Adic space

Definition

A Huber ring is a topological ring $A$ with an open subring $A_0$ and a finitely generated ideal $I$ of $A_0$. The topology of $A_0$ must be the $I$-adic topology. An element $a$ of $A$ is power-bounded if the set $\set{a^n : n \ge 1}$ is bounded, and $A^{\circ}$ denotes the subring of power-bounded elements. A Huber pair $(A, A^+)$ is a Huber ring $A$ with an open subring $A^+ \subseteq A^{\circ}$ that is integrally closed in $A$.

A valuation on $A$ is a map $v \colon A \to \Gamma \cup \set{0}$, where $\Gamma$ is a totally ordered abelian group written multiplicatively, with $v(0) = 0$, $v(1) = 1$, $v(ab) = v(a) v(b)$ and $v(a + b) \le \max(v(a), v(b))$. It is continuous if $\set{a \in A : v(a) < \gamma}$ is open for every $\gamma$ in the value group of $v$. The adic spectrum $\operatorname{Spa}(A, A^+)$ is the set of equivalence classes of continuous valuations $v$ on $A$ with $v(f) \le 1$ for all $f \in A^+$. Its topology is generated by the rational subsets $$U\!\left(\tfrac{f_1, \dots, f_n}{g}\right) = \set{v : v(f_i) \le v(g) \ne 0 \text{ for } i = 1, \dots, n},$$ where $f_1, \dots, f_n$ generate an open ideal of $A$. It carries a structure presheaf $\mathcal{O}$ of complete topological rings, built from completions of the localizations $A[1/g]$.

An adic space is a triple $(X, \mathcal{O}_X, (v_x)_{x \in X})$ such that:

1. $X$ is a topological space and $\mathcal{O}_X$ is a sheaf of complete topological rings on $X$;
2. for each $x \in X$, $v_x$ is an equivalence class of valuations on the stalk $\mathcal{O}_{X,x}$;
3. every point has an open neighbourhood isomorphic, with these structures, to $\operatorname{Spa}(A, A^+)$ for a Huber pair whose structure presheaf is a sheaf.