The rank of a homomorphism of group schemes

Let f : G ⟶ H be a homomorphism of group schemes over a scheme S whose underlying morphism of schemes is finite and flat, so that it has a rank Scheme.Hom.finrank at every point of H. This file proves that the rank of f at every point of its image is the rank at the identity of the fibre: for a point x of G over s ∈ S, the rank of f at f x is the rank of f at e_H(s). The latter is the rank at s of the kernel of f, the fibre of f over the identity section, which is finite and flat over S. In particular, when f is surjective its rank is constant along the fibres of H ⟶ S, and it is a function on the base S: this is how the degree of an isogeny is defined.

The proof is a translation argument after base change to G. The base change f ▷ G : G ×_S G ⟶ H ×_S G of f intertwines left translation by the tautological point on G ×_S G with left translation by its image under f on H ×_S G (leftTranslationProd), because f is a homomorphism. The latter carries the section a ↦ (1, a) of H ×_S G to the section a ↦ (f a, a). Translations are isomorphisms, so the base change of f has the same rank at the images of these two sections at x, and those are the ranks of f at e_H(s) and at f x.

Main results

• TauCeti.GroupScheme.finrank_left_apply: the rank of f at f x is the rank of f at the identity of the fibre through f x.
• TauCeti.GroupScheme.finrank_left_eq: for surjective f, the rank of f at any point y of H is the rank at the identity of the fibre through y.
• TauCeti.GroupScheme.finrank_kernel_hom: the rank of the kernel of f over S at s is the rank of f at the identity of the fibre over s.

References

• N. M. Katz and B. Mazur, Arithmetic Moduli of Elliptic Curves, §1.12 and 2.4.

theorem TauCeti.GroupScheme.finrank_left_apply {S : AlgebraicGeometry.Scheme} {G H : CategoryTheory.Over S} [CategoryTheory.GrpObj G] [CategoryTheory.GrpObj H] (f : G ⟶ H) [CategoryTheory.IsMonHom f] [AlgebraicGeometry.IsFinite (CategoryTheory.Over.Hom.left f)] [AlgebraicGeometry.Flat (CategoryTheory.Over.Hom.left f)] (x : ↥G.left) : AlgebraicGeometry.Scheme.Hom.finrank (CategoryTheory.Over.Hom.left f) ((CategoryTheory.Over.Hom.left f) x) = AlgebraicGeometry.Scheme.Hom.finrank (CategoryTheory.Over.Hom.left f) ((CategoryTheory.Over.Hom.left CategoryTheory.MonObj.one) (G.hom x))

The rank of a homomorphism of group schemes is translation invariant. If f : G ⟶ H is a homomorphism of group schemes over S whose underlying morphism is finite and flat, then for a point x of G over s ∈ S, the rank of f at f x equals its rank at the identity e_H(s) of the fibre of H over s.

theorem TauCeti.GroupScheme.finrank_left_eq {S : AlgebraicGeometry.Scheme} {G H : CategoryTheory.Over S} [CategoryTheory.GrpObj G] [CategoryTheory.GrpObj H] (f : G ⟶ H) [CategoryTheory.IsMonHom f] [AlgebraicGeometry.IsFinite (CategoryTheory.Over.Hom.left f)] [AlgebraicGeometry.Flat (CategoryTheory.Over.Hom.left f)] [AlgebraicGeometry.Surjective (CategoryTheory.Over.Hom.left f)] (y : ↥H.left) : AlgebraicGeometry.Scheme.Hom.finrank (CategoryTheory.Over.Hom.left f) y = AlgebraicGeometry.Scheme.Hom.finrank (CategoryTheory.Over.Hom.left f) ((CategoryTheory.Over.Hom.left CategoryTheory.MonObj.one) (H.hom y))

The rank of a surjective homomorphism of group schemes is constant along fibres. If f : G ⟶ H is a surjective homomorphism of group schemes over S whose underlying morphism is finite and flat, then its rank at a point y of H over s ∈ S is its rank at the identity of the fibre of H over s.

theorem TauCeti.GroupScheme.finrank_kernel_hom {S : AlgebraicGeometry.Scheme} {G H : CategoryTheory.Grp (CategoryTheory.Over S)} (f : G ⟶ H) [AlgebraicGeometry.IsFinite (CategoryTheory.Over.Hom.left f.hom.hom)] [AlgebraicGeometry.Flat (CategoryTheory.Over.Hom.left f.hom.hom)] (s : ↥S) : AlgebraicGeometry.Scheme.Hom.finrank (CategoryTheory.Limits.kernel f).X.hom s = AlgebraicGeometry.Scheme.Hom.finrank (CategoryTheory.Over.Hom.left f.hom.hom) ((CategoryTheory.Over.Hom.left CategoryTheory.MonObj.one) s)

The rank of the kernel. If f : G ⟶ H is a homomorphism of group schemes over S whose underlying morphism is finite and flat, the rank of its kernel over S at s is the rank of f at the identity e_H(s): the kernel is the fibre of f over the identity section.