Kernels of homomorphisms of diagonalizable groups

For a homomorphism p : M →* N of commutative groups, the kernel of D(N) → D(M) has coordinate Hopf algebra R[N / range p]. The comparison respects the quotient coordinate maps, so it identifies the kernel as a closed subgroup, rather than merely identifying its abstract coordinate algebra.

The construction works over any commutative ring and does not require finite generation of either character group. Over a nonzero base the kernel is finite exactly when the character cokernel is finite. This gives the kernel calculation for isogenies of diagonalizable groups, including inseparable isogenies.

References

• J. S. Milne, Algebraic Groups (2017), Theorem 12.9(b).

The ideal calculation uses TauCeti.MonoidAlgebra.map_ker_augmentation_eq_ker_mapDomainRingHom.

theorem TauCeti.DiagonalizableGroup.kernelHopfIdeal_mapDomainBialgHom_toIdeal (R : Type u) [CommRing R] {M N : Type v} [CommGroup M] [CommGroup N] (p : M →* N) : (CommHopfAlgCat.kernelHopfIdeal (CommHopfAlgCat.ofHom (MonoidAlgebra.mapDomainBialgHom R p))).toIdeal = RingHom.ker (MonoidAlgebra.mapDomainRingHom R (QuotientGroup.mk' p.range))

The ideal defining the kernel of D(N) → D(M) is the kernel of the coordinate map from R[N] to the group algebra of the character cokernel.

noncomputable def TauCeti.DiagonalizableGroup.kernelCoordinateIso (R : Type u) [CommRing R] {M N : Type v} [CommGroup M] [CommGroup N] (p : M →* N) : CommHopfAlgCat.quotient (↧(MonoidAlgebra R N)) (CommHopfAlgCat.kernelHopfIdeal (CommHopfAlgCat.ofHom (MonoidAlgebra.mapDomainBialgHom R p))) ≅ ↧(MonoidAlgebra R (N ⧸ p.range))

The coordinate Hopf algebra of the kernel of D(N) → D(M) is the group algebra of the cokernel of M → N.

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem TauCeti.DiagonalizableGroup.mkQuotient_comp_kernelCoordinateIso_hom (R : Type u) [CommRing R] {M N : Type v} [CommGroup M] [CommGroup N] (p : M →* N) : CategoryTheory.CategoryStruct.comp (CommHopfAlgCat.mkQuotient (↧(MonoidAlgebra R N)) (CommHopfAlgCat.kernelHopfIdeal (CommHopfAlgCat.ofHom (MonoidAlgebra.mapDomainBialgHom R p)))) (kernelCoordinateIso R p).hom = CommHopfAlgCat.ofHom (MonoidAlgebra.mapDomainBialgHom R (QuotientGroup.mk' p.range))

The kernel identification commutes with the maps from the ambient coordinate algebra.

theorem TauCeti.DiagonalizableGroup.moduleFinite_kernelCoordinate_iff (R : Type u) [CommRing R] {M N : Type v} [CommGroup M] [CommGroup N] (p : M →* N) [Nontrivial R] : Module.Finite R ↑(CommHopfAlgCat.quotient (↧(MonoidAlgebra R N)) (CommHopfAlgCat.kernelHopfIdeal (CommHopfAlgCat.ofHom (MonoidAlgebra.mapDomainBialgHom R p)))) ↔ Finite (N ⧸ p.range)

Over a nonzero base, the kernel of a diagonalizable-group morphism is finite if and only if the cokernel of its character homomorphism is finite.

theorem TauCeti.DiagonalizableGroup.finrank_kernelCoordinate (R : Type u) [CommRing R] {M N : Type v} [CommGroup M] [CommGroup N] (p : M →* N) [Nontrivial R] : Module.finrank R ↑(CommHopfAlgCat.quotient (↧(MonoidAlgebra R N)) (CommHopfAlgCat.kernelHopfIdeal (CommHopfAlgCat.ofHom (MonoidAlgebra.mapDomainBialgHom R p)))) = Nat.card (N ⧸ p.range)

The finrank of the kernel coordinate algebra equals the natural cardinality of the character cokernel. In particular, this computes the rank of every finite kernel even when its order is divisible by the characteristic.