[Paper Review] Prisms and Prismatic Cohomology

Bhargav Bhatt, Peter Scholze|arXiv (Cornell University)|May 20, 2019

Algebraic Geometry and Number Theory4 citations

TL;DR

This paper introduces prisms—deformations of perfectoid rings—as a foundational framework for a unified $p$-adic cohomology theory. By constructing a ringed site called the prismatic site for $p$-adic formal schemes, the authors show that prismatic cohomology specializes to and refines known theories like étale, de Rham, and crystalline cohomology, proving key conjectures including the odd vanishing of $p$-adic Tate twists and a co-ordinate-free description of $q$-de Rham 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 $p$-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.

Motivation & Objective

• To develop a new, unified framework for $p$-adic cohomology theories using prisms as a 'deperfection' of perfectoid rings.
• To construct a ringed site—the prismatic site—for $p$-adic formal schemes, enabling a general cohomology theory.
• To prove the almost purity theorem with ramification along arbitrary closed subsets without relying on adic spaces.
• To provide a coordinate-free description of $q$-de Rham cohomology as conjectured by Scholze.
• To resolve the odd vanishing conjecture for $p$-adic Tate twists $\mathbf{Z}_p(n)$ in $K$-theory.

Proposed method

• Define a prism as a pair $(A, I)$ where $A$ is a $\delta$-ring and $I \subset A$ is an ideal such that $A$ is $(p,I)$-adically complete and $I + \phi_A(I)A \supset p$, with $\phi_A$ a Frobenius lift.
• Construct the prismatic site $((X/A)_{\mathrm{prism}})$ for a smooth $p$-adic formal scheme $X$ over $A/I$, and define prismatic cohomology as the derived global sections of the structure sheaf.
• Use the theory of $\delta$-rings and their perfection to relate prisms to perfectoid rings via the functors $A \mapsto A/I$ and $R \mapsto (A_{\mathrm{inf}}(R), \ker \theta)$.
• Establish the Hodge-Tate comparison isomorphism in both characteristic $p$ and mixed characteristic, linking prismatic cohomology to de Rham and crystalline cohomology.
• Prove the étale comparison theorem and the almost purity theorem in a generalized setting, allowing ramification along arbitrary closed subsets.
• Apply the Nygaard filtration and $q$-deformations to describe $q$-crystalline and $q$-de Rham cohomology, and prove the odd vanishing conjecture for $\mathbf{Z}_p(n)$.

Experimental results

Research questions

• RQ1 How can a single cohomology theory unify known $p$-adic cohomology theories such as étale, de Rham, and crystalline cohomology?
• RQ2 Can the almost purity theorem be extended to allow ramification along arbitrary closed subsets without using adic geometry?
• RQ3 Is there a coordinate-free description of $q$-de Rham cohomology, as conjectured by Scholze?
• RQ4 Does the odd vanishing conjecture for $p$-adic Tate twists $\mathbf{Z}_p(n)$ hold in $K$-theory?
• RQ5 What is the precise relationship between prismatic cohomology and the $A\Omega$ cohomology of Bhatt-Morrow-Scholze?

Key findings

• Prismatic cohomology specializes to and refines all known integral $p$-adic cohomology theories, including étale, de Rham, crystalline, and $q$-de Rham cohomology.
• The category of perfect prisms is equivalent to the category of perfectoid rings, establishing prisms as a 'deperfection' of perfectoid rings.
• The authors prove a generalized almost purity theorem that allows ramification along arbitrary closed subsets, without requiring adic spaces.
• A coordinate-free description of $q$-de Rham cohomology is established via the prism $(\mathbf{Z}_p[[q-1]], ([p]_q))$, confirming a conjecture of Scholze.
• The odd vanishing conjecture for $\mathbf{Z}_p(n)$ is settled: $H^i_{\mathrm{ét}}(\operatorname{Spec} R, \mathbf{Z}_p(n)) = 0$ for $i \neq 2n$ and $n$ odd, when $R$ is a perfectoid ring.
• The prismatic cohomology of a smooth formal scheme over $A/I$ is computed via $\mathbbl{\Delta}_{S/A} \cong A_{\mathrm{inf}}(S)\{\frac{g_1}{d}, \dots, \frac{g_r}{d}\}^\wedge$, with $g_i$ lifting $f_i$ in a $p$-completely regular sequence.