The Frobenius orbit space as an adic space #
Let R be a perfect ring of characteristic p and give the Witt vectors π R the (p, [Ο])-adic
topology; for R = πͺ_F this is A_inf. The pre-adic spaces π΄ = D(p) β© D([Ο]) and
π³ = π΄ / Ο^β€ of TauCeti.FarguesFontaine.spaYPreAdicSpace and
TauCeti.FarguesFontaine.spaXPreAdicSpace are adic spaces as soon as the structure presheaf
πͺ_π΄ is a sheaf, and π³ is then the quotient of π΄ by Frobenius in the category (Adic) of
adic spaces. When R is the ring of integers of a valuation on a field, πͺ_π΄ is a sheaf
(isSheaf_spaYPresheaf), so π΄ and π³ are adic spaces with no further hypothesis.
π΄is an open subspace of the pre-adic spaceSpa(π R, π R), so it is locally affinoid with no further hypothesis, and it is an adic space exactly whenπͺ_π΄is a sheaf.π³is covered by the chartsV β π³on the wandering opensV β π΄, such as the Frobenius windows. These charts are open immersions, so each open affinoid subspace ofπ΄inside a wandering open is carried onto an open affinoid subspace ofπ³, andπ³is locally affinoid. Its structure presheaf is a sheaf becauseπͺ_π΄is (isSheafy_spaXPreAdicSpace): the sheaf property ofπ³descends fromπ΄, not from the affinoid charts.
For a general perfect ring R, the sheaf condition on πͺ_π΄ is a hypothesis of the results on
π³, as it is for the pre-adic space π³ itself. This file reduces it to windows: π΄ is a
pre-adic space, exhausted by the increasing windows W_n = {v β π΄ : p ^ (-n) β€ ΞΊ(v) β€ p ^ n},
and every open affinoid subspace of π΄ is quasi-compact, so lies in one of them. Hence πͺ_π΄ is
a sheaf as soon as its restriction to each W_n is (isSheaf_spaYPresheaf_of_isSheafy_restrict).
When R is the ring of integers of a valuation v on a field and 0 < v(Ο) < 1,
TauCeti.FarguesFontaine.windowPreAdicSpaceIso identifies W_n with Spa(B^I, B^{I,+}) for the
interval ring B^I of I = [v(Ο) ^ (p ^ n), v(Ο) ^ (p ^ (-n))]. The ring B^I is strongly
noetherian (Kedlaya's Theorem 4.10), so the pair (B^I, B^{I,+}) is sheafy, and πͺ_π΄ is a
sheaf (isSheaf_spaYPresheaf).
Main definitions #
TauCeti.FarguesFontaine.spaYAdicSpaceandTauCeti.FarguesFontaine.spaXAdicSpace:π΄andπ³as objects of(Adic), whenπͺ_π΄is a sheaf.TauCeti.FarguesFontaine.spaYAdicFrobeniusandTauCeti.FarguesFontaine.quotientAdicHom: Frobenius onπ΄and the projectionπ΄ β π³as morphisms of adic spaces.
Main results #
TauCeti.FarguesFontaine.isLocallyAffinoid_spaYPreAdicSpace:π΄is locally affinoid.TauCeti.FarguesFontaine.isPreAdic_spaYPreAdicSpace:π΄is a pre-adic space.TauCeti.FarguesFontaine.isSheaf_spaYPresheaf_of_isSheafy_restrict:πͺ_π΄is a sheaf if its restriction to every windowW_nis.TauCeti.FarguesFontaine.isSheaf_spaYPresheaf:πͺ_π΄is a sheaf whenRis the ring of integers of a valuation on a field.TauCeti.FarguesFontaine.isAdic_spaYPreAdicSpace_iff:π΄is an adic space exactly whenπͺ_π΄is a sheaf.TauCeti.FarguesFontaine.isLocallyAffinoid_spaXPreAdicSpace:π³is locally affinoid.TauCeti.FarguesFontaine.isAdic_spaXPreAdicSpace:π³is an adic space.TauCeti.FarguesFontaine.isColimitQuotientAdicCofork:π³is the coequalizer of the identity and Frobenius onπ΄in(Adic).
References #
- T. Wedhorn, Adic Spaces, arXiv:1910.05934v1, Definition 8.22, Remark 8.27 and Theorem 8.28.
- K. S. Kedlaya, Sheaves, stacks, and shtukas, Arizona Winter School 2017 notes, Β§3.1.
- K. S. Kedlaya, Noetherian properties of FarguesβFontaine curves, IMRN 2016, no. 8, 2544β2567; arXiv:1410.5160v3, Theorem 4.10.
- L. Fargues and J.-M. Fontaine, Courbes et fibrΓ©s vectoriels en thΓ©orie de Hodge p-adique, AstΓ©risque 406 (2018), Chapter 1.
π΄ as an adic space #
π΄ is locally affinoid, as an open subspace of the pre-adic space Spa(π R, π R). No
sheaf condition is needed.
π΄ is a pre-adic space, as the restriction of the pre-adic space Spa(π R, π R) to an
open.
πͺ_π΄ is a sheaf if it is one on every window: if the restriction of π΄ to each window
W_n = {v β π΄ : p ^ (-n) β€ ΞΊ(v) β€ p ^ n} is sheafy, then πͺ_π΄ is a sheaf. The windows are a
directed family of opens exhausting the pre-adic space π΄, so every open affinoid subspace of π΄,
being quasi-compact, lies in one of them.
π΄ is an adic space exactly when πͺ_π΄ is a sheaf, since it is locally affinoid.
π΄ as an adic space, an object of (Adic), when πͺ_π΄ is a sheaf.
Instances For
The pre-adic space underlying the adic space π΄ is spaYPreAdicSpace.
Sheafiness of πͺ_π΄ #
πͺ_π΄ is a sheaf: let R be the ring of integers of a valuation v on a field K,
perfect of characteristic p, and Ο β R with 0 < v(Ο) < 1; for R = πͺ_F this is the
FarguesβFontaine space π΄ β Spa(A_inf, A_inf). Then the structure presheaf πͺ_π΄ is a sheaf, so
π΄ and π³ = π΄ / Ο^β€ are adic spaces (isAdic_spaYPreAdicSpace_iff, isAdic_spaXPreAdicSpace).
The window W_n = {p ^ (-n) β€ ΞΊ β€ p ^ n} is Spa(B^I, B^{I,+}) for
I = [v(Ο) ^ (p ^ n), v(Ο) ^ (p ^ (-n))] (TauCeti.FarguesFontaine.windowPreAdicSpaceIso), and
this pair is sheafy because B^I is strongly noetherian (Kedlaya's Theorem 4.10), so πͺ_π΄ is
sheafy on each window (isSheafy_spaYPreAdicSpace_restrict), and hence a sheaf
(isSheaf_spaYPresheaf_of_isSheafy_restrict).
π³ as an adic space #
π³ is locally affinoid when πͺ_π΄ is a sheaf: every point of π³ lies in the image of an
open affinoid subspace of π΄ contained in a wandering open, and that image is an open affinoid
subspace of π³, since the chart on a wandering open is an open immersion.
π³ = π΄ / Ο^β€ is an adic space when πͺ_π΄ is a sheaf: it is sheafy
(isSheafy_spaXPreAdicSpace) and locally affinoid (isLocallyAffinoid_spaXPreAdicSpace).
The quotient in (Adic) #
π³ = π΄ / Ο^β€ as an adic space, an object of (Adic), when πͺ_π΄ is a sheaf.
Instances For
The pre-adic space underlying the adic space π³ is spaXPreAdicSpace.
Witt Frobenius as an endomorphism of the adic space π΄: the morphism of pre-adic spaces
spaYFrobenius, read in the full subcategory (Adic).
Instances For
The projection π΄ β π³ as a morphism of adic spaces: the morphism of pre-adic spaces
quotientPreAdicHom, read in the full subcategory (Adic).
Instances For
The morphism of pre-adic spaces underlying Frobenius on the adic space π΄ is
spaYFrobenius.
The morphism of pre-adic spaces underlying the projection of adic spaces is
quotientPreAdicHom.
The projection to π³ is Frobenius-invariant as a morphism of adic spaces.
The projection to π³ is Frobenius-invariant as a morphism of adic spaces.
π³ = π΄ / Ο^β€ is the quotient of π΄ by Frobenius in (Adic): when πͺ_π΄ is a sheaf, the
adic space π³ is the coequalizer of the identity and Frobenius on the adic space π΄, with the
projection q as coprojection. The inclusion of (Adic) into π±^pre is fully faithful, so this
is the universal property of π³ in π±^pre (isColimitQuotientPreAdicCofork) restricted to
adic spaces.