The degree of a finite locally free 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. Its scheme-theoretic kernel is the base change of f along the unit
section of H (TauCeti.GroupScheme.isPullback_kernel_left), so it is finite and flat over
S. The degree of f is the function GroupScheme.degree f : S → ℕ whose value at s is the
rank of the kernel at s. When f is also locally of finite presentation it is locally constant,
but over a disconnected base it need not be constant. For an isogeny of elliptic curves, a finite,
flat and surjective homomorphism, this is its degree.
The degree of f controls the rank of f itself. If f is moreover surjective, then the rank of
f at a point x of H is the degree of f at the image of x in S
(GroupScheme.finrank_hom_hom_left), by translation invariance of the rank
(TauCeti.GroupScheme.finrank_left_eq). Hence the degree is multiplicative under composition, and
f is an isomorphism exactly when its degree is 1.
If G is commutative and f is locally of finite presentation, the kernel is a finite locally
free commutative group scheme (TauCeti.AlgebraicGeometry.FiniteFlatCommGroupScheme.ofKernel),
whose rank is the degree. By Deligne's theorem
(TauCeti.AlgebraicGeometry.FiniteFlatCommGroupScheme.pow_eq_one_of_finrank), if f has
constant degree n then its kernel is killed by n. For an isogeny of elliptic curves this says
that its kernel lies in the n-torsion, which is what makes the dual isogeny of a degree-n
isogeny constructible as a factorisation of multiplication by n.
An isogeny (TauCeti.GroupScheme.IsIsogeny: finite, flat and surjective) whose degree is n at
every point of S is an isogeny of degree n (GroupScheme.IsogenyOfDegree f n). Over a
disconnected base this constant-degree case is the one in which a single integer degree makes sense.
Main definitions #
TauCeti.GroupScheme.degree f: the rank of the kernel off, a function on the base.TauCeti.GroupScheme.IsogenyOfDegree f n:fis an isogeny of constant degreen.
Main results #
TauCeti.GroupScheme.isLocallyConstant_degree: the degree is locally constant.TauCeti.GroupScheme.degree_pos: the degree is positive.TauCeti.GroupScheme.degree_pullback_mapGrp_map: the degree is stable under base change.TauCeti.GroupScheme.finrank_hom_hom_left: for surjectivef, the rank offat a point is the degree at its image in the base.TauCeti.GroupScheme.degree_comp: the degree is multiplicative under composition.TauCeti.GroupScheme.isIso_iff_degree_eq_one: a finite, flat and surjective homomorphism is an isomorphism exactly when its degree is1; in particular an isomorphism has degree1(TauCeti.GroupScheme.degree_eq_one_of_isIso).TauCeti.GroupScheme.kernel_ι_pow_eq_one: the kernel of a homomorphism of constant degreenout of a commutative group scheme is killed byn.TauCeti.GroupScheme.IsogenyOfDegree.comp,TauCeti.GroupScheme.isogenyOfDegree_one_iff, andTauCeti.GroupScheme.IsogenyOfDegree.kernel_ι_pow_eq_one: degrees of isogenies multiply, the isogenies of degree1are the isomorphisms, and the kernel of an isogeny of degreenis killed byn.TauCeti.GroupScheme.IsogenyOfDegree.baseChange: isogenies of degreenremain so after arbitrary base change.
References #
- N. M. Katz and B. Mazur, Arithmetic Moduli of Elliptic Curves, Annals of Mathematics Studies 108, Princeton University Press, 1985, §1.12 and Chapter 2.
- The Stacks Project, Tag 02KA, for the rank of a finite locally free morphism.
The degree of a homomorphism of group schemes f : G ⟶ H over S: the function on S
whose value at s is the rank of the kernel of f at s. It is meaningful when f is finite and
flat, so that its kernel is finite and flat over S.
Instances For
The degree of f at s is the rank of the kernel of f at s.
The degree is the rank over the unit section. For a finite flat homomorphism f : G ⟶ H,
the degree of f at s is the rank of f at the image of s under the unit section of H.
The degree is locally constant for a finite locally free homomorphism of group schemes.
The degree of a finite flat homomorphism of group schemes is positive: the unit section of the target lies in the image.
The degree is stable under base change. For a finite flat homomorphism f : G ⟶ H over
S and a morphism s : T ⟶ S, the degree of the base change of f to T at t is the degree
of f at s t: the kernel of the base change is the base change of the kernel
(kernelBaseChangeIso).
The rank of a finite flat surjective homomorphism is its degree, fibrewise. For a finite,
flat and surjective homomorphism f : G ⟶ H of group schemes over S, the rank of f at a point
x of H is the degree of f at the image of x in S.
The degree is multiplicative. If f : G ⟶ H is finite, flat and surjective and
g : H ⟶ K is finite and flat, the degree of f ≫ g at s is the product of the degrees of f
and of g at s.
A homomorphism of degree one is an isomorphism. A finite, flat and surjective
homomorphism of group schemes is an isomorphism exactly when its degree is 1 at every point.
An isomorphism of group schemes has degree 1.
The kernel as a finite locally free commutative group scheme #
The rank of the kernel of f, as a finite locally free commutative group scheme, is the degree
of f.
The kernel of a homomorphism of degree n is killed by n. Let f : G ⟶ H be a finite
locally free homomorphism of group schemes over S, with G commutative, whose degree is n at
every point of S. Then the kernel inclusion, as a point of G with values in the kernel, is
killed by n.
Isogenies of constant degree #
An isogeny of degree n: a homomorphism f : G ⟶ H of group schemes over S that is an
isogeny (finite, flat and surjective) and whose degree is n at every point of S. Over a
disconnected base the degree of an isogeny is only locally constant; this is the constant-degree
case.
- isIsogeny : IsIsogeny f
An isogeny of degree
nis an isogeny. The degree of an isogeny of degree
nisnat every point of the base.
Instances For
Degrees of isogenies multiply. The composite of an isogeny of degree m and an isogeny of
degree n is an isogeny of degree m * n.
Base change preserves isogenies of degree n.
The kernel of an isogeny of degree n out of a commutative group scheme, locally of finite
presentation, is killed by n.
An isomorphism of group schemes is an isogeny of degree 1.
The isogenies of degree 1 are the isomorphisms.