◆ PRISMATIC PERFECTOID SITE

DELTA-PRISMS (Bhatt-Scholze)

A prism is a pair (A, I) where A is a δ-ring and I ⊆ A is an invertible ideal such that A is (p, I)-adically complete and p ∈ I + φ(I)A. The prismatic site (X)_△ organizes p-adic cohomology theories.

  H^n_△(X) = RΓ((X)_△, O_△)

F-CRYSTALS

An F-crystal on the prismatic site is a vector bundle M on (X)_△ with a Frobenius structure φ_M: φ*M[1/I] → M[1/I]. These generalize Fontaine's theory and classify p-divisible groups.

WITT VECTOR TILTS

For a perfectoid ring R, its tilt R^♭ is a perfect F_p-algebra. The map θ: W(R^♭) → R from Witt vectors of the tilt surjects, and (W(R^♭), ker(θ)) is a perfect prism.