Gauss valuations on Witt vectors #
Let O be the ring of integers of a valuation v : Valuation K ℝ≥0 on a field K, perfect of
characteristic p; for instance the ring of integers 𝒪_F of a perfect nonarchimedean field F
of characteristic p, so that 𝕎 O = A_inf. Write a Witt vector as its Teichmüller expansion
x = ∑ₙ [xₙ] pⁿ (WittVector.teichmullerCoeff). For ρ < 1 the Gauss valuation
λ_ρ(∑ₙ [xₙ] pⁿ) = supₙ v(xₙ) ρⁿ
is a valuation on 𝕎 O. It takes the value v(a) ρ ^ m on [a] p ^ m, so λ_ρ(p) = ρ and
λ_ρ([a]) = v(a), and for 0 < ρ it vanishes only at 0. These are the rank-one valuations from
which the Gauss points of Spa(A_inf, A_inf) and the norms of the interval rings of the
Fargues–Fontaine curve are built.
The ultrametric inequality rests on a description of the ideals ([c], p ^ (n + 1)) of 𝕎 O:
they consist exactly of the Witt vectors whose first n + 1 Teichmüller coordinates have valuation
at most v(c) (Valuation.Integers.mem_span_teichmuller_pow_iff), which uses that O is a
valuation ring. Multiplicativity follows by comparing the leading Teichmüller terms of a product
modulo a suitable power of p.
Main definitions #
TauCeti.WittVector.gaussValuation: the Gauss valuationλ_ρon𝕎 O.TauCeti.WittVector.gaussDegree: Kedlaya's degree forλ_ρ, the largest indexdwithv(x_d) ρ ^ d = λ_ρ(x).
Main results #
Valuation.Integers.mem_span_teichmuller_pow_iff: the ideal([c], p ^ (n + 1))in terms of Teichmüller coordinates.TauCeti.WittVector.gaussValuation_apply:λ_ρ(x) = supₙ v(xₙ) ρⁿ.TauCeti.WittVector.gaussValuation_teichmuller_mul_pow:λ_ρ([a] p ^ m) = v(a) ρ ^ m.TauCeti.WittVector.gaussValuation_eq_zero_iff: for0 < ρ,λ_ρvanishes only at0.TauCeti.WittVector.gaussValuation_pos: for0 < ρ,λ_ρis positive away from0.TauCeti.WittVector.gaussValuation_teichmuller_add_teichmuller_sub_le: Teichmüller representatives are additive up to multiples ofp, withλ_ρ([a] + [b] - [a + b]) ≤ ρ max (v(a), v(b)).TauCeti.WittVector.isContinuous_gaussValuation: ifv(ϖ) < 1, thenλ_ρis continuous for the(p, [ϖ])-adic topology.TauCeti.WittVector.eventually_gaussValuation_eq_mul_pow_gaussDegree: forρ'slightly larger thanρ,λ_ρ'(x) = v(x_d) ρ' ^ dfor the degreedofxforλ_ρ.TauCeti.WittVector.gaussDegree_eq_of_gaussValuation_sub_lt: the degree only depends on the leading part (Kedlaya, Remark 2.7).TauCeti.WittVector.gaussDegree_mul: the degree is additive on products (Kedlaya, Lemma 2.6).
References #
- L. Fargues and J.-M. Fontaine, Courbes et fibrés vectoriels en théorie de Hodge p-adique,
Astérisque 406 (2018), Chapter 1, for the Gauss norms on
A_inf. - K. S. Kedlaya, Nonarchimedean geometry of Witt vectors, Nagoya Math. J. 209 (2013).
- K. S. Kedlaya, Noetherian properties of Fargues–Fontaine curves, IMRN 2016, no. 8, 2544–2567; arXiv:1410.5160v3, Definition 2.5, Lemma 2.6 and Remark 2.7, for the degree.
- K. S. Kedlaya, Sheaves, stacks, and shtukas, lecture notes, Arizona Winter School 2017, §3.1.
Let R be the ring of integers of a valuation v. Then the ideal ([c], p ^ (n + 1)) of
𝕎 R consists exactly of the Witt vectors whose first n + 1 Teichmüller coordinates have
valuation at most v(c).
The Gauss valuation λ_ρ(∑ₙ [xₙ] pⁿ) = supₙ v(xₙ) ρⁿ on the Witt vectors of the ring of
integers O of a valuation v : Valuation K ℝ≥0, for O perfect of characteristic p and
ρ < 1.
Instances For
The Gauss valuation is the supremum of the weighted valuations of the Teichmüller coordinates.
Each weighted valuation v(xₙ) ρⁿ of a Teichmüller coordinate is bounded by the Gauss
valuation.
Increasing the radius increases the Gauss valuation of a Witt vector.
The Hadamard three circles inequality for Gauss valuations. For radii
0 < ρ₁ ≤ ρ ≤ ρ₂ and m : ℕ, the value λ_ρ(x) / ρ ^ m, which is the Gauss valuation at ρ of
x / p ^ m once p is inverted, is at most the larger of its values at ρ₁ and ρ₂: each term
v(xₙ) ρ ^ (n - m) is monotone in ρ, increasing if m ≤ n and decreasing otherwise.
The Gauss valuation is at most one.
The Gauss valuation of [a] p ^ m is v(a) ρ ^ m.
The Gauss valuation of a Teichmüller representative [a] is v(a).
In the splitting y = ∑_{i < m} [yᵢ] pⁱ + pᵐ b of the Teichmüller expansion
(WittVector.exists_eq_sum_teichmullerCoeff_add_pow_mul), the tail satisfies
λ_ρ(b) ρᵐ ≤ λ_ρ(y).
A Witt vector whose 0-th coordinate vanishes, that is, a multiple of p, has Gauss
valuation at most ρ.
Teichmüller representatives are additive up to p, quantitatively:
λ_ρ([a] + [b] - [a + b]) ≤ ρ max (v(a), v(b)).
For 0 < ρ, the Gauss valuation vanishes only at zero.
For 0 < ρ, the Gauss valuation of a nonzero Witt vector is positive.
The Gauss valuations are continuous for the (p, [ϖ])-adic topology on 𝕎 O, when
v(ϖ) < 1: λ_ρ is at most 1 everywhere, and at most max ρ v(ϖ) < 1 at p and [ϖ].
The degree #
For 0 < ρ < 1 and x ≠ 0, the weighted valuations v(xₙ) ρⁿ attain their supremum λ_ρ(x) at
finitely many indices. Kedlaya calls the largest of them the degree of x. Viewing x as a
power series in p with Teichmüller coefficients, it is the degree of the leading part of x for
λ_ρ, and it behaves like the degree of a polynomial: it is additive on products and only depends
on x up to terms of smaller Gauss valuation.
Both facts come from the radii slightly larger than ρ. There λ_ρ'(x) = v(x_d) ρ' ^ d for the
degree d, because enlarging the radius favours the larger indices among those attaining
λ_ρ(x). Since every λ_ρ' is a valuation, the degree is the exponent of this power law, and it
is computed from the valuations λ_ρ' alone, with no reference to Witt vector arithmetic.
The degree of a Witt vector for the Gauss valuation λ_ρ: the largest index d with
v(x_d) ρ ^ d = λ_ρ(x), for x ≠ 0 and 0 < ρ < 1.
Instances For
The degree of 0 is 0: every index attains its Gauss valuation 0.
Every Witt vector attains its Gauss valuation at its degree, including at radius zero:
λ_ρ(x) = v(x_d) ρ ^ d for d = gaussDegree p hv ρ hρ x.
Every index at which a nonzero Witt vector attains its Gauss valuation is at most its degree.
Beyond its degree, the weighted valuations of a nonzero Witt vector are smaller than its Gauss valuation.
The Gauss valuations near ρ follow a power law of the degree. For ρ' slightly larger
than ρ, the Gauss valuation λ_ρ'(x) of a Witt vector is attained at its degree d for
λ_ρ: λ_ρ'(x) = v(x_d) ρ' ^ d.
The degree is the exponent of the power law. If, for all ρ' slightly larger than ρ,
the Gauss valuation λ_ρ'(x) of a nonzero Witt vector is c ρ' ^ n, then n is the degree of
x for λ_ρ.
The degree only depends on the leading part (Kedlaya, Remark 2.7): if
λ_ρ(x - y) < λ_ρ(x), then x and y have the same degree.
The degree is additive on products (Kedlaya, Lemma 2.6).
The degree of [a] p ^ m is m, for a ≠ 0.
Multiplying by a Teichmüller representative [c] with c ≠ 0 does not change the degree,
including at radius zero.
A nonzero finite Teichmüller sum ∑_{j < n} [aⱼ] pʲ has degree less than n.