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  • #subjects/arithmetic-geometry
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• Links:
  • prism
  • tilting
  • Faltings almost purity
  • semistable reduction
  • weight monodromy conjecture
  • perfect field
  • tilting
  • solid mathematics
  • condensed sets

perfectoid

Idea: the mixed characteristic analogs of \({{{\mathbb{F}}_p}{\hbox{-}}\mathsf{Alg}}\).

Definition: the perfection or perfect closure of a field: \begin{align*} k^{\mathrm{perf}}\coloneqq\displaystyle\bigcup_{n\geq 1} k^{1\over p^n} \subseteq \mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu .\end{align*} # References

• https://www.youtube.com/watch?v=3YF6fCFbymk&list=PLCe-H2N8-ny5O8svc5I4RAhPFj7AmH9Jq
• Lots of notes from a 2015 learning seminar: https://math.stanford.edu/~conrad/Perfseminar/

Topics

• Perfectoid ring
• smooth algebra
• Formal spectrum
• Crystalline cohomology
• Prismatic cohomology
• What does it mean for an algebra over \({ {\mathbb{Q}}_p }\) to be ramified at \(p\)?