Windows of π΄ are the affinoid spectra Spa(B^I, B^{I,+}) #
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; for instance
O = πͺ_F for a perfect nonarchimedean field F of characteristic p, so that π O = A_inf.
Give π O its (p, [Ο])-adic topology. For nonnegative rationals q = a / b β 0 and
q' = a' / b' in lowest terms, the window W = {v β π΄ : q β€ ΞΊ(v) β€ q'} of the FarguesβFontaine
curve is a rational subset of Spa(π O, π O), and this file identifies the pre-adic space it
carries with the affinoid pre-adic space of the interval ring B^I, for the radii
Οβ = v(Ο) ^ (1 / q) and Οβ = v(Ο) ^ (1 / q'):
W β
Spa(B^I, B^{I,+}).
The Frobenius windows U_n and V_n are the windows q = p ^ n, q' = c p ^ n and
q = c p ^ n, q' = p ^ (n + 1), with c = (p + 1) / 2.
- The window is the rational subset
R(T/s)presented byT = {[Ο] ^ (b + b'), p ^ (a + a')}ands = p ^ a [Ο] ^ b'(TauCeti.WittVector.windowNumerators,TauCeti.WittVector.windowDenominator): overs, the two numerators give the fractions[Ο] ^ b / p ^ aandp ^ a' / [Ο] ^ b', which are at most1exactly on the window. Hence the open embeddingj : Spa(π Oβ¨T/sβ©, A_UβΊ) β Spa(π O, π O)induced by the rational localisation has imageW. - Wedhorn's Remark 8.8 (
TauCeti.ValuationSpectrum.presentationLimitPreAdicSpaceLocIso) identifies the restriction ofSpa(π O, π O)alongjwithSpa(π Oβ¨T/sβ©, A_UβΊ). - The isomorphism
π Oβ¨T/sβ© β+* B^Iof complete topological rings (TauCeti.WittVector.completionLocRingEquivIntervalRing) carriesA_UβΊontoB^{I,+}(TauCeti.WittVector.map_completedPlusSubring_windowNumerators), so it is an isomorphism of Huber pairs and induces an isomorphism of pre-adic spaces (TauCeti.ValuationSpectrum.presentationLimitPreAdicSpaceComapIso). Neither step needs a sheaf condition.
The pair of definition of B^I used to form its pre-adic space is arbitrary; any one of them gives
the same space up to this isomorphism.
Sheafiness transfers along the isomorphism. The interval ring B^I is strongly noetherian
(TauCeti.WittVector.isStronglyNoetherian_intervalRing, Kedlaya's Theorem 4.10), so by Wedhorn's
Theorem 8.28(b) the pair (B^I, B^{I,+}) is sheafy and, by Wedhorn's Corollary 8.35, B^I is
stably sheafy. Hence the window is an adic space, and the structure presheaf πͺ_π΄ of π΄ is a
sheaf on every open of π΄ contained in the window. The sheaf property of π΄, and so of the
FarguesβFontaine curve π³ = π΄ / Ο^β€, is assembled from these charts.
Main results #
TauCeti.FarguesFontaine.rationalSubset_windowNumerators: the window{q β€ ΞΊ β€ q'}is the rational subsetR(T/s).TauCeti.FarguesFontaine.range_spaComapLoc_windowNumerators: the open embeddingj : Spa(π Oβ¨T/sβ©, A_UβΊ) β Spa(π O, π O)has image the window.TauCeti.FarguesFontaine.windowPreAdicSpaceIso: the restriction ofSpa(π O, π O)alongjis isomorphic inπ±^pretoSpa(B^I, B^{I,+}); on points it is pullback alongB^I β+* π Oβ¨T/sβ©(windowPreAdicSpaceIso_hom_base_apply).TauCeti.WittVector.isStablySheafyRing_intervalRing: the interval ringB^Iis stably sheafy.TauCeti.FarguesFontaine.isAdic_spaPreAdicSpace_restrict_windowNumerators: the restriction ofSpa(π O, π O)alongjis an adic space.TauCeti.FarguesFontaine.isSheafy_spaYPreAdicSpace_restrict: the restriction ofπ΄to any open inside the window is sheafy.
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. - K. S. Kedlaya, Noetherian properties of FarguesβFontaine curves, IMRN 2016, no. 8,
2544β2567; arXiv:1410.5160v3, Theorem 4.10, for the strong noetherianness of
B^I. - T. Wedhorn, Adic Spaces (arXiv:1910.05934v1), Remark 8.8, Definition 8.22, Theorem 8.28(b) and Corollary 8.35.
The window as a rational subset #
The window {v β π΄ : q β€ ΞΊ(v) β€ q'} is the rational subset R(T/s), for q β 0, where
T = {[Ο] ^ (b + b'), p ^ (a + a')} and s = p ^ a [Ο] ^ b' for q = a / b and q' = a' / b' in
lowest terms. Off the support of s, the inequalities v([Ο] ^ (b + b')) β€ v(s) and
v(p ^ (a + a')) β€ v(s) are the radius bounds v([Ο]) ^ b β€ v(p) ^ a and
v(p) ^ a' β€ v([Ο]) ^ b'.
The rational localisation at the window has image the window: for q β 0, the open
embedding j : Spa(π Rβ¨T/sβ©, A_UβΊ) β Spa(π R, π R) induced by the structure map has image
{v β π΄ : q β€ ΞΊ(v) β€ q'}.
Stable sheafiness of the interval rings #
The interval ring B^I is stably sheafy for all radii 0 < Οβ β€ Οβ < 1: every complete
Hausdorff Huber ring topologically of finite type over B^I is sheafy. The ring B^I is a
complete Hausdorff Tate ring that is strongly noetherian by Kedlaya's Theorem 4.10
(TauCeti.WittVector.isStronglyNoetherian_intervalRing), so this is Wedhorn's Corollary 8.35
(TauCeti.Huber.isStablySheafyRing_of_isStronglyNoetherian). In particular B^I is sheafy
(TauCeti.Huber.IsStablySheafyRing.isSheafyRing).
The window as an affinoid pre-adic space #
A window of π΄ is the affinoid pre-adic space Spa(B^I, B^{I,+}): the restriction of
Spa(π O, π O) along the open embedding j : Spa(π Oβ¨T/sβ©, A_UβΊ) β Spa(π O, π O), whose image
is the window {q β€ ΞΊ β€ q'} (range_spaComapLoc_windowNumerators), is isomorphic in π±^pre to
the pre-adic space of the interval ring B^I with plus ring B^{I,+} = (B^I)Β°, for the radii
Οβ ^ a = v(Ο) ^ b, Οβ ^ a' = v(Ο) ^ b', Οβ β€ Οβ matching the window. It is Wedhorn's
Remark 8.8 for the window followed by the isomorphism of Huber pairs
(π Oβ¨T/sβ©, A_UβΊ) β
(B^I, B^{I,+}). Here P' is any pair of definition of B^I. The hypothesis
that π O[1/(p [Ο])] is a localisation away from s = p ^ a [Ο] ^ b' holds when q β 0
(TauCeti.WittVector.isLocalization_away_windowDenominator).
Instances For
On points, windowPreAdicSpaceIso is pullback along B^I β+* π Oβ¨T/sβ©: a point x of
Spa(π Oβ¨T/sβ©, A_UβΊ), the source of j, goes to the valuation x β eβ»ΒΉ of B^I, where
e : π Oβ¨T/sβ© β+* B^I is completionLocRingEquivIntervalRing.
On points, the inverse window chart is pullback along π Oβ¨T/sβ© β+* B^I.
This is the inverse formula to windowPreAdicSpaceIso_hom_base_apply.
A window of π΄ is an adic space: the window {q β€ ΞΊ β€ q'} of Spa(π O, π O), the
restriction along the open embedding j : Spa(π Oβ¨T/sβ©, A_UβΊ) β Spa(π O, π O), is isomorphic to
Spa(B^I, B^{I,+}) (windowPreAdicSpaceIso), for the interval ring B^I with its plus ring
B^{I,+} = (B^I)Β°. The structure presheaf of Spa(B^I, B^{I,+}) is a sheaf by Wedhorn's
Theorem 8.28(b)
(TauCeti.ValuationSpectrum.isSheaf_presentationLimitPresheaf_of_isStronglyNoetherian), since
B^I is a strongly noetherian Tate ring (TauCeti.WittVector.isStronglyNoetherian_intervalRing).
Sheafiness of π΄ over a window #
πͺ_π΄ is a sheaf over every open contained in a window: let q = a / b and q' = a' / b'
in lowest terms, with radii Οβ ^ a = v(Ο) ^ b, Οβ ^ a' = v(Ο) ^ b' and Οβ β€ Οβ. Then the
restriction of the pre-adic space π΄ along any open embedding whose image lies in the window
{q β€ ΞΊ β€ q'} is sheafy. The window is an adic space
(isAdic_spaPreAdicSpace_restrict_windowNumerators), and sheafiness passes to its open
subspaces.