# Perfectoid Spaces

@article{Scholze2020PerfectoidS, title={Perfectoid Spaces}, author={Peter Scholze}, journal={Publications math{\'e}matiques de l'IH{\'E}S}, year={2020}, volume={116}, pages={245-313} }

We introduce a certain class of so-called perfectoid rings and spaces, which give a natural framework for Faltings’ almost purity theorem, and for which there is a natural tilting operation which exchanges characteristic 0 and characteristic p. We deduce the weight-monodromy conjecture in certain cases by reduction to equal characteristic.

## 277 Citations

PERFECTOID SPACES AND THE HOMOLOGICAL CONJECTURES

- MathematicsProceedings of the International Congress of Mathematicians (ICM 2018)
- 2019

This is a survey of recent advances in commutative algebra, especially in mixed characteristic, obtained by using the theory of perfectoid spaces. An explanation of these techniques and a short…

Another proof of the almost purity theorem for perfectoid valuation rings

- MathematicsJournal of Pure and Applied Algebra
- 2021

Perfectoid Shimura varieties of abelian type

- Mathematics
- 2015

We prove that Shimura varieties of abelian type with infinite level at $p$ are perfectoid. As a corollary, the moduli spaces of polarized K3 surfaces with infinite level at $p$ are also perfectoid.

Logarithmic adic spaces: some foundational results

- Mathematics
- 2019

We develop a theory of log adic spaces by combining the theories of adic spaces and log schemes, and study the Kummer \'etale and pro-Kummer \'etale topology for such spaces. We also establish the…

Stably uniform affinoids are sheafy

- MathematicsJournal für die reine und angewandte Mathematik (Crelles Journal)
- 2018

We develop some of the foundations of affinoid pre-adic spaces without
Noetherian or finiteness hypotheses. We give explicit examples of
non-adic affinoid pre-adic spaces, and also a new condition…

On the Witt Vectors of Perfect Rings in Positive Characteristic

- Mathematics
- 2014

The purpose of this article is to prove some results on the Witt vectors of perfect F p -algebras. Let A be a perfect F p -algebra for a prime integer p, and assume that A has the property P. Then…

$p$-adic vanishing cycles as Frobenius-fixed points

- Mathematics
- 2018

Given a smooth formal scheme over the ring of integers of a mixed-characteristic perfectoid field, we study its $p$-adic vanishing cycles via de Rham--Witt and $q$-de Rham complexes.

Some recent advances in topological Hochschild homology

- MathematicsBulletin of the London Mathematical Society
- 2022

We give an account of the construction of the Bhatt–Morrow–Scholze motivic filtration on topological cyclic homology and related invariants, focusing on the case of equal characteristic p$p$ and the…

Note on weight-monodromy conjecture for $p$-adically uniformized varieties

- MathematicsProceedings of the American Mathematical Society
- 2019

We prove the weight-monodromy conjecture for varieties which are p-adically uniformized by a product of the Drinfeld upper half spaces. It is an easy consequence of Dat's work on the cohomology…

On the p-adic weight-monodromy conjecture for complete intersetions in toric varieties

- Mathematics
- 2022

A BSTRACT . We give a proof of the p -adic weight monodromy conjecture for scheme-theoretic complete intersections in projective smooth toric varieties. The strategy is based on Scholze’s proof in…

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