nLab adic space

Idea

Adic spaces are the basic objects in Huber’s approach to non-archimedean analytic geometry. They are built by gluing valuation spectra? of a certain class of topological rings. Unlike Berkovich analytic spectra the points of adic spaces correspond to valuations of arbitrary rank, not only rank one. If a Berkovich space is corresponding to a separated rigid analytic space then it can be obtained as the largest Hausdorff quotient of the corresponding adic space.

The framework of adic spaces are used to build perfectoid spaces out of perfectoid rings.

Definitions

Definition

Let $(A,A^{+})$ be a Huber pair, i.e. $A$ is a Huber ring and $A^{+}\subseteq A$ is a ring of integral elements. The adic spectrum $\mathrm{Spa}(A,A^{+})$ is the set of equivalence classes of continuous valuations $\vert\cdot\vert$ on $A$ such that $\vert A^{+}\vert\leq 1$.

If $x$ is a valuation, and $g\in A$, we also suggestively write $g\mapsto\vert g(x)\vert$ for the valuation $x$ applied to $g$. The topology on $\mathrm{Spa}(A,A^{+})$ is the one generated by open sets of the form

$$\lbrace x:\vert f(x)\vert\leq\vert g(x)\vert\neq 0\rbrace$$

where $f,g\in A$.

Examples

• The final object in the category of adic spaces is $\mathrm{Spa}(\mathbb{Z},\mathbb{Z})$.

• The adic closed disc over $\mathbb{Q}_{p}$ is given by $\mathrm{Spa}(A,A^{+})$ where $A=\mathbb{Q}_{p}\langle T\rangle$ and $A^{+}=\mathbb{Z}_{p}\langle T\rangle$.

• The adic open disc over $\mathbb{Q}_{p}$ is the generic fiber of $\mathrm{Spa}(A,A)\to\mathrm{Spa}(\mathbb{Z}_{p},\mathbb{Z}_{p})$, where $A=\mathbb{Z}_{p}[[T]]$.

Related concepts

• perfectoid space

• p-adic geometry

• analytic geometry

References

• R. Huber, Étale cohomology of rigid analytic varieties and adic spaces, Aspects of Mathematics, E30. Friedr. Vieweg & Sohn, Braunschweig, 1996. x+450 pp. (MR2001c:14046)

• Sophie Morel, Adic spaces (pdf)

• Torsten Wedhorn, Adic spaces (arXiv:1910.05934)

• Brian Conrad: A brief introduction to adic spaces [pdf, pdf]