Abhyankar's perfectoid lemma
The article under review provides an extension of the almost purity theorem of \textit{G. Faltings} [J. Am. Math. Soc. 1, No. 1, 255--299 (1988; Zbl 0764.14012)]. Let $A$ be a perfectoid algebra over a perfectoid field $K$. In the form given by \textit{P. Scholze} [Publ. Math., Inst. Hautes Étud. Sci. 116, 245--313 (2012; Zbl 1263.14022)] and \textit{K. S. Kedlaya} and \textit{R. Liu} [Relative $p$-adic Hodge theory: foundations. Paris: Société Mathématique de France (SMF) (2015; Zbl 1370.14025)], the later theorem claims that each finite étale $A$-algebra $B$ is perfectoid and its set $B^\circ$ of power-bounded elements is almost finite étale over $A^\circ$ in the basic setting $(K^\circ, K^{\circ\circ})$. Here, ``almost'' is to be understood is the sense of Faltings' almost algebra (see [\textit{O. Gabber} and \textit{L. Ramero}, Almost ring theory. Berlin: Springer (2003; Zbl 1045.13002)] for a general reference), roughly meaning that we work up to $K^{\circ\circ}$-torsion. The author proves a similar result but allowing ramification. The basic setting here is $(K^\circ [T^{1/p^\infty}], T^{1/p^\infty} K^{\circ\circ} [T^{1/p^\infty}])$. Assume that $A$ has residue characteristic $p$ and contains a sequence of compatible $p^n$ roots of some $g \in A^\circ$, so as to see $A^\circ$ as a $K^\circ [T^{1/p^\infty}]$-algebra. Then, for each finite étale $A[1/g]$-algebra $B'$, there exists a perfectoid sub-algebra $B$ of $B'$ with $B[1/g] = B'$, $B^\circ$ is almost isomorphic to the integral closure of $g^{-1/p^\infty} A^\circ$ in $B'$ and, for each $m$, $B^\circ/p^m$ is almost finite étale over $A^\circ/p^m$. (The author notes that it is even plausible that $B^\circ$ be almost finite étale over $A^\circ$.) In the sense that it makes ramification vanish, this result is reminiscent of Abhyankar's lemma with some major differences however in the assumptions, in particular the fact that no tameness is required. Let us describe the contents of the paper. The first two sections are rather introductory. Section 1 contains some definitions and results of almost algebra, following mostly Gabber-Ramero [loc. cit.] Section 2 contains results on uniform Banach algebra, \textit{i.e.} Banach algebras whose set of power-bounded elements is bounded, including almost algebra in this setting. The point of view is categorical In Section 3, the author gives a presentation of perfectoid algebras. He recalls the tilting equivalence and gives a new proof of the almost purity theorem, relying on Galois theoretic arguments. Section 4 contains an avatar of Riemann's extension theorem is the perfectoid setting. Fix the same setting as above. The author considers the algebras $A\{p^j/g\}^\circ$, which are associated to the complement of some tubulars neighborhoods of the vanishing locus of $g$ and proves that their limit over $j$ coincides with the totally integral closure of $A^\circ$ in $A[1/g]$ and with $g^{-1/p^\infty} A^\circ$. In Section 5, the author proves a version of the almost purity theorem in a ramified case: for each finite étale $A[1/g]$-algebra $B$, $B^\circ$ is almost finite étale modulo any power of $p$ and almost finite étale after inverting $g$. The proof uses Riemann's extension theorem to get away from the ramification locus and the classical almost purity theorem then applies. The analogue of Abhyankar's lemma is deduced at the end of the paper. The paper is very well written. The author made a great effort to explain the meaning of the main statements in down-to-earth terms and also included several examples (and even ``prophylactic'' examples!) in the first sections to help understand the theory.
scientific article; zbMATH DE number 6914158
scientific article