The Frobenius orbit space as a pre-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 adic FarguesβFontaine curve is the orbit space
π³ = π΄ / Ο^β€ of the open subset π΄ = D(p) β© D([Ο]) of Spa(π R, π R) under Witt Frobenius, with
the structure presheaf πͺ_π³(U) = πͺ_π΄(qβ»ΒΉU)^{Ο=1} (TauCeti.FarguesFontaine.spaXPresheaf). This
file makes π³ an object of Wedhorn's category π±^pre of pre-adic spaces and identifies its charts
on wandering opens, such as the Frobenius windows, with the corresponding pieces of π΄.
π΄is the pre-adic space obtained by restrictingSpa(π R, π R), and Frobenius acts on it by an endomorphism inπ±^pre, the restriction of the automorphism ofSpa(π R, π R)induced byΟ; on sections it isspaYPresheafFrobenius.- The projection
q : π΄ β π³, with the inclusion of the Frobenius-invariant sections, is a morphism of presheafed spaces, and it is invariant under Frobenius. - Suppose that
πͺ_π΄is a sheaf. On an openV β π΄meeting none of its nontrivial Frobenius translates,qis an open embedding along whichπͺ_π³restricts toπͺ_π΄(TauCeti.FarguesFontaine.isIso_spaXPresheafRestrict), soV β π³is an open immersion. Every point ofπ΄lies in such an open, a Frobenius window, so the stalk maps ofqare isomorphisms: the stalks ofπͺ_π³are local, and the stalk atq yreceives the valuation ofπ΄aty. Frobenius preserves these valuations, so they do not depend on the pointyof the orbit, andPreAdicSpace.descend, applied toqalone, makesπ³a pre-adic space withqa morphism ofπ±^pre. π³is the quotient ofπ΄by Frobenius: a morphism out ofπ΄withΟ β« g = gfactors uniquely throughq, soqis the coequalizer of the identity and Frobenius, among presheafed spaces and, whenπͺ_π΄is a sheaf, inπ±^pre. On points the factorization is the induced map of orbit spaces; on sections it takes values in the Frobenius-invariant sections ofπͺ_π΄, which are those ofπͺ_π³.
The sheaf condition on πͺ_π΄ is a hypothesis for the window charts and the stalks of π³, for π³
as a pre-adic space and for the universal property in π±^pre, as for isSheaf_spaXPresheaf and
isIso_spaXPresheafRestrict. It is not proved here; TauCeti.FarguesFontaine.isSheaf_spaYPresheaf
proves it when R is the ring of integers of a valuation on a field. Frobenius on π΄, the
projection q and its universal property among presheafed spaces do not need it.
Main definitions #
TauCeti.FarguesFontaine.spaYPreAdicSpace:π΄as a pre-adic space.TauCeti.FarguesFontaine.spaYFrobenius: Witt Frobenius as an endomorphism ofπ΄inπ±^pre.TauCeti.FarguesFontaine.quotientPresheafedHom: the projectionq : π΄ β π³of presheafed spaces.TauCeti.FarguesFontaine.spaXPreAdicSpace:π³as a pre-adic space, whenπͺ_π΄is a sheaf.TauCeti.FarguesFontaine.quotientPreAdicHom: the projectionqas a morphism of pre-adic spaces.TauCeti.FarguesFontaine.quotientPresheafedDescandTauCeti.FarguesFontaine.quotientPreAdicDesc: the factorization throughqof a Frobenius-invariant morphism out ofπ΄.
Main results #
TauCeti.FarguesFontaine.spaYFrobenius_toHom_comp_quotientPresheafedHom:q β Ο = qas morphisms of presheafed spaces.TauCeti.FarguesFontaine.isOpenImmersion_ofRestrict_comp_quotientPresheafedHom:V β π³is an open immersion for every wandering openV.TauCeti.FarguesFontaine.isIso_stalkMap_quotientPresheafedHom: the stalk maps ofqare isomorphisms.TauCeti.FarguesFontaine.isSheafy_spaXPreAdicSpace: the pre-adic spaceπ³is sheafy.TauCeti.FarguesFontaine.isOpenImmersion_ofRestrict_comp_quotientPreAdicHom: the window chartsV β π³are open immersions of pre-adic spaces.TauCeti.FarguesFontaine.spaYFrobenius_comp_ofRestrict: Frobenius onπ΄is the restriction of Frobenius onSpa(π R, π R).TauCeti.FarguesFontaine.isColimitQuotientPresheafedCoforkandTauCeti.FarguesFontaine.isColimitQuotientPreAdicCofork:qis the coequalizer of the identity and Frobenius onπ΄, among presheafed spaces and inπ±^pre; uniqueness isquotientPresheafedHom_hom_extandquotientPreAdicHom_hom_ext.
References #
- T. Wedhorn, Adic Spaces, arXiv:1910.05934v1, Β§8.1, for the category
π±^pre. - K. S. Kedlaya, Sheaves, stacks, and shtukas, Arizona Winter School 2017 notes, Β§3.1.
- 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 a pre-adic space #
Spa(π R, π R) with its presentation-limit structure presheaf, as a pre-adic space. Every
element of the (p, [Ο])-adic ring π R is power-bounded, so π R is a plus ring of itself.
Instances For
π΄ as a pre-adic space: the restriction of the pre-adic space Spa(π R, π R) to the open
subset π΄ = D(p) β© D([Ο]). Its presheaf is πͺ_π΄ = spaYPresheaf P Ο.
Instances For
The presheaf of the pre-adic space π΄ is πͺ_π΄.
Frobenius on π΄ #
Witt Frobenius on π΄ as a morphism of presheafed spaces: v β¦ v β Ο on points, and
spaYPresheafFrobenius on sections. It underlies the morphism of pre-adic spaces
spaYFrobenius.
Instances For
Witt Frobenius as an endomorphism of the pre-adic space π΄: on points it is v β¦ v β Ο
and on sections it is spaYPresheafFrobenius (spaYFrobenius_toHom). It is compatible with the
stalk valuations, being the restriction to π΄ of the automorphism of Spa(π R, π R) induced by
Ο.
Instances For
The underlying morphism of presheafed spaces of spaYFrobenius.
Frobenius on π΄ is the restriction of Frobenius on Spa(π R, π R): the inclusion of π΄
intertwines spaYFrobenius with the automorphism of Spa(π R, π R) induced by Ο.
Frobenius on π΄ is the restriction of Frobenius on Spa(π R, π R): the inclusion of π΄
intertwines spaYFrobenius with the automorphism of Spa(π R, π R) induced by Ο.
The projection to π³ #
The Frobenius orbit space π³ with its structure presheaf πͺ_π³, as a presheafed space of
complete separated topological rings.
Instances For
The projection q : π΄ β π³ as a morphism of presheafed spaces: on sections it is the
inclusion πͺ_π³(U) β πͺ_π΄(qβ»ΒΉU) of the Frobenius-invariant sections.
Instances For
The projection to π³ is Frobenius-invariant, as a morphism of presheafed spaces: on
points because q β Ο = q, on sections because the sections of πͺ_π³ are Frobenius-invariant.
The universal property of the projection #
Morphisms out of π³: a morphism of presheafed spaces g : π΄ β W with Ο β« g = g
factors through the projection q : π΄ β π³
(quotientPresheafedHom_comp_quotientPresheafedDesc), and uniquely so
(quotientPresheafedHom_hom_ext). On points it is the map induced on Frobenius orbits; on
sections it lands in the Frobenius-invariant sections of πͺ_π΄, which are those of πͺ_π³.
Instances For
The factorization of a Frobenius-invariant morphism through q sends the orbit of a point to
the image of that point.
A Frobenius-invariant morphism out of π΄ factors through the projection q : π΄ β π³.
A Frobenius-invariant morphism out of π΄ factors through the projection q : π΄ β π³.
Morphisms out of π³ are determined by their composites with q: the projection is an
epimorphism of presheafed spaces. On points this is surjectivity of q; on sections it is
injectivity of the inclusion of the Frobenius-invariant sections.
π³ is the coequalizer of the identity and Frobenius on π΄ in presheafed spaces of complete
separated topological rings, with the projection q as coprojection.
Instances For
Charts on wandering opens #
The projection is an open immersion on every wandering open. If πͺ_π΄ is a sheaf and the
open V β π΄ meets none of its nontrivial integer Frobenius translates, then V β π³, the
inclusion followed by the projection, is an open immersion of presheafed spaces: it is an open
embedding, and over each open of its image it identifies the sections of πͺ_π³ with those of
πͺ_π΄.
The stalk maps of the projection are isomorphisms when πͺ_π΄ is a sheaf: every point of
π΄ lies in a wandering window, on which the projection is an open immersion.
The pre-adic space π³ #
The Frobenius orbit space π³ = π΄ / Ο^β€ as a pre-adic space, when πͺ_π΄ is a sheaf. Its
presheaf is πͺ_π³, the Frobenius-invariant sections of πͺ_π΄; the stalk of πͺ_π³ at q y is the
stalk of πͺ_π΄ at y through the projection, hence local, and carries the valuation of π΄ at y,
which does not depend on the point y of the orbit.
Instances For
The projection π΄ β π³ as a morphism of pre-adic spaces, when πͺ_π΄ is a sheaf.
Instances For
The presheafed space underlying the pre-adic space π³ is (π³, πͺ_π³).
The morphism of presheafed spaces underlying the pre-adic projection is q.
The pre-adic space π³ is sheafy when πͺ_π΄ is a sheaf.
The projection to π³ is Frobenius-invariant as a morphism of pre-adic spaces.
The projection to π³ is Frobenius-invariant as a morphism of pre-adic spaces.
The window charts of π³. If πͺ_π΄ is a sheaf and the open V β π΄ meets none of its
nontrivial integer Frobenius translates, as every open subset of a Frobenius window does, then
V β π³, the inclusion followed by the projection, is an open immersion of pre-adic spaces, so
that PreAdicSpace.IsOpenImmersion.isoRestrict identifies π΄ restricted to V with π³
restricted to the open image q V, as objects of π±^pre.
The universal property of π³ in π±^pre #
Morphisms out of the pre-adic space π³: when πͺ_π΄ is a sheaf, a morphism of pre-adic
spaces g : π΄ β W with Ο β« g = g factors through the projection q : π΄ β π³
(quotientPreAdicHom_comp_quotientPreAdicDesc), and uniquely so (quotientPreAdicHom_hom_ext).
Its underlying morphism of presheafed spaces is quotientPresheafedDesc; it is compatible with the
stalk valuations because q is surjective and those of π³ are descended from π΄.
Instances For
The morphism of presheafed spaces underlying quotientPreAdicDesc is
quotientPresheafedDesc.
A Frobenius-invariant morphism of pre-adic spaces out of π΄ factors through the projection
q : π΄ β π³.
A Frobenius-invariant morphism of pre-adic spaces out of π΄ factors through the projection
q : π΄ β π³.
Morphisms of pre-adic spaces out of π³ are determined by their composites with q.
π³ = π΄ / Ο^β€ is the quotient of π΄ by Frobenius in π±^pre: when πͺ_π΄ is a sheaf, the
pre-adic space π³ is the coequalizer of the identity and Frobenius on π΄, with the projection q
as coprojection.