Independence of the pseudouniformiser #
The space π΄ = D(p) β© D([Ο]) β Spa(π R, π R), its Frobenius action and the orbit space
π³ = π΄ / Ο^β€ are defined in terms of a chosen element Ο : R, which for R = πͺ_F is a
pseudouniformiser of a complete perfect nonarchimedean field F of characteristic p. None of
these depends on that choice. Let Ο, Ο' : R generate ideals with the same radical, as any two
pseudouniformisers of πͺ_F do (Valuation.Integers.radical_span_singleton_eq_of_norm_lt_one).
Then the ideals (p, [Ο]) and (p, [Ο']) define the same adic topology on π R
(TauCeti.WittVector.isAdic_span_p_teichmuller_iff), and this file shows the following.
- The subsets
π΄defined byΟand byΟ'coincide, since a prime ideal contains[Ο]exactly when it contains[Ο']. - The Frobenius actions on them agree, since both are pullback along the Witt-vector Frobenius.
- Consequently the two orbit spaces
π³are homeomorphic, compatibly with the projections fromπ΄. - As pre-adic spaces, the two restrictions
π΄ofSpa(π R, π R)are isomorphic inπ±^pre, compatibly with their Frobenius endomorphisms. Whenπͺ_π΄is a sheaf, so thatπ³is a pre-adic space, this descends through the universal property of the quotient (TauCeti.FarguesFontaine.isColimitQuotientPreAdicCofork) to an isomorphism inπ±^preof the two pre-adic spacesπ³, which on points is the homeomorphism of orbit spaces. - Both isomorphisms lift to the category
(Adic)of adic spaces, where they identify the adic spacesπ΄compatibly with Frobenius and the adic spacesπ³compatibly with the projectionsπ΄ β π³. WhenRis perfect of characteristicpand is the ring of integers of a valuationvon a field, and0 < v(Ο) < 1,πͺ_π΄is a sheaf (TauCeti.FarguesFontaine.isSheaf_spaYPresheaf), so the adic FarguesβFontaine curveπ³ = π΄ / Ο^β€does not depend on the pseudouniformiser.
Main results #
TauCeti.FarguesFontaine.spaY_eq_of_radical_eq:π΄depends only on the radical of(Ο).TauCeti.FarguesFontaine.frobeniusHomeomorph_zpow_apply_val_eq: the integer powers of Frobenius onπ΄do not depend on the element used to defineπ΄.TauCeti.FarguesFontaine.spaXCongr: the homeomorphism between the orbit spaces defined by two elements with the sameπ΄.TauCeti.FarguesFontaine.spaYPreAdicSpaceCongrandTauCeti.FarguesFontaine.spaYPreAdicSpaceCongr_hom_comp_spaYFrobenius: the isomorphism of the pre-adic spacesπ΄, which intertwines Frobenius.TauCeti.FarguesFontaine.isSheaf_spaYPresheaf_of_spaY_eq: the sheaf condition onπͺ_π΄depends only on the subsetπ΄.TauCeti.FarguesFontaine.spaXPreAdicSpaceCongr: the isomorphism inπ±^preof the pre-adic spacesπ³, withquotientPreAdicHom_comp_spaXPreAdicSpaceCongr_homandspaXPreAdicSpaceCongr_hom_basedescribing it.TauCeti.FarguesFontaine.spaYAdicSpaceCongrandTauCeti.FarguesFontaine.spaYAdicSpaceCongr_hom_comp_spaYAdicFrobenius: the isomorphism of the adic spacesπ΄, which intertwines Frobenius.TauCeti.FarguesFontaine.spaXAdicSpaceCongr: the isomorphism of the adic spacesπ³, withquotientAdicHom_comp_spaXAdicSpaceCongr_homdescribing it.
References #
- L. Fargues and J.-M. Fontaine, Courbes et fibrΓ©s vectoriels en thΓ©orie de Hodge p-adique, AstΓ©risque 406 (2018).
- K. S. Kedlaya, Sheaves, stacks, and shtukas, lecture notes, Arizona Winter School 2017, Β§3.1.
- T. Wedhorn, Adic Spaces, arXiv:1910.05934v1, Β§8.1 and Definition 8.22, for the categories
π±^preand(Adic).
π΄ depends only on the radical of (Ο). If Ο and Ο' generate ideals with the same
radical, for instance if they are two pseudouniformisers of πͺ_F, then
D(p) β© D([Ο]) = D(p) β© D([Ο']) in Spa(π R, π R).
π΄ as a pre-adic space depends only on the subset π΄. If Ο and Ο' define the same
subset π΄ of Spa(π R, π R), the two restrictions of the pre-adic space Spa(π R, π R) to it
are isomorphic in π±^pre, compatibly with the inclusions into Spa(π R, π R)
(spaYPreAdicSpaceCongr_hom_ofRestrict).
Instances For
The isomorphism spaYPreAdicSpaceCongr commutes with the inclusions into
Spa(π R, π R).
The isomorphism spaYPreAdicSpaceCongr commutes with the inclusions into
Spa(π R, π R).
On points, spaYPreAdicSpaceCongr does not change the underlying valuation.
The sheaf condition on πͺ_π΄ depends only on the subset π΄. Both presheaves restrict the
structure presheaf of Spa(π R, π R) to the same open subset.
The adic spaces π΄ #
The adic spaces π΄ for two pseudouniformisers are isomorphic. If Ο and Ο' define the
same subset π΄ of Spa(π R, π R) and πͺ_π΄ is a sheaf, the isomorphism spaYPreAdicSpaceCongr of
pre-adic spaces is an isomorphism in the full subcategory (Adic). For perfect R it
intertwines Frobenius (spaYAdicSpaceCongr_hom_comp_spaYAdicFrobenius).
Instances For
The morphism of pre-adic spaces underlying spaYAdicSpaceCongr is spaYPreAdicSpaceCongr.
The morphism of pre-adic spaces underlying the inverse of spaYAdicSpaceCongr is the inverse
of spaYPreAdicSpaceCongr.
The Frobenius actions on π΄ do not depend on the pseudouniformiser. For points of the
subsets π΄ defined by Ο and by Ο' with the same underlying valuation, every integer power of
Frobenius gives the same valuation: on either subset, Frobenius is pullback along the Witt-vector
Frobenius.
The orbit spaces π΄ / Ο^β€ for two pseudouniformisers are homeomorphic. If Ο and Ο'
define the same subset π΄ of Spa(π R, π R), as they do when they generate ideals with the same
radical (spaY_eq_of_radical_eq), then the identity of π΄ descends to a homeomorphism of the
Frobenius orbit spaces, because the two Frobenius actions agree.
Instances For
The homeomorphism of orbit spaces sends the orbit of a point of π΄ to its orbit.
The inverse of the homeomorphism of orbit spaces is the homeomorphism in the other direction.
The pre-adic spaces π³ #
The identification of the two pre-adic spaces π΄ intertwines their Frobenius
endomorphisms: both are restrictions of the automorphism of Spa(π R, π R) induced by Ο.
The identification of the two pre-adic spaces π΄ intertwines their Frobenius
endomorphisms: both are restrictions of the automorphism of Spa(π R, π R) induced by Ο.
The inverse of spaYPreAdicSpaceCongr also intertwines the Frobenius endomorphisms.
The inverse of spaYPreAdicSpaceCongr also intertwines the Frobenius endomorphisms.
The pre-adic spaces π³ = π΄ / Ο^β€ for two pseudouniformisers are isomorphic. If Ο and
Ο' define the same subset π΄ of Spa(π R, π R), as they do when they generate ideals with the
same radical (spaY_eq_of_radical_eq), and πͺ_π΄ is a sheaf, then the identification
spaYPreAdicSpaceCongr of the two pre-adic spaces π΄ descends to an isomorphism in π±^pre of the
Frobenius quotients, by their universal property (isColimitQuotientPreAdicCofork). On points it is
the homeomorphism spaXCongr (spaXPreAdicSpaceCongr_hom_base).
Instances For
The isomorphism of the pre-adic spaces π³ is induced by the identification of the pre-adic
spaces π΄.
The isomorphism of the pre-adic spaces π³ is induced by the identification of the pre-adic
spaces π΄.
On points, the isomorphism of the pre-adic spaces π³ is the homeomorphism spaXCongr of the
orbit spaces.
Frobenius on the adic spaces π΄, and the adic spaces π³ #
The identification of the two adic spaces π΄ intertwines Frobenius.
The identification of the two adic spaces π΄ intertwines Frobenius.
The inverse of spaYAdicSpaceCongr also intertwines Frobenius.
The inverse of spaYAdicSpaceCongr also intertwines Frobenius.
The adic FarguesβFontaine curves π³ = π΄ / Ο^β€ for two pseudouniformisers are isomorphic.
If Ο and Ο' define the same subset π΄ of Spa(π R, π R), as they do when they generate ideals
with the same radical (spaY_eq_of_radical_eq), and πͺ_π΄ is a sheaf, then the isomorphism
spaXPreAdicSpaceCongr of pre-adic spaces is an isomorphism in the full subcategory (Adic). It is
the isomorphism induced by spaYAdicSpaceCongr on the quotients by Frobenius
(quotientAdicHom_comp_spaXAdicSpaceCongr_hom), and on points it is the homeomorphism
spaXCongr of orbit spaces (spaXPreAdicSpaceCongr_hom_base). When R is perfect of
characteristic p and is the ring of integers of a valuation v on a field, and 0 < v(Ο) < 1,
πͺ_π΄ is a sheaf by isSheaf_spaYPresheaf.
Instances For
The morphism of pre-adic spaces underlying spaXAdicSpaceCongr is spaXPreAdicSpaceCongr.
The morphism of pre-adic spaces underlying the inverse of spaXAdicSpaceCongr is the inverse
of spaXPreAdicSpaceCongr.
The isomorphism of the adic spaces π³ is induced by the identification of the adic spaces
π΄: it commutes with the projections π΄ β π³. Together with the universal property of π³
(isColimitQuotientAdicCofork) this determines it.
The isomorphism of the adic spaces π³ is induced by the identification of the adic spaces
π΄: it commutes with the projections π΄ β π³. Together with the universal property of π³
(isColimitQuotientAdicCofork) this determines it.