Adic space - Knowlpedia

Definition

A Huber pair $(A,A^+)$ consists of a topological ring $A$ admitting an open subring with a finitely generated ideal of definition, together with an open integrally closed subring $A^+\subseteq A^\circ$ of power-bounded elements. Its adic spectrum

\operatorname{Spa}(A,A^+)

is the set of equivalence classes of continuous valuations on $A$ that are bounded by $1$ on $A^+$ , equipped with the topology generated by rational subsets and with Huber's structure presheaf.

An adic space is a locally ringed space locally isomorphic to affinoid adic spectra for which the structure presheaves are sheaves.

Why two rings appear

The ring $A$ records analytic functions, while $A^+$ records an integral structure. Different choices of $A^+$ can give the same broad analytic geometry but different integral data.

Relation to other geometries

Adic spaces include the spaces associated with Tate rigid-analytic varieties and generic fibers of formal schemes. Unlike a scheme spectrum, their points are valuations of many ranks, so specialization and integral information are visible simultaneously.

Perfectoid spaces are special adic spaces whose affinoid rings satisfy a strong Frobenius condition.

References

Roland Huber, “A generalization of formal schemes and rigid analytic varieties,” Mathematische Zeitschrift 217 (1994), 513–551. EuDML.
Katharina Hübner, “Adic spaces,” 2024. arXiv.