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  "body": "\n## Definition\n\nLet $K$ be a field complete for a nontrivial [[non-archimedean absolute value|1H5F1BP6XS]] $\\abs{\\cdot}$.\nThe *Tate algebra* in $n$ variables is\n$$T_n = K\\langle x_1, \\dots, x_n \\rangle = \\set{\\textstyle\\sum_{\\nu \\in \\N^n} a_\\nu x^\\nu : a_\\nu \\in K,\\ \\abs{a_\\nu} \\to 0 \\text{ as } \\abs{\\nu} \\to \\infty},$$\nthe algebra of power series that converge on the closed unit polydisc.\nAn *affinoid algebra* is a $K$-algebra $A$ isomorphic to a quotient $T_n / I$ for some $n$ and some ideal $I$.\nThe *affinoid space* $\\operatorname{Sp} A$ is the set of maximal ideals of $A$ with a Grothendieck topology and a structure [[sheaf|TB6TM6R4MD]] $\\mathcal{O}$.\nIn this topology the admissible opens are the finite unions of affinoid subdomains.\nThe admissible coverings are the finite coverings by such sets.\nA *rigid analytic space* over $K$ is a set $X$ with a Grothendieck topology and a sheaf of $K$-algebras, as a [[ringed space|KNDXDE7K8M]].\nIt must have an admissible covering by open subsets each isomorphic, with these structures, to an affinoid space.\n\n## Remarks\n\nThat $\\mathcal{O}$ is a sheaf for finite affinoid coverings is Tate's acyclicity theorem.\nThe general admissible opens and coverings are defined from these by a standard extension of the topology.\n\nExample: $\\operatorname{Sp} T_1$ is the closed unit disc.\nIts points are the maximal ideals of $T_1$, which correspond to elements $x$ of an algebraic closure of $K$ with $\\abs{x} \\le 1$, up to conjugation over $K$.\n\nNon-example of an admissible covering: for $K = \\Q_p$, the closed disc $\\operatorname{Sp} T_1$ is the disjoint union of the open disc $\\set{\\abs{x} \u003c 1}$ and the circle $\\set{\\abs{x} = 1}$.\nThis covering is not admissible.\nEvery affinoid subdomain of the open disc lies in a disc $\\set{\\abs{x} \\le r}$ with $r \u003c 1$.\nSo no finite family of affinoid subdomains of the two pieces covers the points with $r \u003c \\abs{x} \u003c 1$.\nThis is what keeps the closed disc connected in rigid geometry.\n\nThe analytification of a variety of finite type over $K$ is a rigid analytic space, and a rigid GAGA theorem compares coherent sheaves on both sides for proper varieties.\n",
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