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  "body": "\n## Definition\n\nA *Huber ring* is a [[topological ring|TSF8RZDD5B]] $A$ with an open subring $A_0$ and a finitely generated ideal $I$ of $A_0$.\nThe topology of $A_0$ must be the [[$I$-adic topology|2VF2TSJ3PY]].\nAn 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.\nA *Huber pair* $(A, A^+)$ is a Huber ring $A$ with an open subring $A^+ \\subseteq A^{\\circ}$ that is [[integrally closed|G0V86SJBQW]] in $A$.\n\nA *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 $v(a + b) \\le \\max(v(a), v(b))$.\nIt is *continuous* if $\\set{a \\in A : v(a) \u003c \\gamma}$ is open for every $\\gamma$ in the value group of $v$.\nThe *adic spectrum* $\\operatorname{Spa}(A, A^+)$ is the set of equivalence classes of continuous [[valuations|B85T5VKR3M]] $v$ on $A$ with $v(f) \\le 1$ for all $f \\in A^+$.\nIts topology is generated by the rational subsets\n$$U\\!\\left(\\tfrac{f_1, \\dots, f_n}{g}\\right) = \\set{v : v(f_i) \\le v(g) \\ne 0 \\text{ for } i = 1, \\dots, n},$$\nwhere $f_1, \\dots, f_n$ generate an open ideal of $A$.\nIt carries a structure presheaf $\\mathcal{O}$ of complete topological rings, built from completions of the localizations $A[1/g]$.\n\nAn *adic space* is a triple $(X, \\mathcal{O}_X, (v_x)_{x \\in X})$ such that:\n\n1. $X$ is a topological space and $\\mathcal{O}_X$ is a [[sheaf|TB6TM6R4MD]] of complete topological rings on $X$;\n2. for each $x \\in X$, $v_x$ is an equivalence class of valuations on the stalk $\\mathcal{O}_{X,x}$;\n3. 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.\n\n## Remarks\n\nExample: $\\operatorname{Spa}(\\Q_p, \\Z_p)$ is a single point.\nThe $p$-adic valuation is continuous and bounded by $1$ on $\\Z_p$.\nThe trivial valuation, with $v(a) = 1$ for $a \\ne 0$, is not continuous, since $\\set{a : v(a) \u003c 1} = \\set{0}$ is not open in $\\Q_p$.\nOther valuations of $\\Q_p$ bounded on $\\Z_p$ are equivalent to one of these two.\n\nNon-example of a Huber ring: $\\R$ with its usual topology.\nAn open subring contains an interval $(-\\varepsilon, \\varepsilon)$, which generates $\\R$ as an additive group, so the only open subring is $\\R$.\nIts topology is neither discrete nor indiscrete, so it is not the $I$-adic topology for any ideal $I$ of $\\R$.\n\nThe structure presheaf of $\\operatorname{Spa}(A, A^+)$ need not be a sheaf in general; this is why clause 3 asks for it.\nHuber proved that it is a sheaf when, for example, $A$ is a strongly Noetherian Tate ring, which covers the affinoid algebras of rigid geometry.\n",
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