Minimal polynomials and the supremum seminorm

Let B be an affinoid algebra over a complete nonarchimedean field K, integral over a K-algebra A, and let f : B be a root of a monic polynomial p = X ^ d + ∑_{i < d} aᵢ X ^ i over A which divides every polynomial over A vanishing at f. Then the supremum seminorm of f is the spectral value of p for the supremum seminorm of A:

|f|sup = max_{i < d} |aᵢ|sup ^ (1 / (d - i)).

For d = 0 both sides vanish, since then B = 0.

The inequality ≤ holds for every monic equation of f, by TauCeti.IsAffinoidAlgebra.supSeminorm_le_iSup_coeff_rpow_of_eval_eq_zero. For ≥, fix a classical point x of A and a root α of the reduction of p in an algebraic closure of A ⧸ x. Evaluation at α factors through A[f] ≅ A[X] ⧸ (p), and its kernel is a maximal ideal of A[f] lying under a classical point y of B, at which |f(y)| = |α|. Taking α of largest absolute value, the coefficients of the reduction satisfy |aᵢ(x)| ≤ |f(y)| ^ (d - i), and in particular |aᵢ(x)| ^ (1 / (d - i)) ≤ |f|sup.

When A is an integrally closed domain and B is a torsion-free algebra, the minimal polynomial of f over A satisfies these hypotheses. This applies to a finite injective map from a Tate algebra into an affinoid algebra which is torsion-free over it, possibly with several components or nilpotents, since Tate algebras are unique factorisation domains and hence integrally closed. The statement with a divisibility hypothesis also applies without torsion-freeness when an annihilating polynomial with the required divisibility property is supplied. Nilpotents in B are retained throughout.

Main results

• TauCeti.IsAffinoidAlgebra.exists_pointNorm_coeff_le_pointNorm_pow: over each classical point x of A lies a classical point y of B with |aᵢ(x)| ≤ |f(y)| ^ (d - i) for all i.
• TauCeti.IsAffinoidAlgebra.supSeminorm_eq_iSup_coeff_rpow_of_dvd: |f|sup is the spectral value of a monic polynomial generating the annihilator of f.
• TauCeti.IsAffinoidAlgebra.supSeminorm_eq_iSup_minpoly_coeff_rpow: |f|sup is the spectral value of the minimal polynomial of f over an integrally closed domain.

References

• [Bosch, Güntzer, Remmert, Non-Archimedean Analysis][bosch-guntzer-remmert], §3.8.1, Proposition 7.

theorem TauCeti.IsAffinoidAlgebra.exists_pointNorm_coeff_le_pointNorm_pow {K : Type u_1} [NontriviallyNormedField K] [IsUltrametricDist K] [CompleteSpace K] {A : Type u_2} [CommRing A] [Algebra K A] {B : Type u_3} [CommRing B] [Algebra K B] [Algebra A B] [IsScalarTower K A B] (hB : IsAffinoidAlgebra K B) [Algebra.IsIntegral A B] {f : B} {p : Polynomial A} (hp : p.Monic) (hpos : 0 < p.natDegree) (hdvd : ∀ (q : Polynomial A), (Polynomial.aeval f) q = 0 → p ∣ q) (x : MaximalSpectrum A) : ∃ (y : MaximalSpectrum B), ∀ (i : ℕ), pointNorm K x (p.coeff i) ≤ pointNorm K y f ^ (p.natDegree - i)

Coefficients of a minimal equation at a classical point. Let B be an affinoid algebra integral over A, and let f : B be a root of a monic polynomial p = X ^ d + ∑_{i < d} aᵢ X ^ i over A of positive degree dividing every polynomial over A which vanishes at f. Over each classical point x of A there is a classical point y of B with |aᵢ(x)| ≤ |f(y)| ^ (d - i) for every i.

theorem TauCeti.IsAffinoidAlgebra.supSeminorm_eq_iSup_coeff_rpow_of_dvd {K : Type u_1} [NontriviallyNormedField K] [IsUltrametricDist K] [CompleteSpace K] {A : Type u_2} [CommRing A] [Algebra K A] {B : Type u_3} [CommRing B] [Algebra K B] [Algebra A B] [IsScalarTower K A B] (hB : IsAffinoidAlgebra K B) [Algebra.IsIntegral A B] {p : Polynomial A} (hp : p.Monic) {f : B} (hf : (Polynomial.aeval f) p = 0) (hdvd : ∀ (q : Polynomial A), (Polynomial.aeval f) q = 0 → p ∣ q) : supSeminorm K f = ⨆ (i : Fin p.natDegree), supSeminorm K (p.coeff ↑i) ^ (1 / (↑p.natDegree - ↑↑i))

The supremum seminorm is the spectral value of a minimal equation. Let B be an affinoid algebra integral over A, and let f : B be a root of a monic polynomial p = X ^ d + ∑_{i < d} aᵢ X ^ i over A dividing every polynomial over A which vanishes at f. Then |f|sup = max_{i < d} |aᵢ|sup ^ (1 / (d - i)).

theorem TauCeti.IsAffinoidAlgebra.supSeminorm_eq_iSup_minpoly_coeff_rpow {K : Type u_1} [NontriviallyNormedField K] [IsUltrametricDist K] [CompleteSpace K] {A : Type u_2} [CommRing A] [Algebra K A] {B : Type u_3} [CommRing B] [Algebra K B] [Algebra A B] [IsScalarTower K A B] (hB : IsAffinoidAlgebra K B) [IsDomain A] [IsIntegrallyClosed A] [Module.IsTorsionFree A B] [Algebra.IsIntegral A B] (f : B) : supSeminorm K f = ⨆ (i : Fin (minpoly A f).natDegree), supSeminorm K ((minpoly A f).coeff ↑i) ^ (1 / (↑(minpoly A f).natDegree - ↑↑i))

The supremum seminorm is the spectral value of the minimal polynomial. Let B be an affinoid algebra which is integral and torsion-free over an integrally closed domain A. For f : B with minimal polynomial X ^ d + ∑_{i < d} aᵢ X ^ i over A, |f|sup = max_{i < d} |aᵢ|sup ^ (1 / (d - i)).