Point norms and the supremum seminorm of an affinoid algebra #
Let A be a K-algebra over a complete nonarchimedean field K. For a maximal ideal x of A
and f : A, the value |f(x)| is the spectral norm over K of the residue of f in A ⧸ x.
When A is affinoid this residue field is finite over K, and the spectral norm is the unique
extension of the norm of K to it; no embedding into a chosen algebraic closure enters. The
supremum seminorm |f|sup is the supremum of |f(x)| over all classical points x, with value
zero for the zero algebra.
For an affinoid algebra the point norms are multiplicative, ultrametric and compatible with
scalars, and they are bounded by every complete ultrametric algebra norm on A. In particular
|f|sup is a finite, power-multiplicative ring seminorm, its kernel is the nilradical, algebra
homomorphisms of affinoid algebras do not increase it, and integral injective ones preserve it.
It is unchanged by passing to the reduction A ⧸ nilradical A, it is the maximum of its values on
the quotients by the finitely many minimal primes of A, and a monic equation
f ^ d + ∑_{i < d} aᵢ f ^ i = 0 bounds it by max_{i < d} |aᵢ|sup ^ (1 / (d - i)).
The bound by a complete algebra norm uses automatic continuity,
TauCeti.IsAffinoidAlgebra.continuous_algHom, of the residue map into A ⧸ x with its spectral
norm, followed by the power-multiplicativity of the spectral norm. The kernel computation uses
that affinoid algebras are Jacobson rings.
Main definitions #
TauCeti.pointNorm: the absolute value|f(x)|off : Aat a maximal idealx.TauCeti.supSeminorm: the supremum seminorm|f|sup.TauCeti.IsAffinoidAlgebra.supRingSeminorm:|·|supbundled as a ring seminorm.
Main results #
TauCeti.IsAffinoidAlgebra.pointNorm_le_norm:|f(x)| ≤ ‖f‖for every complete ultrametric algebra norm on an affinoid algebra.TauCeti.IsAffinoidAlgebra.isUnit_iff_forall_pointNorm_ne_zero:fis a unit if and only if|f(x)| ≠ 0at every classical pointx.TauCeti.IsAffinoidAlgebra.bddAbove_range_pointNorm: the point norms offare bounded.TauCeti.isUnit_one_sub_of_supSeminorm_lt_one: if the point norms ofaare bounded and|a|sup < 1, then1 - ais a unit.TauCeti.isPowMul_supSeminorm:|f ^ n|sup = |f|sup ^ nforn ≥ 1.TauCeti.IsAffinoidAlgebra.supSeminorm_eq_zero_iff:|f|sup = 0if and only iffis nilpotent.TauCeti.IsAffinoidAlgebra.isReduced_iff_supSeminorm_eq_zero:|·|supseparates points exactly whenAis reduced.TauCeti.IsAffinoidAlgebra.supSeminorm_map_le: algebra homomorphisms contract|·|sup.TauCeti.IsAffinoidAlgebra.supSeminorm_map_of_isIntegral: integral injective algebra homomorphisms preserve|·|sup.TauCeti.supSeminorm_mk_nilradical:|·|supis unchanged by reduction.TauCeti.IsAffinoidAlgebra.supSeminorm_eq_iSup_minimalPrimes:|f|supis the supremum of the supremum seminorms of the residues offmodulo the minimal primes ofA.TauCeti.IsAffinoidAlgebra.supSeminorm_le_iSup_coeff_rpow_of_eval_eq_zero: the bound of|f|supby the coefficients of a monic polynomial vanishing atf.
References #
- [Bosch, Güntzer, Remmert, Non-Archimedean Analysis][bosch-guntzer-remmert], §§3.8 and 6.2.1.
The absolute value |f(x)| of f : A at a maximal ideal x: the spectral norm over K of
the residue of f in the field A ⧸ x. For an affinoid algebra this residue field is finite
over K, and the spectral norm is the unique extension of the norm of K to it.
Instances For
The supremum seminorm |f|sup of f : A: the supremum of its absolute values |f(x)| at
the maximal ideals x of A. It is zero on the zero algebra, which has no maximal ideals.
Instances For
The absolute value at a classical point can be computed in any field L over K that
receives the residue field: if a K-algebra map ψ : A → L vanishes on x, then |f(x)| is
the spectral norm of ψ f.
Elements congruent modulo a classical point have the same absolute value there.
The absolute value at a classical point is unchanged by pulling the point back along an algebra map, whenever the pullback is again a maximal ideal.
The absolute value at a classical point is unchanged by pulling the point back along an algebra map into an affinoid algebra.
The absolute value at a classical point is homogeneous for scalars of K:
|(c • f)(x)| = ‖c‖ * |f(x)|.
|(f ^ n)(x)| = |f(x)| ^ n at every classical point x.
The absolute value |f(x)| vanishes exactly when f lies in the maximal ideal x.
The absolute value at a classical point is multiplicative:
|(f * g)(x)| = |f(x)| * |g(x)|.
The absolute value at a classical point is ultrametric.
The units of an affinoid algebra are the functions without zeros: f is a unit if and
only if |f(x)| ≠ 0 at every classical point x.
The absolute value of the inverse of a unit is the inverse absolute value:
|u⁻¹(x)| = |u(x)|⁻¹.
Point norms are bounded by complete algebra norms. On an affinoid algebra with a complete
ultrametric algebra norm, |f(x)| ≤ ‖f‖ at every classical point x.
A nonnegative bound for all point norms bounds the supremum seminorm.
If the absolute values of f at the classical points are bounded, then each of them is at
most the supremum seminorm: |f(x)| ≤ |f|sup.
The supremum seminorm vanishes on the zero algebra, which has no maximal ideals.
Perturbations of 1 by elements of supremum seminorm < 1 are units. If the absolute
values of a at the classical points are bounded and |a|sup < 1, then 1 - a is a unit.
The supremum seminorm is homogeneous for scalars of K: |c • f|sup = ‖c‖ * |f|sup.
|f ^ n|sup = |f|sup ^ n for n ≠ 0, including on the zero algebra. For n = 0 the
statement fails on the zero algebra.
The supremum seminorm is power-multiplicative: |f ^ n|sup = |f|sup ^ n for n ≥ 1.
A surjective algebra map whose kernel consists of nilpotent elements preserves supremum seminorms, since every classical point of the source contains the kernel and so comes from a classical point of the target.
The supremum seminorm is unchanged by reduction: |f|sup is the same in A and in its
reduction A ⧸ nilradical A.
Boundedness of point norms. The absolute values of an element of an affinoid algebra at its classical points are bounded.
Each absolute value |f(x)| is bounded by the supremum seminorm |f|sup.
The supremum seminorm is bounded by every complete ultrametric algebra norm.
The supremum seminorm is submultiplicative: |f * g|sup ≤ |f|sup * |g|sup.
The supremum seminorm is ultrametric.
The supremum seminorm is nonarchimedean.
The supremum seminorm of an affinoid algebra vanishes exactly on the elements lying in every maximal ideal.
The kernel of the supremum seminorm is the nilradical. An element of an affinoid algebra has supremum seminorm zero exactly when it is nilpotent, since an affinoid algebra is a Jacobson ring.
The supremum seminorm is a norm exactly on reduced affinoid algebras.
An algebra homomorphism of affinoid algebras does not increase supremum seminorms.
An integral injective algebra homomorphism into an affinoid algebra preserves supremum seminorms, since every classical point of the source lifts to the target.
At a classical point x of A containing the kernel of a surjective algebra map
φ : A → B onto an affinoid algebra, |a(x)| ≤ |φ a|sup: the point x comes from a classical
point of B.
The supremum seminorm is computed on minimal-prime quotients. On an affinoid algebra,
|f|sup is the supremum, over the finitely many minimal primes 𝔭 of A, of the supremum
seminorm of the residue of f in the affinoid domain A ⧸ 𝔭.
On a nonzero affinoid algebra the supremum seminorm |f|sup is attained on the quotient by
one of its minimal primes.
The supremum seminorm of an affinoid algebra, bundled as a ring seminorm.
Instances For
Integral equations bound the supremum seminorm. If f is a root of a monic polynomial
p = X ^ d + ∑_{i < d} aᵢ X ^ i over an affinoid algebra, then
|f|sup ≤ max_{i < d} |aᵢ|sup ^ (1 / (d - i)).