The sheaf condition for standard rational covers #
Let A be a Tate ring, A⁺ a subring of power-bounded elements, W a rational subset of
X = Spa(A, A⁺) and T ⊆ A a nonempty finite set. The standard rational cover of W generated
by T consists of the pieces
W ∩ R(T/t) = W ∩ {v | v(t') ≤ v(t) ≠ 0 for t' ∈ T}, t ∈ T.
When every t ∈ T is a unit, the Laurent cover generated by the ratios t t'⁻¹ refines this
cover (exists_rationalSubset_superset_laurentPiece_mul_inverse, Wedhorn's Lemma 8.34(iii)). So a
presheaf of sets satisfying the sheaf condition for Laurent covers of rational subsets satisfies it
for the standard cover too (isSheafFor_ofArrows_inf_spaBasicOpen_of_isUnit_of_isSheafFor).
In Wedhorn's proof of Lemma 8.34, part (iii) is applied to standard covers of a rational subset W
generated by elements that are units of the coordinate ring of W rather than of A: they vanish
nowhere on W, as the generators of sign ≥ do on a piece of the Laurent cover of part (ii)
(not_vle_zero_of_mem_laurentPiece_inv_mul). Writing W = R(U/s) and B = A⟨U/s⟩, the elements
of T become units of B (isUnit_toCompletionLoc_iff_forall_notMem_supp), the pieces pull back
to the standard cover of Spa(B, A_U⁺) generated by their images, and the sheaf condition is
transported back along Wedhorn's Remark 8.4 (isSheafFor_ofArrows_iff_locOpensComap). This needs
Laurent gluing over B rather than over A, so it is proved for the presentation-limit presheaf
of a Tate ring satisfying a predicate C with LaurentGluing C, which passes to B
(isSheafFor_ofArrows_inf_spaBasicOpen_of_laurentGluing).
Part (ii) of Lemma 8.34 removes the restriction on T: it suffices that T generate the unit
ideal of A (isSheafFor_ofArrows_inf_spaBasicOpen_of_span_eq_top_of_laurentGluing). Choose a
unit ϖ strictly dominated at every point by an element of T
(exists_unit_forall_mem_spa_exists_vlt). On a rational subset of a piece of the Laurent cover
generated by the ϖ⁻¹ t, the standard cover is generated by the elements of sign ≥, which vanish
nowhere there, so it satisfies the sheaf condition
(isSheafFor_ofArrows_inf_spaBasicOpen_of_subset_laurentPiece_of_isSheafFor). Since the Laurent
cover itself satisfies the sheaf condition, sections over the pieces of the standard cover glue
first on each Laurent piece and then over W (TauCeti.TopologicalSpace.Opens.isSheafFor_trans).
Strong noetherianness satisfies LaurentGluing (laurentGluing_isStronglyNoetherian), which gives
the statements for a strongly noetherian Tate ring.
All these statements concern the sheaf condition in degree zero, for the presentation-limit presheaf as a presheaf of sets; higher Čech cohomology is not treated here.
Main results #
TauCeti.ValuationSpectrum.isSheafFor_ofArrows_inf_spaBasicOpen_of_isUnit_of_isSheafFor: a presheaf of sets satisfying the sheaf condition for Laurent covers of rational subsets satisfies it for the standard cover of a rational subset generated by units ofA.TauCeti.ValuationSpectrum.isSheafFor_ofArrows_inf_spaBasicOpen_of_laurentGluing: the standard cover of a rational subsetWgenerated by elements vanishing nowhere onWsatisfies the sheaf condition.TauCeti.ValuationSpectrum.isSheafFor_ofArrows_inf_spaBasicOpen_of_span_eq_top_of_laurentGluing: the standard cover of a rational subset generated by a finite set generating the unit ideal satisfies the sheaf condition.TauCeti.ValuationSpectrum.isSheafFor_ofArrows_inf_spaBasicOpen_of_isUnit,TauCeti.ValuationSpectrum.isSheafFor_ofArrows_inf_spaBasicOpen,TauCeti.ValuationSpectrum.isSheafFor_ofArrows_inf_spaBasicOpen_of_subset_laurentPieceandTauCeti.ValuationSpectrum.isSheafFor_ofArrows_inf_spaBasicOpen_of_span_eq_top: the corresponding statements over a strongly noetherian Tate ring.
References #
- T. Wedhorn, Adic Spaces (arXiv:1910.05934v1), Remark 8.4 and Lemma 8.34(ii), (iii).
- S. Bosch, U. Güntzer, R. Remmert, Non-Archimedean Analysis, §8.2.2, Lemmas 3 and 4, the rigid-analytic originals of the two reductions.
Wedhorn's Lemma 8.34(iii) in degree zero for a presheaf of sets. Let W be a rational
subset of Spa(A, A⁺) and T a nonempty finite set of units of A, and let F be a presheaf of
sets that satisfies the sheaf condition for the Laurent cover of every rational subset generated by
a finite set. Then F satisfies the sheaf condition for the standard rational cover
(W ∩ R(T/t))_{t ∈ T} of W, which the Laurent cover generated by the ratios t t'⁻¹ refines.
Standard covers generated by elements vanishing nowhere. Let C be a predicate on Tate
rings satisfying LaurentGluing, and A a Tate ring satisfying C. Let P be a pair of
definition whose ring of definition lies in A⁺, A⁺ a subring of power-bounded elements, W a
rational subset of Spa(A, A⁺) and T a nonempty finite subset of A that vanishes at no point
of W. The presentation-limit presheaf, as a presheaf of sets, satisfies the sheaf condition for
the standard rational cover (W ∩ R(T/t))_{t ∈ T} of W.
The elements of T are units of the coordinate ring B of W, though not necessarily of A.
The cover pulls back to the standard cover of Spa(B, A_U⁺) generated by these units, to which
isSheafFor_ofArrows_inf_spaBasicOpen_of_isUnit_of_isSheafFor applies since B again satisfies
C.
Standard covers generated by the unit ideal #
The sieve-theoretic reduction of a standard cover on a Laurent piece to the subcover whose generators have positive sign. This is independent of the target functor: it only requires the sheaf condition for standard covers whose generators vanish nowhere.
The sieve-theoretic reduction from standard covers generated by the unit ideal to standard covers on Laurent pieces. This is independent of the target functor: it only requires the sheaf conditions for Laurent covers and for standard covers whose generators vanish nowhere.
Wedhorn's Lemma 8.34 in degree zero for standard covers. Let C be a predicate on Tate
rings satisfying LaurentGluing, and A a Tate ring satisfying C. Let P be a pair of
definition whose ring of definition lies in A⁺, A⁺ a subring of power-bounded elements, W a
rational subset of Spa(A, A⁺) and T a nonempty finite subset of A generating the unit ideal.
The presentation-limit presheaf, as a presheaf of sets, satisfies the sheaf condition for the
standard rational cover (W ∩ R(T/t))_{t ∈ T} of W: sections over the pieces that agree on the
overlaps glue uniquely to a section over W.
Following Wedhorn, choose a unit ϖ strictly dominated at every point by an element of T
(exists_unit_forall_mem_spa_exists_vlt). The Laurent cover of W generated by the ϖ⁻¹ t
satisfies the sheaf condition (isSheafFor_laurentSieve_of_isSheafFor), and on each of its
pieces, and on their overlaps, the restricted standard cover is generated by elements vanishing
nowhere (isSheafFor_ofArrows_inf_spaBasicOpen_of_laurentGluing).
The strongly noetherian case #
Wedhorn's Lemma 8.34(iii) in degree zero: standard covers generated by units. Let A be a
strongly noetherian Tate ring, A⁺ a subring of power-bounded elements, W a rational subset of
Spa(A, A⁺) and T a nonempty finite set of units of A. The presentation-limit presheaf, as a
presheaf of sets, satisfies the sheaf condition for the standard rational cover
(W ∩ R(T/t))_{t ∈ T} of W: sections over the pieces that agree on the overlaps glue uniquely
to a section over W. A itself need not be complete.
Standard covers generated by elements vanishing nowhere. Let A be a strongly noetherian
Tate ring, P a pair of definition whose ring of definition lies in A⁺, A⁺ a subring of
power-bounded elements, W a rational subset of Spa(A, A⁺) and T a nonempty finite subset of
A that vanishes at no point of W. The presentation-limit presheaf, as a presheaf of sets,
satisfies the sheaf condition for the standard rational cover (W ∩ R(T/t))_{t ∈ T} of W.
The elements of T are units of the coordinate ring of W, though not necessarily of A; when
they are units of A, see isSheafFor_ofArrows_inf_spaBasicOpen_of_isUnit, which needs no
hypothesis on P.
Wedhorn's Lemma 8.34(ii) in degree zero: a standard cover restricted to a Laurent piece.
Let A be a strongly noetherian Tate ring, P a pair of definition whose ring of definition lies
in A⁺, A⁺ a subring of power-bounded elements, T ⊆ A a nonempty finite set and ϖ a unit of
A strictly dominated at every point of Spa(A, A⁺) by some element of T. Let V be a rational
subset contained in the piece with sign set J of the Laurent cover generated by the ϖ⁻¹ t,
t ∈ T. The presentation-limit presheaf, as a presheaf of sets, satisfies the sheaf condition for
the restriction (V ∩ R(T/t))_{t ∈ T} of the standard cover generated by T to V.
On V, the pieces with t ∉ J are empty (rationalSubset_inter_laurentPiece_inv_mul_eq_empty),
and the others form the standard cover of V generated by the elements of J
(mem_rationalSubset_iff_of_mem_laurentPiece_inv_mul), which vanish nowhere on V
(not_vle_zero_of_mem_laurentPiece_inv_mul).
Wedhorn's Lemma 8.34 in degree zero for standard covers. Let A be a strongly noetherian
Tate ring, P a pair of definition whose ring of definition lies in A⁺, A⁺ a subring of
power-bounded elements, W a rational subset of Spa(A, A⁺) and T a nonempty finite subset of
A generating the unit ideal. The presentation-limit presheaf, as a presheaf of sets, satisfies
the sheaf condition for the standard rational cover (W ∩ R(T/t))_{t ∈ T} of W: sections over
the pieces that agree on the overlaps glue uniquely to a section over W.
This is isSheafFor_ofArrows_inf_spaBasicOpen_of_span_eq_top_of_laurentGluing for strong
noetherianness (laurentGluing_isStronglyNoetherian).