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  "body": "\n## Definition\n\nLet $p$ be a prime.\nLet $K$ be a field, complete for a [[non-archimedean absolute value|1H5F1BP6XS]] $\\abs{\\cdot}$, and let $\\mathcal{O}_K = \\set{x : \\abs{x} \\le 1}$.\nThe field $K$ is a *perfectoid field* if:\n\n1. The value group $\\abs{K^\\times}$ is not discrete.\n2. The residue field of $\\mathcal{O}_K$ has characteristic $p$.\n3. The [[Frobenius map|0T3WVTD675]] $x \\mapsto x^p$ on $\\mathcal{O}_K / p\\mathcal{O}_K$ is surjective.\n\n\nLet $K$ be a perfectoid field.\nA *perfectoid $K$-algebra* is a Banach $K$-algebra $R$ such that the subring $R^\\circ$ of power-bounded elements is bounded and the Frobenius map $R^\\circ / p R^\\circ \\to R^\\circ / p R^\\circ$ is surjective.\nA *perfectoid affinoid $K$-algebra* is a pair $(R, R^+)$ with $R$ a perfectoid $K$-algebra and $R^+ \\subseteq R^\\circ$ an open and integrally closed subring.\nA *perfectoid space* over $K$ is an adic space over $K$ that is covered by open subspaces isomorphic to $\\operatorname{Spa}(R, R^+)$ with $(R, R^+)$ perfectoid affinoid.\n\n## Remarks\n\nExample: the [[completion|W168DS966V]] $K$ of $\\Q_p(p^{1/p^\\infty}) = \\bigcup_n \\Q_p(p^{1/p^n})$ is a perfectoid field.\nIts value group is $p^{\\Z[1/p]}$, which is not discrete, and every element of $\\mathcal{O}_K / p$ is a polynomial in the $p^{1/p^n}$, each of which is the $p$-th power of $p^{1/p^{n+1}}$.\nThe completed algebraic closure $\\C_p$ of the [[$p$-adic numbers|BT6T702C28]] is also perfectoid.\n\nNon-example: $\\Q_p$ is not perfectoid.\nThe Frobenius map on $\\Z_p / p = \\F_p$ is surjective, but the value group $p^{\\Z}$ is discrete, so the first condition fails.\n\nThe tilt of $K$ is $K^\\flat$, the fraction field of $\\varprojlim_{x \\mapsto x^p} \\mathcal{O}_K / p$; it is a perfectoid field of characteristic $p$.\nScholze's tilting equivalence identifies perfectoid spaces over $K$ with perfectoid spaces over $K^\\flat$.\n",
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