Abstract
These notes give a quick overview of recent work by the speaker and of some subsequent works introducing perfectoid geometry into homological commutative algebra and singularity theory. The emphasis is on big Cohen-Macaulay algebras and applications. The progresses take place primarily in mixed characteristic, but sometimes provide a bridge between characteristic p and characteristic 0.
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Notes
Given an equation involving any number of fluent quantities to find the fluxions, and vice versa.
It turns out that this property is equivalent to deep ramification: \({\varOmega }_{\bar K^{o}/K^{o}} = 0\) (Gabber-Ramero).
The p-root closure of a p-adic ring R: elements r of R[1/p] such that \(r^{p^{j}}\in R\) for some j > 0.
Or at least, in the Smith version, for some p big enough, but with an ineffective lower bound.
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Lecture at the Annual Meeting 2019 of the Vietnam Institute for Advanced Study in Mathematics
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André, Y. Singularities and Perfectoid Geometry. Acta Math Vietnam 46, 1–8 (2021). https://doi.org/10.1007/s40306-020-00378-y
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DOI: https://doi.org/10.1007/s40306-020-00378-y