The radius rings A^r, their degree, and Euclidean division #
Let O be the ring of integers of a valuation v : Valuation K ℝ≥0 on a field K, perfect of
characteristic p, and let ϖ ∈ O be a pseudouniformiser, 0 < v(ϖ) < 1. Inverting the
Teichmüller representative [ϖ] in 𝕎 O adjoins the Teichmüller representatives of all elements
of K: the localisation 𝕎 O[1/[ϖ]] is Kedlaya's ring A_L = W(𝒪_L)[[x] : x ∈ L] for
L = K. For 0 < ρ < 1 the Gauss valuation λ_ρ(∑ₙ [xₙ] pⁿ) = supₙ v(xₙ) ρⁿ
(TauCeti.WittVector.gaussValuation) does not vanish at [ϖ], so it extends to 𝕎 O[1/[ϖ]],
where it is a multiplicative ultrametric norm. The radius ring A^r is the completion of
𝕎 O[1/[ϖ]] for this norm. It is a complete Hausdorff Tate ring with a multiplicative norm,
with pseudouniformiser [ϖ].
Kedlaya writes λ_r(∑ₙ pⁿ [xₙ]) = maxₙ p^(-n) |xₙ|^r for r > 0. This is λ_ρ ^ r for
ρ = p^(-1/r), so both norms define the same completion A^r; here the ring is indexed by ρ.
The degree of a nonzero element of A^r extends the degree of a Witt vector for λ_ρ
(TauCeti.WittVector.gaussDegree), the largest index at which its Teichmüller expansion attains
the Gauss norm. Every nonzero x ∈ A^r is approximated by some [ϖ]^(-k) y with y ∈ 𝕎 O up to
an error smaller than ‖x‖, and the degree of x is the degree of any such y. It only depends
on the leading part of x (Kedlaya, Remark 2.7) and is additive on products (Kedlaya,
Lemma 2.6).
A^r admits Euclidean division for this degree (Kedlaya, Proposition 2.9): for x ≠ 0, every
y is z x + w with ‖w‖ ≤ ‖y‖ and either w = 0 or deg w < deg x. Consequently the rings
A^r{T₁/ρ₁, …, Tₙ/ρₙ} of restricted power series are noetherian (Kedlaya, Theorem 3.2,
TauCeti.WittVector.RadiusRing.isNoetherianRing_isRestricted_subring in
TauCeti.RingTheory.MvPowerSeries.TateAlgebra.RadiusRing). This is the input for the strong
noetherianity of the interval rings B^I (TauCeti.WittVector.IntervalRing, Kedlaya's
Theorem 4.10).
Euclidean division #
Let x ≠ 0 have degree m. Up to a power of [ϖ], x is close to a Teichmüller sum
∑_{l ≤ m} [tₗ] pˡ whose weighted coefficients v(tₗ) ρˡ are at most v(t_m) ρᵐ, and the
error is bounded by ε v(t_m) ρᵐ for some ε < 1 with ρ ≤ ε. Given y close to a Teichmüller
sum ∑_{l ≤ n} [aₗ] pˡ with n ≥ m, multiplying by the unit [t_m] and
subtracting [a_n] p^(n - m) x cancels the term of index n. Witt vector addition only carries
into higher indices, and its carries are smaller by the factor ρ ≤ ε
(TauCeti.WittVector.gaussValuation_teichmuller_add_teichmuller_sub_le), so the result is again
close to a Teichmüller sum, now of length n. Descending to length m gives Kedlaya's Lemma 2.8:
a remainder of norm at most ‖y‖ that either has degree less than m or is smaller than
ε ‖y‖. Iterating on the remainders (TauCeti.subset_idealSpan_of_approx) gives Proposition 2.9.
Main definitions #
TauCeti.WittVector.RadiusLocalization:𝕎 O[1/[ϖ]], normed by the Gauss normλ_ρ.TauCeti.WittVector.radiusNorm: the extension ofλ_ρto𝕎 O[1/[ϖ]], as an absolute value.TauCeti.WittVector.RadiusRing: the radius ringA^r, the completion ofRadiusLocalization.TauCeti.WittVector.RadiusRing.degree: Kedlaya's degree onA^r.
Main results #
TauCeti.WittVector.isNonarchimedean_radiusNorm: the Gauss norm on𝕎 O[1/[ϖ]]is nonarchimedean.TauCeti.WittVector.norm_algebraMap_radiusRing: on𝕎 O, the norm ofA^risλ_ρ.TauCeti.WittVector.RadiusLocalization.instIsTateRing:𝕎 O[1/[ϖ]]is a Tate ring, with pseudouniformiser[ϖ], so its completionA^ris a complete Hausdorff Tate ring.TauCeti.WittVector.RadiusRing.exists_norm_teichmuller_pow_mul_sub_lt: every nonzerox ∈ A^rhas somey ∈ 𝕎 Owith‖[ϖ]^k x - y‖ < ‖[ϖ]^k x‖.TauCeti.WittVector.RadiusRing.degree_eq_gaussDegree: the degree ofx ∈ A^ris the degree of anyy ∈ 𝕎 Owith‖[ϖ]^k x - y‖ < ‖[ϖ]^k x‖.TauCeti.WittVector.RadiusRing.degree_eq_of_norm_sub_lt: if‖x - y‖ < ‖x‖, thenxandyhave the same degree.TauCeti.WittVector.RadiusRing.degree_mul: the degree is additive on products.TauCeti.WittVector.RadiusRing.exists_eq_mul_add: Euclidean division inA^r.TauCeti.WittVector.RadiusRing.isUnit_iff_ne_zero_and_degree_eq_zero: the units ofA^rare its nonzero elements of degree0.
References #
- K. S. Kedlaya, Noetherian properties of Fargues–Fontaine curves, IMRN 2016, no. 8, 2544–2567; arXiv:1410.5160v3, Definitions 2.2, 2.4 and 2.5, Lemmas 2.6 and 2.8, Remark 2.7, Proposition 2.9 and Theorem 3.2.
- K. S. Kedlaya, Sheaves, stacks, and shtukas, lecture notes, Arizona Winter School 2017, §3.1.
𝕎 O[1/[ϖ]] with the Gauss norm. This is the localisation of 𝕎 O away from the
Teichmüller representative [ϖ] of a pseudouniformiser ϖ (0 < v(ϖ) < 1), normed by the
extension of the Gauss valuation λ_ρ (TauCeti.WittVector.radiusNorm); its completion is the
radius ring A^r.
Instances For
[ϖ] is a unit of 𝕎 O[1/[ϖ]].
The Gauss norm on 𝕎 O[1/[ϖ]]: the extension of the Gauss valuation λ_ρ to the
localisation, an absolute value because λ_ρ is a valuation vanishing only at 0.
Instances For
The Gauss norm on 𝕎 O[1/[ϖ]] extends λ_ρ.
The Gauss norm of a fraction x / s in 𝕎 O[1/[ϖ]] is λ_ρ(x) / λ_ρ(s).
The Gauss norm on 𝕎 O[1/[ϖ]] is nonarchimedean.
The norm of 𝕎 O[1/[ϖ]] is the extension of the Gauss valuation λ_ρ.
On 𝕎 O, the norm of 𝕎 O[1/[ϖ]] is the Gauss valuation λ_ρ.
The norm of a fraction x / s in 𝕎 O[1/[ϖ]] is λ_ρ(x) / λ_ρ(s).
𝕎 O[1/[ϖ]] with the Gauss norm is a Tate ring, with pseudouniformiser [ϖ], a unit of
norm v(ϖ) < 1.
Scalar multiplication by 𝕎 O is uniformly continuous, so that the 𝕎 O-algebra structure
extends to the completion A^r.
The radius ring A^r: the completion of 𝕎 O[1/[ϖ]] for the Gauss norm λ_ρ, for a
radius ρ ∈ (0, 1) and a pseudouniformiser ϖ (0 < v(ϖ) < 1). It is a complete Hausdorff
normed commutative ring with a multiplicative ultrametric norm, and a Tate ring.
Instances For
On 𝕎 O, the norm of A^r is the Gauss valuation λ_ρ.
On 𝕎 O, the norm of A^r is the Gauss valuation λ_ρ, as a nonnegative real.
The norm of p in A^r is ρ.
A^r is a complete Hausdorff Tate ring with a multiplicative ultrametric norm: completeness
and separatedness hold for every completion, multiplicativity and the ultrametric inequality pass
to the completion by continuity, and the Tate structure of 𝕎 O[1/[ϖ]] is carried to the
completion by TauCeti.Huber.IsTateRing.completion (Wedhorn, Remark 6.8).
A^r is nontrivial, as ‖1‖ = 1.
The degree #
The Teichmüller representative [b] of a nonzero b ∈ O is a unit of A^r: it divides a
power of [ϖ].
Every nonzero x ∈ A^r is approximated by some [ϖ]^(-k) y with y ∈ 𝕎 O up to an
arbitrarily small relative error ε: elements of 𝕎 O[1/[ϖ]] are dense in A^r.
Every nonzero x ∈ A^r is approximated by some [ϖ]^(-k) y with y ∈ 𝕎 O, up to an error
smaller than ‖x‖. Such an approximation computes the degree of x
(TauCeti.WittVector.RadiusRing.degree_eq_gaussDegree).
Kedlaya's degree on A^r. For x ≠ 0 it is the degree
(TauCeti.WittVector.gaussDegree) of any y ∈ 𝕎 O with ‖[ϖ]^k x - y‖ < ‖[ϖ]^k x‖ for some
k, that is, of any approximation [ϖ]^(-k) y of x with an error smaller than ‖x‖
(TauCeti.WittVector.RadiusRing.degree_eq_gaussDegree). It is 0 for x = 0.
Instances For
The degree of x ∈ A^r is the degree of any approximation of x by a Witt vector: if
‖[ϖ]^k x - y‖ < ‖[ϖ]^k x‖ with y ∈ 𝕎 O, then the degree of x is the degree of y for
λ_ρ.
The degree of 0 is 0.
On a Witt vector, the degree of A^r is its degree for the Gauss valuation λ_ρ.
The degree of 1 is 0.
The degree of p is 1.
The degree only depends on the leading part (Kedlaya, Remark 2.7): if ‖x - y‖ < ‖x‖,
then x and y have the same degree.
The degree is additive on products (Kedlaya, Lemma 2.6).
Units of A^r have degree 0.
Euclidean division #
Euclidean division in A^r (Kedlaya, Proposition 2.9). For x ≠ 0, every y ∈ A^r can be
written y = z x + w with ‖w‖ ≤ ‖y‖ and either w = 0 or w of smaller degree than x.
A nonzero element of A^r of degree 0 is a unit: dividing 1 by it leaves no remainder.
The units of A^r are its nonzero elements of degree 0.