of an affine group G of finite type over a field of characteristic zero,
restriction to the invariant functions identifies the fppf homogeneous quotient G/N with the
sheaf of points of Spec O(G)^N.
Matsushima's criterion for a closed immersion, in characteristic zero. Let
f : O(G) ⟶ O(N) be a surjection of finite-type commutative Hopf algebras over a field of
characteristic zero, representing a closed immersion of a reductive group N into G. Then
restriction to the functions on G invariant under right translation by N identifies the fppf
homogeneous quotient G/N with the sheaf of points of Spec O(G)^N.