Tracking issue: The Tilting Equivalence of Perfectoid Fields

[jjdishere]

This issue is meant to track PRs on the development of the theory of perfectoid fields. This issue belongs to a broader project of proving the Fontaine-Wintenberger theorem.

Rough roadmap

• define the topological nilpotent elements and bounded elements
• define the pefectoid fields
• show that the tilt of a perfectoid field is still perfectoid (to be decomposed later)
• define the tilt of the morphism
• define the category of perfectoid fields and the tilting functor
• show that char p perfectoid field is just a rank 1 valued perfect char p field

Open or already closed PRs on this topic

Krasner's lemma

Witt vectors

• Teichmuller represenatives
• Fontaine's theta map

Integral perfectoid rings

• Integral perfectoid ring and perfectoid pseudo-uniformizer

Perfectoid fields

• Valued perfectoid field and perfectoid field, char p case

Almost mathemetics

The final result