The Laurent cover for the presentation-limit presheaf #
For f ∈ A the rational opens R({f, 1}/1) = {|f| ≤ 1} and R({1}/f) = {|f| ≥ 1} cover
X = Spa(A, A⁺). When A is a complete Hausdorff strongly noetherian Tate ring and A⁺ consists
of power-bounded elements, the augmented two-piece Čech sequence of the presentation-limit
presheaf is exact: sections glue uniquely, and every section on the overlap is a difference of
restrictions. This is Wedhorn's Lemma 8.33 and the Laurent-cover case of Lemma 8.34(i), stated
for presentationLimit. The degree-zero injectivity and gluing statements are transported from
the corresponding statements for A and the completed rational localisations of the pieces by
lemmas that take those ring-level statements as hypotheses, so they also apply when A is uniform
(Buzzard--Verberkmoes, Corollary 4).
Main definitions #
TauCeti.ValuationSpectrum.laurentCoverOpen: the two piecesR({f, 1}/1)andR({1}/f)of the Laurent cover, indexed byBool. Both are rational opens (TauCeti.ValuationSpectrum.laurentCoverOpen_mem_spaRationalOpens).
Main results #
TauCeti.ValuationSpectrum.injective_presentationLimitMap_laurentCoverOpen: restriction fromXto the two pieces is injective.TauCeti.ValuationSpectrum.exists_presentationLimitMap_eq_of_laurentCoverOpen: sections over the two pieces that agree on their overlap come from a section overX.TauCeti.ValuationSpectrum.surjective_presentationLimitMap_sub_laurentCoverOpen: the difference of restrictions from the two pieces onto their overlap is surjective.injective_presentationLimitMap_laurentCoverOpen_of_injective_toCompletionLocandexists_presentationLimitMap_eq_of_laurentCoverOpen_of_exact_toCompletionLoc: the degree-zero statements for any complete HausdorffA, assuming their ring-level counterparts.
References #
- T. Wedhorn, Adic Spaces (arXiv:1910.05934v1), Lemma 8.33 and Lemma 8.34(i).
- K. Buzzard, A. Verberkmoes, Stably uniform affinoids are sheafy, J. reine angew. Math. 740 (2018), 25--39, Corollary 4.
Elements under equality transports #
Restriction between rational opens #
Restriction from the whole spectrum #
The Laurent cover of f: the rational opens R({f, 1}/1) = {|f| ≤ 1} (at true) and
R({1}/f) = {|f| ≥ 1} (at false) of Spa(A, A⁺), indexed by Bool so that they form one
family. They cover the adic spectrum (spa_subset_iUnion_laurentCover). Since this is an
abbrev for spaBasicOpen, the spaBasicOpen API (such as mem_spaBasicOpen) applies to it
directly.
Equations
- TauCeti.ValuationSpectrum.laurentCoverOpen Aplus f b = TauCeti.ValuationSpectrum.spaBasicOpen Aplus (bif b then {f, 1} else {1}) (bif b then 1 else f)
Instances For
Each piece of the Laurent cover is a rational open: its numerators contain 1, so they span
the unit ideal, which is open.
Transport from the completed rational localisations #
Laurent injectivity transported to the presentation-limit presheaf. Let A be a complete
Hausdorff Huber ring, A⁺ a subring of power-bounded elements and f ∈ A. If the map
a ↦ (a, a) from A into the completed rational localisations of R({f, 1}/1) and R({1}/f)
is injective, then a section of presentationLimit over Spa(A, A⁺) is determined by its
restrictions to the two pieces laurentCoverOpen Aplus f b of the Laurent cover. The hypothesis
holds for strongly noetherian Tate rings (laurentCover_injective) and for uniform Tate rings
(isClosedEmbedding_laurentCover_of_isUniform).
Laurent gluing transported to the presentation-limit presheaf. Let A be a complete
Hausdorff Huber ring, A⁺ a subring of power-bounded elements and f ∈ A. If the ring-level
sequence A → A⟨U₁⟩ × A⟨U₂⟩ → A⟨U₁ ∩ U₂⟩ of the Laurent cover of f is exact in the middle,
then sections x b of presentationLimit over the two pieces laurentCoverOpen Aplus f b that
agree on their overlap are the restrictions of one section over Spa(A, A⁺). The hypothesis
holds for strongly noetherian Tate rings (laurentCover_exact) and for uniform Tate rings
(laurentCover_exact_of_isUniform).
Strongly noetherian Tate rings #
Wedhorn's Lemma 8.33, injectivity, for the presentation-limit presheaf. Let A be a
complete Hausdorff strongly noetherian Tate ring, A⁺ a subring of power-bounded elements and
f ∈ A. A section of presentationLimit over Spa(A, A⁺) is determined by its restrictions to
the two pieces laurentCoverOpen Aplus f b of the Laurent cover. The corresponding statement for
the map a ↦ (a, a) from A into the completed rational localisations of the two pieces is
laurentCover_injective. Sections over the pieces that agree on their overlap do come from a
section over Spa(A, A⁺): exists_presentationLimitMap_eq_of_laurentCoverOpen.
Wedhorn's Lemma 8.33, exactness in the middle, for the presentation-limit presheaf. Let A
be a complete Hausdorff strongly noetherian Tate ring, A⁺ a subring of power-bounded elements and
f ∈ A. Sections x b of presentationLimit over the two pieces laurentCoverOpen Aplus f b of
the Laurent cover that agree on their overlap are the restrictions of one section over
Spa(A, A⁺), which is unique by injective_presentationLimitMap_laurentCoverOpen. The
corresponding statement for A and the completed rational localisations of the two pieces and of
their overlap is laurentCover_exact.
Wedhorn's Lemma 8.33, degree-one surjectivity, for the presentation-limit presheaf. Every section on the intersection of the two Laurent pieces is a difference of restrictions of sections on the pieces. Together with the degree-zero results above, this gives exactness of the augmented two-piece Čech complex.