---
date: 2022-01-15 21:49
modification date: Monday 24th January 2022 19:56:37
title: perfectoid
aliases: [perfectoid, perfection, perfect closure, Scholze, adic space, adic, perfectoid space, perfectoid spaces]

---

Last modified date: <%+ tp.file.last_modified_date() %>


---

- Tags 
	- #arithmetic-geometry 
- Refs:
	- Course notes on p-adic geometry: <https://www.dropbox.com/s/j53etb62pfjcp57/sntsnotes.pdf?dl=0> #resources/course-notes 
	- Lectures by Kedlaya: <https://www.youtube.com/watch?v=yyLLemzaCRQ>
	- Course on adic spaces: <http://perso.ens-lyon.fr/sophie.morel/adic_notes.pdf#page=1> #resources/full-courses 
	- <https://www.youtube.com/watch?v=3YF6fCFbymk&list=PLCe-H2N8-ny5O8svc5I4RAhPFj7AmH9Jq> #resources/videos 
	- Lots of notes from a 2015 learning seminar:
	<https://math.stanford.edu/~conrad/Perfseminar/> #resources/websites 

- Links:
	- [prism](Unsorted/prismatic%20cohomology.md)
	- [prismatic cohomology](Unsorted/prismatic%20cohomology.md)
	- [tilting](Unsorted/tilting.md)
	- [Faltings almost purity](Faltings%20almost%20purity)
	- [semistable reduction](Unsorted/semistable%20reduction.md)
	- [weight monodromy conjecture](weight%20monodromy%20conjecture)
	- [Unsorted/perfect field](Unsorted/perfect%20field.md)
	- [Unsorted/tilting](Unsorted/tilting.md)
	- [solid mathematics](solid%20mathematics.md)
	- [condensed sets](condensed sets)
	- [affine Grassmannian](affine%20Grassmannian)
	- [Huber ring](Huber%20ring)
	- Perfectoid ring
	- [Unsorted/smooth algebra](Unsorted/smooth%20algebra.md)
	- [Formal spectrum](Formal%20spectrum.md)
	- [Crystalline cohomology](Crystalline%20cohomology.md)
	- [Prismatic cohomology](Prismatic%20cohomology.md)
	- What does it mean for an algebra over $\QQpadic$ to be ramified at $p$?
	- [shtuka](Unsorted/moduli%20of%20shtukas.md)

---

# Adic spaces

In [Shimura varieties](Unsorted/Shimura%20variety.md):

![](attachments/2023-03-15.png)

# perfectoid

![](attachments/Pasted%20image%2020220515001143.png)

Idea: the [mixed characteristic](Unsorted/mixed%20characteristic.md) analogs of $\algs{\FF_p}$.

Definition: the **perfection** or **perfect closure** of a field:
$$
k^\perf \da \Union_{n\geq 1} k^{1\over p^n} \subseteq \bar k
.$$
![](attachments/Pasted%20image%2020220129013015.png)
![](attachments/Pasted%20image%2020220129012959.png)

Recent result: used by Yves André to prove [[Hochster's Direct Summand Conjecture]].
# Perfection

![](attachments/Pasted%20image%2020220502151754.png)

# References



# Examples

![](attachments/Pasted%20image%2020220515001214.png)