Mathematics
#olivine #adic-spacesThis is a demo for writing mathematics on Olivine.
MathJax can be enabled by setting the option
extra.olivine.mathjax = true and diagrams can be enable by setting
extra.olivine.tikzjax = true. The version of TikZJax that we use comes from
benrbray/tikzjax. The mathematics itself
is based on Ian Gleason’s thesis.
The olivine spectrum
A point in the adic spectrum is by definition a multiplicative valuation on the ring. A point in the olivine spectrum is supposed to be a pair of two valuations $(v^h, v^a)$ where
- $v^a$ is rank-$1$,
- $v^a$ is a vertical generalization of $v^h$.
Warning. A trivial valuation is also regarded as a rank-$1$ valuation.
By definition, there is a continuous map $$ \operatorname{Spo}(R, R^+) \to \operatorname{Spa}(R, R^+); \quad (v^h, v^a) \mapsto v^h. $$
Theorem. Let $(R, R^+)$ be a complete Huber pair over $\mathbb{Z}_ p$. Then there exists a canonical continuous bijection $\lvert \operatorname{Spd}(R, R^+) \rvert \to \operatorname{Spo}(R, R^+)$. This map is a homeomorphism when either (i) $R$ is a Tate ring, or (ii) $R = R^+$ is an adic ring.
Such Huber pairs $(R, R^+)$ for which this map is a homeomorphism is called an olivine Huber pair.
The reduction functor
Definition. We say that a morphism $S \to T$ of (perfect) affine schemes is a v-cover when for every valuation ring $V$ and a map $\operatorname{Spec} V \to T$ there is an extension $V \subseteq W$ of valuation rings and a commutative diagram
Fact. A map $S \to T$ of perfect affine schemes is a v-cover if and only if the induced map $S^\diamond \to T^\diamond$ is v-surjective. Here, we are using the definition that $(\operatorname{Spec} A)^\diamond = \operatorname{Spd}(A, A)$.
This allows us to define for each v-sheaf $\mathscr{F}$ (on perfectoid spaces) its reduction $$ \mathscr{F}^\mathrm{red}(A) = \Hom(\operatorname{Spd}(A, A), \mathscr{F}) $$ as a sheaf on perfect schemes with the v-topology.