Étale kernels of diagonalizable-group morphisms #
The kernel of D(N) → D(M) associated to a character homomorphism p : M →* N
is D(N / range p). When the character cokernel is finite, this kernel is étale if
its order is invertible in the base ring; over a field the converse holds as well.
Thus the criterion detects separable versus infinitesimal kernels of isogenies,
without assuming either ambient diagonalizable group is smooth.
References #
- J. S. Milne, Algebraic Groups (2017), §12, diagonalizable groups.
The comparison uses TauCeti.DiagonalizableGroup.kernelCoordinateIso.
theorem
TauCeti.DiagonalizableGroup.etale_kernelCoordinate_of_isUnit_card
(R : Type u)
[CommRing R]
{M N : Type v}
[CommGroup M]
[CommGroup N]
(p : M →* N)
[Finite (N ⧸ p.range)]
(h : IsUnit ↑(Nat.card (N ⧸ p.range)))
:
A diagonalizable-group kernel is étale when the order of the character cokernel is invertible in the base ring.
theorem
TauCeti.DiagonalizableGroup.etale_kernelCoordinate_iff_isUnit_card
{M N : Type v}
[CommGroup M]
[CommGroup N]
(p : M →* N)
[Finite (N ⧸ p.range)]
(k : Type u)
[Field k]
:
Algebra.Etale k
↑(CommHopfAlgCat.quotient (↧(MonoidAlgebra k N))
(CommHopfAlgCat.kernelHopfIdeal (CommHopfAlgCat.ofHom (MonoidAlgebra.mapDomainBialgHom k p)))) ↔ IsUnit ↑(Nat.card (N ⧸ p.range))
Over a field, a finite diagonalizable-group kernel is étale exactly when the order of the character cokernel is invertible in the field.