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• 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
  • prismatic cohomology
  • tilting
  • Faltings almost purity
  • semistable reduction
  • weight monodromy conjecture
  • Unsorted/perfect field
  • Unsorted/tilting
  • solid mathematics
  • condensed sets
  • affine Grassmannian
  • Huber ring
  • Perfectoid ring
  • Unsorted/smooth algebra
  • Formal spectrum
  • Crystalline cohomology
  • Prismatic cohomology
  • What does it mean for an algebra over ${ {\mathbf{Q}}_p }$ to be ramified at $p$?

perfectoid

Idea: the mixed characteristic analogs of ${ {}_{{ \mathbf{F} }_p} \mathsf{Alg} }$.

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

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

References

Examples