The window subring of π O[1/(p [Ο])] and the interval norm #
Let O be the ring of integers of a valuation v : Valuation K ββ₯0 on a field K, perfect of
characteristic p, and let Ο β O be a pseudouniformiser, 0 < v(Ο) < 1. In a localisation B
of π O away from p [Ο], for nonnegative rationals q = a / b and q' = a' / b' in lowest
terms, consider the fractions
[Ο] ^ b / p ^ a, p ^ a' / [Ο] ^ b'
(TauCeti.WittVector.radiusLowerFrac, TauCeti.WittVector.radiusUpperFrac) and the subring
π O[[Ο] ^ b / p ^ a, p ^ a' / [Ο] ^ b']
of B they generate over π O (TauCeti.WittVector.windowSubring). A valuation w on B is
at most 1 at the first fraction exactly when w([Ο]) ^ b β€ w(p) ^ a, the lower radius bound
q β€ ΞΊ(w), and at the second exactly when w(p) ^ a' β€ w([Ο]) ^ b', the upper radius bound
ΞΊ(w) β€ q'. So the window subring is the ring of definition Aβ[T / s], for Aβ = π O, of the
rational localisation of π O at the window {q β€ ΞΊ β€ q'} of the FarguesβFontaine curve,
presented by these two fractions.
The interval ring B^I is instead the completion of π O[1/(p [Ο])] for the interval norm
Ξ»_I = max(Ξ»_{Οβ}, Ξ»_{Οβ}) (TauCeti.WittVector.IntervalLocalization). The main theorem compares
the two topologies. If the radii are the ones matching the window,
Οβ ^ a = v(Ο) ^ b, Οβ ^ a' = v(Ο) ^ b', Οβ β€ Οβ,
that is Οβ = v(Ο) ^ (1 / q) and Οβ = v(Ο) ^ (1 / q'), then the window subring is open and
bounded for Ξ»_I, hence a ring of definition of π O[1/(p [Ο])] with the interval norm (Wedhorn,
Lemma 6.2). This is the analytic input for identifying B^I with the completed coordinate ring of
the window.
For the same radii, the integral closure of the window subring in π O[1/(p [Ο])] is the unit
ball of Ξ»_I, and so the closure of its image in B^I is the unit ball B^{I,+}. The plus ring
A_UβΊ of a rational localisation is the closure of the image of the integral closure of
AβΊ[T / s] (TauCeti.Huber.PairOfDefinition.completedPlusSubring). With AβΊ = π O, this
equality is the analytic input for identifying the plus ring of the window with B^{I,+}, which
transporting it along the isomorphism between the coordinate ring of the window and B^I
achieves (TauCeti.WittVector.map_completedPlusSubring_windowNumerators).
Proof outline #
Boundedness holds because both fractions have interval norm at most 1. Openness comes from the
unit ball: every x with Ξ»_I(x) β€ 1 lies in the window subring after multiplication by
[Ο] ^ (b + b'). Write x = y / (p [Ο]) ^ m and expand y = ββ [yβ] pβΏ in TeichmΓΌller
coordinates. The bound Ξ»_{Οβ}(x) β€ 1 controls the terms with n < m, which carry a negative
power of p and are reached through [Ο] ^ b / p ^ a. The bound Ξ»_{Οβ}(x) β€ 1 controls the
terms with n β₯ m, which carry a negative power of [Ο] and are reached through
p ^ a' / [Ο] ^ b'. In each case the valuation inequality makes a power of Ο divide yβ in
O. The tail p ^ (M + 1) z of the expansion is absorbed by a power of p ^ a' / [Ο] ^ b'.
Main definitions #
TauCeti.WittVector.radiusLowerFrac: the fraction[Ο] ^ b / p ^ aforq = a / b.TauCeti.WittVector.radiusUpperFrac: the fractionp ^ a' / [Ο] ^ b'forq' = a' / b'.TauCeti.WittVector.windowSubring: the subringπ O[[Ο] ^ b / p ^ a, p ^ a' / [Ο] ^ b'].
Main results #
TauCeti.WittVector.radiusLowerFrac_mul_algebraMapandTauCeti.WittVector.radiusUpperFrac_mul_algebraMap: the defining equations of the fractions.TauCeti.WittVector.windowSubring_le_iff: the universal property of the window subring.TauCeti.WittVector.algebraMap_teichmuller_pow_mul_mem_windowSubring_of_norm_le_one: the unit ball ofΞ»_Ilies in[Ο] ^ -(b + b')times the window subring.TauCeti.WittVector.isOpen_windowSubring: the window subring is open forΞ»_I.TauCeti.WittVector.windowSubring_le_powerBoundedSubringandTauCeti.WittVector.isBounded_windowSubring: the window subring lies in the unit ball ofΞ»_I, so it is bounded.TauCeti.WittVector.exists_pairOfDefinition_ringOfDefinition_eq_windowSubring: for the radii of the window, the window subring is a ring of definition ofπ O[1/(p [Ο])]with the interval norm.TauCeti.WittVector.isIntegral_windowSubring_of_norm_le_one: the unit ball ofΞ»_Iis integral over the window subring.TauCeti.WittVector.integralClosure_windowSubring_eq_powerBoundedSubring: for the radii of the window, the integral closure of the window subring is the unit ball ofΞ»_I.TauCeti.WittVector.topologicalClosure_map_coeRingHom_integralClosure_windowSubring: for the radii of the window, the closure inB^Iof the image of that integral closure isB^{I,+}.
References #
- K. S. Kedlaya, Sheaves, stacks, and shtukas, lecture notes, Arizona Winter School 2017,
Β§3.1, for the interval rings
B^Iand the windows. - T. Wedhorn, Adic Spaces (arXiv:1910.05934v1), Lemma 6.2 and Proposition and Definition 5.51, and Β§8.1 for the plus ring of a rational localisation.
The fraction [Ο] ^ b / p ^ a of a localisation B of π O away from p [Ο], for
q = a / b in lowest terms. Since p is a unit of B, a valuation w on B has w(x) β€ 1 at
this fraction x exactly when w([Ο]) ^ b β€ w(p) ^ a, the lower radius bound q β€ ΞΊ(w).
Instances For
The fraction p ^ a' / [Ο] ^ b' of a localisation B of π O away from p [Ο], for
q' = a' / b' in lowest terms. Since [Ο] is a unit of B, a valuation w on B has
w(x) β€ 1 at this fraction x exactly when w(p) ^ a' β€ w([Ο]) ^ b', the upper radius bound
ΞΊ(w) β€ q'.
Instances For
The defining equation of [Ο] ^ b / p ^ a: its product with p ^ a is [Ο] ^ b. It
determines the fraction, since p is a unit of B.
The defining equation of p ^ a' / [Ο] ^ b': its product with [Ο] ^ b' is p ^ a'. It
determines the fraction, since [Ο] is a unit of B.
The window subring π O[[Ο] ^ b / p ^ a, p ^ a' / [Ο] ^ b'] of a localisation B of
π O away from p [Ο], for q = a / b and q' = a' / b' in lowest terms: the subring generated
over π O by radiusLowerFrac p Ο B q and radiusUpperFrac p Ο B q'. This is the ring
Aβ[T / s], for Aβ = π O, that the rational localisation of π O at the window
{q β€ ΞΊ β€ q'}, presented by these two fractions, carries as its ring of definition.
Instances For
The window subring is the subring generated by the two fractions over π O.
The image of π O lies in the window subring.
The fraction [Ο] ^ b / p ^ a lies in the window subring.
The fraction p ^ a' / [Ο] ^ b' lies in the window subring.
The universal property of the window subring: it lies in a subring S of B exactly
when S contains the image of π O and the two fractions.
The unit ball of the interval norm lies in the window subring, up to [Ο] ^ (b + b').
If Οβ ^ a β€ v(Ο) ^ b and v(Ο) ^ b' β€ Οβ ^ a' for q = a / b and q' = a' / b' β 0 in lowest
terms, then [Ο] ^ (b + b') x β π O[[Ο] ^ b / p ^ a, p ^ a' / [Ο] ^ b'] for every x with
Ξ»_I(x) β€ 1.
The window subring is open for the interval norm, if Οβ ^ a β€ v(Ο) ^ b and
v(Ο) ^ b' β€ Οβ ^ a' for q = a / b and q' = a' / b' β 0 in lowest terms: it contains every x
with Ξ»_I(x) small.
The window subring consists of power-bounded elements, if v(Ο) ^ b β€ Οα΅’ ^ a and
Οα΅’ ^ a' β€ v(Ο) ^ b' at both radii, for q = a / b and q' = a' / b' in lowest terms: both
fractions, and so the whole window subring, lie in the unit ball of Ξ»_I, which is the
power-bounded subring of π O[1/(p [Ο])].
The window subring is bounded for the interval norm, if v(Ο) ^ b β€ Οα΅’ ^ a and
Οα΅’ ^ a' β€ v(Ο) ^ b' at both radii, for q = a / b and q' = a' / b' in lowest terms: it lies in
the unit ball of Ξ»_I by TauCeti.WittVector.windowSubring_le_powerBoundedSubring.
The window subring is a ring of definition for the interval norm. If the radii are those
of the window {q β€ ΞΊ β€ q'}, that is Οβ ^ a = v(Ο) ^ b, Οβ ^ a' = v(Ο) ^ b' and Οβ β€ Οβ for
q = a / b and q' = a' / b' in lowest terms, then π O[[Ο] ^ b / p ^ a, p ^ a' / [Ο] ^ b'] is
a ring of definition of π O[1/(p [Ο])] with the interval norm Ξ»_I. So Ξ»_I defines the
topology of the rational localisation of π O at the window.
The integral closure of the window subring #
The unit ball of the interval norm is integral over the window subring. If
Οβ ^ a β€ v(Ο) ^ b and v(Ο) ^ b' β€ Οβ ^ a' for q = a / b and q' = a' / b' β 0 in lowest
terms, then every x β π O[1/(p [Ο])] with Ξ»_I(x) β€ 1 is integral over
π O[[Ο] ^ b / p ^ a, p ^ a' / [Ο] ^ b'].
The integral closure of the window subring is the unit ball of the interval norm. If the
radii are those of the window {q β€ ΞΊ β€ q'}, that is Οβ ^ a = v(Ο) ^ b, Οβ ^ a' = v(Ο) ^ b'
and Οβ β€ Οβ for q = a / b and q' = a' / b' in lowest terms, then the integral closure of
π O[[Ο] ^ b / p ^ a, p ^ a' / [Ο] ^ b'] in π O[1/(p [Ο])] is the power-bounded subring, the
set of x with Ξ»_I(x) β€ 1 (TauCeti.WittVector.IntervalLocalization.coe_powerBoundedSubring).
B^{I,+} is the closure of the integral closure of the window subring. If the radii are
those of the window {q β€ ΞΊ β€ q'}, as in
TauCeti.WittVector.integralClosure_windowSubring_eq_powerBoundedSubring, then the closure in
B^I of the image of the integral closure of π O[[Ο] ^ b / p ^ a, p ^ a' / [Ο] ^ b'] in
π O[1/(p [Ο])] is the power-bounded subring of B^I, its unit ball B^{I,+}
(TauCeti.WittVector.coe_powerBoundedSubring_intervalRing). The left-hand side has the shape of
the plus ring A_UβΊ of a rational localisation
(TauCeti.Huber.PairOfDefinition.completedPlusSubring), with AβΊ = π O and AβΊ[T/s] the window
subring.