Analytic points and the analytic locus of a pre-adic space #
A point x of a pre-adic space X is analytic if some open neighbourhood U of x has a
pseudouniformiser, a topologically nilpotent unit, in 𝒪_X(U). This is the intrinsic form of
Huber's analytic points, which on an affinoid Spa(A, A⁺) are the points whose support is not
open. The analytic points form an open subset by definition, and they are compatible with
restriction to open subspaces.
On the adic spectrum Spa(A, A⁺) with its presentation-limit structure presheaf the two notions
agree. A section of a neighbourhood of x which is a pseudouniformiser restricts to one of a
completed rational localisation A⟨T/s⟩ with x ∈ R(T/s), and A → A⟨T/s⟩ is adic, so x is
analytic in Huber's sense
(TauCeti.ValuationSpectrum.rationalSubset_subset_spaAnalytic_of_isTateRing).
Conversely an analytic point lies in a chart R(G/g) of the standard rational cover of the
analytic locus, g running over generators of an ideal of definition, and the image of g is a
pseudouniformiser of the sections A⟨G/g⟩ over that chart.
Every morphism of pre-adic spaces pulls analytic points back to analytic points, since its maps on sections are continuous and so preserve pseudouniformisers; the converse inclusion is what a morphism must satisfy to restrict to analytic loci.
The analytic locus X_a is the open subspace of X on its analytic points (Wedhorn,
Remark 8.37), with its inclusion analyticLocus.ι : X_a ⟶ X, an open immersion. Every point of
X_a is analytic, ι is an isomorphism exactly when every point of X is analytic, and a
morphism into X factors through X_a exactly when its image consists of analytic points. A
morphism sending analytic points to analytic points restricts to the analytic loci, functorially;
by Wedhorn, Proposition 8.39, adic morphisms are of this kind. The analytic locus of an adic space
is an open adic subspace, TauCeti.AdicSpace.analyticLocus. It is the setting in which
constructions on Tate affinoids, such as completed tensor products, apply to non-Tate spaces like
Spa(ℤ_p⟦T⟧).
Main definitions #
TauCeti.PreAdicSpace.analyticPoints: the analytic points of a pre-adic space.TauCeti.PreAdicSpace.analyticLocus,TauCeti.PreAdicSpace.analyticLocus.ι: the analytic locusX_aand its inclusion intoX.TauCeti.PreAdicSpace.analyticLocus.lift: the factorisation throughX_aof a morphism whose image consists of analytic points.TauCeti.PreAdicSpace.analyticLocusMap: the restrictionY_a ⟶ X_aof a morphismY ⟶ Xsending analytic points to analytic points.TauCeti.AdicSpace.analyticLocus,TauCeti.AdicSpace.analyticLocus.ι: the analytic locus of an adic space, as an adic space, and its inclusion.
Main results #
TauCeti.PreAdicSpace.isOpen_analyticPoints: the analytic points form an open subset.TauCeti.PreAdicSpace.analyticPoints_restrict: the analytic points of the restriction ofXalong an open embeddingfare the preimage underfof those ofX.TauCeti.ValuationSpectrum.analyticPoints_presentationLimitPreAdicSpace: the analytic points ofSpa(A, A⁺)are the points of its analytic locusSpa(A, A⁺)ᵃ.TauCeti.PreAdicSpace.preimage_analyticPoints_subset: a point over an analytic point is analytic.TauCeti.PreAdicSpace.preimage_analyticPoints_of_isOpenImmersion: open immersions identify analytic points.TauCeti.PreAdicSpace.mapsTo_analyticPoints_comp_iff_of_isOpenImmersion: composing with an open immersion neither creates nor destroys the preservation of analytic points.TauCeti.ValuationSpectrum.analyticPoints_presentationLimitPreAdicSpace_of_isTateRing: every point of the adic spectrum of a Tate ring is analytic.TauCeti.ValuationSpectrum.exists_mem_propOpens_of_mem_analyticPointsandTauCeti.PreAdicSpace.exists_mem_propOpens_of_mem_analyticPoints: covering analytic points by standard analytic rational opens inpropOpens Q, onSpa(A, A⁺)and across an affinoid chart.TauCeti.ValuationSpectrum.mapsTo_analyticPoints_presentationLimitPreAdicSpaceComap_iff: the morphism of adic spectra induced by a morphism of Huber pairs with complete target sends analytic points to analytic points exactly when the ring homomorphism is adic (Wedhorn, Lemma 7.46).TauCeti.PreAdicSpace.analyticPoints_analyticLocus: every point ofX_ais analytic.TauCeti.PreAdicSpace.isIso_analyticLocus_ι_iff:X_a ⟶ Xis an isomorphism exactly when every point ofXis analytic.TauCeti.PreAdicSpace.analyticLocus.exists_comp_ι_eq_iff: a morphism factors throughX_aexactly when its image consists of analytic points.TauCeti.PreAdicSpace.analyticLocusMap_id,TauCeti.PreAdicSpace.analyticLocusMap_comp: restriction to analytic loci is functorial.
References #
- T. Wedhorn, Adic Spaces (arXiv:1910.05934v1), Definition 7.39, Lemma 7.46, Proposition 7.49, Proposition and Definition 8.36, Remark 8.37 and Proposition 8.39.
- [R. Huber, A generalization of formal schemes and rigid analytic varieties], §3.
Analytic points of a pre-adic space (Wedhorn, Proposition and Definition 8.36): the points
x having an open neighbourhood U such that 𝒪_X(U) contains a pseudouniformiser, that is, a
topologically nilpotent unit. On Spa(A, A⁺) these are the points whose support is not open
(TauCeti.ValuationSpectrum.analyticPoints_presentationLimitPreAdicSpace).
Instances For
Unfolding lemma for TauCeti.PreAdicSpace.analyticPoints.
The analytic points form an open subset of a pre-adic space.
Analytic points of an open subspace (Wedhorn, Remark 8.37(2)): the analytic points of the
restriction of X along an open embedding f are the points that f sends to analytic points of
X.
Analytic points along morphisms #
A point over an analytic point is analytic: a morphism of pre-adic spaces pulls the analytic points of its target back into those of its source, because its maps on sections are continuous ring homomorphisms, which send pseudouniformisers to pseudouniformisers.
A morphism of pre-adic spaces sends analytic points to analytic points exactly when the analytic points of its source are the preimage of those of its target.
Open immersions send analytic points to analytic points: an open immersion f : Y ⟶ X
identifies Y with the open subspace of X on its image, whose analytic points are the points
that are analytic in X.
Open immersions identify analytic points: a point of the source of an open immersion is analytic exactly when its image is.
A morphism followed by an open immersion sends analytic points to analytic points exactly when the morphism itself does.
The analytic locus #
The analytic locus X_a of a pre-adic space X (Wedhorn, Remark 8.37): the open
subspace of X on its analytic points.
Instances For
The inclusion X_a ⟶ X of the analytic locus, an open immersion.
Instances For
The image of the analytic locus is the set of analytic points.
The analytic locus is analytic: every point of X_a is analytic.
The inclusion of the analytic locus is an isomorphism exactly when every point of X is
analytic.
The universal property of the analytic locus: a morphism whose image consists of analytic
points factors through X_a, uniquely (analyticLocus.lift_uniq).
Instances For
The factorisation through the analytic locus composes with the inclusion to the original morphism.
The factorisation through the analytic locus composes with the inclusion to the original morphism.
A factorisation through the analytic locus is the lift.
A morphism factors through the analytic locus exactly when its image consists of analytic points.
Restriction of a morphism to analytic loci: a morphism sending analytic points to
analytic points, for instance an adic morphism (Wedhorn, Proposition 8.39), restricts to a
morphism Y_a ⟶ X_a, the unique one compatible with the inclusions (analyticLocusMap_ι).
Instances For
The restriction to analytic loci commutes with the inclusions.
The restriction to analytic loci commutes with the inclusions.
The restriction of the identity to the analytic locus is the identity.
The restriction of a composite to analytic loci is the composite of the restrictions.
The analytic locus of an adic space is an adic space, an open adic subspace.
The analytic points of Spa(A, A⁺) are its analytic locus (Wedhorn, Proposition and
Definition 8.36): a point of the adic spectrum has a neighbourhood whose sections contain a
pseudouniformiser exactly when its support is not open. The ring A need not be complete.
Every point of the adic spectrum of a Tate ring is analytic: over a Tate ring the analytic locus is the whole adic spectrum (Wedhorn, Remark 7.40(3)).
Covering analytic points of Spa(A, A⁺) by standard analytic rational opens. Let Q be a
property of objects of 𝒱^pre that holds on every rational open R(T/s) of Spa(A, A⁺) with T
spanning an open ideal and s topologically nilpotent. Then every analytic point of Spa(A, A⁺)
lies in an open of propOpens Q, namely one of the charts R(G/g) where G is a finite generating
set of P.idealOfDefinition.
Transporting a standard analytic rational open from an affinoid chart X|_V ≅ Spa(A, A⁺) back
to X: if x is an analytic point of X lying in V and Q holds on rational opens of
Spa(A, A⁺) with topologically nilpotent denominator, then x lies in an open of X.propOpens Q.
Wedhorn Lemma 7.46(1), on adic spectra: if φ is adic, the morphism Spa(φ) it induces
sends analytic points to analytic points.
Wedhorn Lemma 7.46, on adic spectra: for a continuous φ : A → B with B complete, the
morphism Spa(φ) it induces sends analytic points to analytic points exactly when φ is adic.
The analytic locus X_a of an adic space X (Wedhorn, Remark 8.37): the open adic
subspace of X on its analytic points.
Instances For
The pre-adic space underlying the analytic locus of an adic space is the analytic locus of the underlying pre-adic space.
The inclusion X_a ⟶ X of the analytic locus of an adic space, transported along
analyticLocus_obj from the inclusion of the analytic locus of the underlying pre-adic space.
Instances For
The underlying morphism of pre-adic spaces of the inclusion of the analytic locus.
The image of the analytic-locus inclusion is the set of analytic points.
A map whose image consists of analytic points factors through the analytic locus.
Instances For
Composing the factorization with the open inclusion recovers the original map.
Composing the factorization with the open inclusion recovers the original map.
The factorization through the analytic locus is unique.
A morphism factors through the analytic locus exactly when its image is analytic.