The sheaf condition for rational covers of rational subsets, and on all opens #
Let C be a predicate on Tate rings satisfying TauCeti.ValuationSpectrum.LaurentGluing: it
passes to completed rational localisations, and on rings satisfying it two-piece Laurent covers of
rational subsets glue. Strong noetherianness is the basic example. Let A be a Tate ring
satisfying C, P a pair of definition whose ring of definition lies in A⁺, and A⁺ a subring
of power-bounded elements. This file proves that the presentation-limit presheaf of
X = Spa(A, A⁺), as a presheaf of sets, satisfies the sheaf condition for every cover of a
rational subset W ⊆ X by rational subsets U i ⊆ W
(isSheafFor_ofArrows_spaRationalOpens_of_iSup_eq_of_laurentGluing): sections over the U i that
agree on the pairwise overlaps glue uniquely to a section over W. The cover may be infinite.
This is Wedhorn's Lemma 8.34 in degree zero for an arbitrary rational cover. Following Wedhorn, the
statement is reduced to standard rational covers, for which it is
isSheafFor_ofArrows_inf_spaBasicOpen_of_span_eq_top_of_laurentGluing.
- Write
W = R(T/s)andB = A⟨T/s⟩. Wedhorn's Remark 8.4 (isSheafFor_ofArrows_iff_locOpensComap) transports the sheaf condition to the pullbacks of theU ialongj : Spa(B, A_U⁺) → Spa(A, A⁺), which cover all ofSpa(B, A_U⁺). The ringBis again a Tate ring satisfyingC, now complete and Hausdorff, andA_U⁺is a ring of integral elements ofB. - Over a complete Hausdorff Tate ring, Wedhorn's Lemma 7.54
(
exists_span_eq_top_forall_rationalSubset_subset_of_isTateRing) refines a cover of the whole adic spectrum by a standard rational cover(R(S/f))_{f ∈ S}, withSgenerating the unit ideal. That standard cover satisfies the sheaf condition, and so does its restriction to eachU i, which is the standard cover ofU igenerated byS. Hence so does the original cover.
Intersections of rational opens are rational, so gluing along rational covers of rational opens is
the sheaf condition on the basis of rational opens (isSheaf_rational_comp_of_isSheafFor_ofArrows).
Since the presentation-limit presheaf is the limit of its values on that basis, the presheaf of
sets underlying it is then a sheaf on all opens
(isSheaf_underlying_presentationLimitPresheaf_of_laurentGluing).
The statements concern the sheaf condition in degree zero, for the presentation-limit presheaf as a presheaf of sets; neither the topology on the sections nor higher Čech cohomology is treated here.
Main results #
TauCeti.ValuationSpectrum.isSheafFor_ofArrows_spaRationalOpens_of_iSup_eq_of_laurentGluing: a cover of a rational subset by rational subsets satisfies the sheaf condition, over a Tate ring satisfying a predicate withLaurentGluing.TauCeti.ValuationSpectrum.isSheafFor_ofArrows_spaRationalOpens_of_iSup_eq: the same over a strongly noetherian Tate ring.TauCeti.ValuationSpectrum.isSheaf_rational_comp_of_isSheafFor_ofArrows: gluing along rational covers of rational opens is the sheaf condition on the basis of rational opens.TauCeti.ValuationSpectrum.isSheaf_underlying_presentationLimitPresheaf_of_laurentGluing: over a Tate ring satisfying a predicate withLaurentGluing, the presheaf of sets underlying the presentation-limit presheaf is a sheaf.
References #
- T. Wedhorn, Adic Spaces (arXiv:1910.05934v1), Remark 8.4, Lemma 7.54, Lemma 8.34 and Theorem 8.28.
- R. Huber, A generalization of formal schemes and rigid analytic varieties, Math. Z. 217 (1994), Lemma 2.6 and Theorem 2.5.
Wedhorn's Lemma 8.34 in degree zero: rational covers of rational subsets. 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 and W a rational subset of Spa(A, A⁺). Let (U i) be a family of rational subsets of
W whose union is W. The presentation-limit presheaf, as a presheaf of sets, satisfies the sheaf
condition for this cover: sections over the U i that agree on the pairwise overlaps glue uniquely
to a section over W.
A itself need not be complete, and the family may be infinite or empty. An empty family covers
the empty open, whose sections form a singleton.
The sheaf condition on all opens #
The sheaf condition on the rational basis. Let G be a functor from complete separated
topological rings into types. If the presentation-limit presheaf followed by G satisfies the
sheaf condition for every cover of a rational open by rational opens, then its restriction to the
rational opens is a sheaf for the topology restricted from Spa(A, A⁺). Intersections of
rational opens are rational, which is what lets compatibility on the basis stand in for
compatibility on pairwise intersections.
The structure presheaf of sets is a sheaf under Laurent gluing. Let C be a predicate on
Tate rings satisfying LaurentGluing, and A a Tate ring satisfying C. If the ring of
definition of P lies in A⁺ and A⁺ consists of power-bounded elements, the presheaf of sets
underlying the presentation-limit structure presheaf is a sheaf on all opens of Spa(A, A⁺). The
ring need not be complete or Hausdorff.
The strongly noetherian case #
Wedhorn's Lemma 8.34 in degree zero for a strongly noetherian Tate ring. 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 and W a rational subset of Spa(A, A⁺). The
presentation-limit presheaf, as a presheaf of sets, satisfies the sheaf condition for every family
of rational subsets of W whose union is W. This is
isSheafFor_ofArrows_spaRationalOpens_of_iSup_eq_of_laurentGluing for strong noetherianness
(laurentGluing_isStronglyNoetherian).