Perfectoid Spaces

By Peter Scholze. Berkeley Lectures on p-adic Geometry, 2011

331 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/454137cf65e9bc4cab52fa59770d54bb7bc091ab

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

1. Dustin Clausen
2. Lue Pan
3. M. Morrow
4. A. Mathew

In Geometry and Topology

1. Dustin Clausen
2. A. Mathew
3. Lue Pan
4. M. Morrow
5. B. Bhatt
6. Peter Scholze
7. Lennart Meier
8. N. Axel Naumann

Citations in the corpus

• Formal Mathematics Statement Curriculum Learning (2022, background): “Also, Lean has recently received a lot of attention from the mathematical community, thanks to projects such as the Perfectoid Spaces (Buzzard et al., 2019) and the Liquid Tensor experiment (Scholze, 2020), and benefits from a vibrant community of hundreds of contributors to its main mathematical lib”
• K-theory and topological cyclic homology of henselian pairs (2018, background): “Given a perfectoid field C (in the sense of [66]), the ring of integers OC has perfect residue field, so the above conclusion holds.”
• Cohen-Macaulayness of absolute integral closures (2020, unknown): “outlined above for proving Theorem 1.1 emerged in various discussions with Peter Scholze over a long period of time; indeed, the prospect of approaching Theorem 1.1 by tilting (as in Scholze’s thesis [Sch12]) to [HH92] was one of the first questions we ever discussed, back in 2011. Both Lurie and Sch”
• On locally analytic vectors of the completed cohomology of modular curves (2020, methodology): “cover U⊂B. Let C•(U,Oe),C•(U,Oela) be the Cech complex forˇ Oe and Oela with respect to this cover. Then B→C•(U,Oe) is strictly exact by the almost vanishing of higher cohomology (Theorem 1.8.(iv) of [Sch12]). Note that each term in this complex is LA-acyclic by Theorem 3.6.1. By passing to locally ”