Sections on the empty open of an adic spectrum #
The presentation-limit presheaf has exactly one section on the empty open when the plus subring
consists of power-bounded elements. The rational presentation R({1}/0) identifies this value
with the completion of the zero ring. This supplies the empty-cover case of the sheaf condition,
without completeness, Tate, or noetherian hypotheses on the original ring.
References #
- T. Wedhorn, Adic Spaces (arXiv:1910.05934v1), §8.1.
theorem
TauCeti.ValuationSpectrum.subsingleton_presentationLimit_bot
{A : Type v}
[CommRing A]
[TopologicalSpace A]
[IsTopologicalRing A]
(P : Huber.PairOfDefinition A)
{Aplus : Subring A}
(hAplus : ∀ ⦃a : A⦄, a ∈ Aplus → Huber.IsPowerBounded a)
:
Subsingleton (presentationLimit Aplus ⊥).obj.α
There is exactly one presentation-limit section on the empty open if A⁺ consists of
power-bounded elements.