Perfectoid spaces
We closely follow the very clear and instructive introduction to the paper. From the introduction: ``In commutative algebra and algebraic geometry, some of the most subtle problems arise in the context of mixed characteristic, i.e. over local fields such as \({\mathbb Q}_p\) which are of characteristic \(0\), but whose residue field \({\mathbb F}_p\) is of characteristic \(p\). The aim of this paper is to establish a general framework for reducing certain problems about mixed characteristic rings to problems about rings in characteristic \(p\). We will use this framework to establish a generalization of Faltings's almost purity theorem, and new results on Deligne's weight-monodromy conjecture.'' The point of departure is the following theorem of \textit{J.-M. Fontaine} and \textit{J.-P. Wintenberger} [C. R. Acad. Sci., Paris, Sér. A 288, 441--444 (1979; Zbl 0403.12018)]: Theorem. The absolute Galois groups of \({\mathbb Q}_p (p^{1/p^{\infty}})\) and \({\mathbb F}_p((t))\) are canonically isomorphic. The autzor defines a perfectoid field to be a complete topological field \(K\) whose topology is induced by a nondiscrete valuation of rank \(1\), such that the Frobenius is surjective on \(K^0/p\), where \(K^0\) denotes the subring of powerbounded elements in \(K\). Following the construction of Fontaine, to any such \(K\) one associates another perfectoid field \(K^{\flat}\) of characteristic \(p\). The above theorem is then generalized to: Theorem. The absolute Galois groups of \(K\) and \(K^{\flat}\) are canonically isomorphic. The objective is then to generalize this to a comparison of geometric objects over \(K\) with geometric objects over \(K^{\flat}\). In order to adequately formulate such a generalization in turns out that Huber's \textit{adic spaces} are a suitable framework. For example, any variety over \(K\) has an associated adic space \(X^{\text{ad}}\) over \(K\) with underlying topological space \(|X^{\text{m ad}}|\). Theorem. There is a homeomorphism of topological spaces \[ |({\mathbb{A}}^1_{K^{\flat}})^{\text{ad}}|\cong\text{lim}_{\leftarrow\atop T\mapsto T^p} |({\mathbb{A}}^1_{K})^{\text{ad}}| \] (where \(T\) is a coordinate on the affine line). Next it is argued that also the respective structure sheaves can be compared --- although their characteristics (i.e., \(\text{Char}(K)\in\{0,p\}\) versus \(\text{Char}(K^{\flat})=p\)) may differ! The author defines a perfectoid \(K\)-algebra (over the perfectoid field \(K\)) to be a Banach \(K\)-algebra \(R\) such that the set of power bounded elements \(R^0\subset R\) is bounded, and such that the Frobenius is surjective on \(R^0/p\). He proves: Theorem. There is a natural equivalence of categories, called the tilting equivalence, between the category of perfectoid \(K\)-algebras and the category of perfectoid \(K^{\flat}\)-algebras. An example of a perfectoid algebra is \(K\langle T^{\frac{1}{p^{\infty}}}\rangle\), its tilt is \(K^{\flat}\langle T^{\frac{1}{p^{\infty}}}\rangle\). Now affinoid perfectoid spaces are defined --- they are associated with pairs \((R,R^+)\), where \(R\) is a perfectoid \(K\)-algebra and \(R^+\subset R^0\) is open and integrally closed (often \(R^+=R^0\)). Proceeding as in Huber's definition of adic spaces, the author now defines perfectoid spaces, built from affinoid perfectoid spaces. Theorem. The category of perfectoid spaces over \(K\) and the category of perfectoid spaces over \(K^{\flat}\) are equivalent. In order to define and investigate the étale site \(X_{\text{ét}}\) of a perfectoid space \(X\), a generalization of Faltings's almost purity theorem is needed: Theorem. Let \(R\) be a perfectoid \(K\)-algebra. Let \(S/R\) be finite étale. Then \(S\) is a perfectoid algebra, and \(S^0\) is almost finite étale over \(R^0\). This theorem essentially compares finite étale covers of a perfectoid space \(X\) with those of \(X^{\flat}\) and leads to Theorem. Let \(X\) be a perfectoid space over \(K\), with tilt \(X^{\flat}\) over \(K^{\flat}\). Then tilting induces an equivalence of sites \(X_{\text{ét}}\cong X^{\flat}_{\text{ét}}\). Theorem. The étale topos \(({\mathbb{P}}^{n,\text{ad}}_{K^{\flat}})^{\widetilde{\,}}_{\text{ét}}\) is equivalent to the inverse limit \(\text{lim}_{\leftarrow \varphi}({\mathbb{P}}^{n,\text{ad}}_{K})^{\widetilde{\,}}_{\text{ét}}\). Finally, applications of these results to the weight monodromy conjecture are given. Theorem. Let \(k\) be a local field of characteristic \(0\). Let \(X\) be a geometrically connected proper smooth variety over \(k\) such that \(X\) is a set theoretic complete intersection in a projective smooth toric variety. Then the weight monodromy conjecture is true for \(X\).
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