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Prisms and Prismatic Cohomology

Bhargav Bhatt, Peter Scholze

Abstract

We introduce the notion of a prism, which may be regarded as a "deperfection" of the notion of a perfectoid ring. Using prisms, we attach a ringed site -- the prismatic site -- to a $p$-adic formal scheme. The resulting cohomology theory specializes to (and often refines) most known integral $p$-adic cohomology theories. As applications, we prove an improved version of the almost purity theorem allowing ramification along arbitrary closed subsets (without using adic spaces), give a co-ordinate free description of $q$-de Rham cohomology as conjectured by the second author, and settle a vanishing conjecture for the $p$-adic Tate twists $\mathbf{Z}_p(n)$ introduced in previous joint work with Morrow.

Prisms and Prismatic Cohomology

Abstract

We introduce the notion of a prism, which may be regarded as a "deperfection" of the notion of a perfectoid ring. Using prisms, we attach a ringed site -- the prismatic site -- to a -adic formal scheme. The resulting cohomology theory specializes to (and often refines) most known integral -adic cohomology theories. As applications, we prove an improved version of the almost purity theorem allowing ramification along arbitrary closed subsets (without using adic spaces), give a co-ordinate free description of -de Rham cohomology as conjectured by the second author, and settle a vanishing conjecture for the -adic Tate twists introduced in previous joint work with Morrow.

Paper Structure

This paper contains 46 sections, 115 theorems, 247 equations.

Table of Contents

  1. Introduction
  2. Leitfaden of the paper
  3. Notation
  4. $\delta$-rings
  5. Definition and basic properties
  6. Extending $\delta$-structures
  7. Distinguished elements
  8. Perfect $\delta$-rings
  9. Relation to divided power algebras
  10. Flatness for certain pd-envelopes and prismatic-envelopes
  11. Prisms
  12. The prismatic site and formulation of the Hodge-Tate comparison
  13. The prismatic site
  14. The Hodge-Tate comparison theorem: formulation
  15. Generalities on computing prismatic cohomology
  16. ...and 31 more sections

Key Result

Proposition 1.5

If $(A,I)\to (B,J)$ is a map of prisms, then $J=IB$.

Theorems & Definitions (200)

  • Definition 1.1
  • Remark 1.2
  • Example 1.3
  • Definition 1.4
  • Proposition 1.5: Proposition \ref{['PrismMapTaut']}
  • Definition 1.6
  • Remark 1.7
  • Theorem 1.8
  • Example 1.9
  • Remark 1.11
  • ...and 190 more