nLab prismatic cohomology

Contents

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Algebraic topology

(,1)(\infty,1)-Topos Theory

(∞,1)-topos theory

Background

Definitions

Characterization

Morphisms

Extra stuff, structure and property

Models

Constructions

structures in a cohesive (∞,1)-topos

Contents

Idea

Prismatic cohomology is a cohomology theory which can specialize into various pp-adic cohomology theories, including étale cohomology, de Rham cohomology and crystalline cohomology, as well as the so far conjectural qq-de Rham cohomology of Peter Scholze. It is a geometric approach to integral p-adic Hodge theory.

Prisms

A prism is a pair (A,I)(A,I) where AA is a δ\delta-ring and II is an ideal defining a Cartier divisor on Spec(A)\mathrm{Spec}(A), such that such that AA is derived (p,I)(p,I)-complete, and pI+ϕ(I)Ap \in I + \phi(I)A.

Examples of Prisms

  • A= p[[u]]A=\mathbb{Z}_{p}[[u]] and I=(up)I=(u-p)
  • A=A inf(R)A=A_{\mathrm{inf}}(R) and I=ker(θ)I=\mathrm{ker}(\theta), where θ:A inf(R)R\theta:A_{\mathrm{inf}}(R)\to R is the canonical surjection

Relation to Perfectoid Rings

A prism (A,I)(A,I) is perfect if AA is perfect. The category of perfect prisms is equivalent to the category of integral perfectoid rings which are related to, but not the same as, the perfectoid rings in perfectoid spaces. The definition of integral perfectoid ring can be found in Definition 3.5 of BhattScholze19 and the relation between the two notions of perfectoid can be found in Lemma 3.20 of the same paper.

Definition of Prismatic Cohomology

Let (A,I)(A,I) be a prism as defined above. Let RR be a formally smooth A/IA/I-algebra. The prismatic site (R/A) Δ(R/A)_{\Delta} has objects which are prisms (B,IB)(B,I B) over (A,I)(A,I) together with a map RB/IBR\to B/I B over A/IA/I. Such an object is written (RB/IBB)(R\to B/I B\leftarrow B).

We have functors 𝒪 Δ\mathcal{O}_{\Delta} and 𝒪¯ Δ\overline{\mathcal{O}}_{\Delta} which sends (RB/IBB)(R\to B/I B\leftarrow B) to BB and B/IBB/I B respectively. The prismatic cohomology of RR is defined to be Δ R/A:=RΓ((R/A) Δ,𝒪 Δ)\Delta_{R/A}:=R\Gamma((R/A)_{\Delta},\mathcal{O}_{\Delta}).

Comparison theorems

We have the following comparison theorems relating prismatic cohomology to the crystalline and étale cohomology:

Theorem

Let A,(p)A,(p) be a bounded prism and let RR be a smooth A/pA/p-algebra. There exists a canonical isomorphism

ϕ A *Δ R/A RΓ crys(R/A)\phi_{A}^{*}\Delta_{R/A}^{\wedge}\cong R\Gamma_{crys}(R/A)

in D(A)D(A), compatible with the action of Frobenius on both sides.

Theorem

Let (A,I)(A,I) be a perfect prism and let RR be a pp-complete A/IA/I-algebra. For all n1n\geq 1, there exists a canonical isomorphism

RΓ et(Spec(R[1/p]),/p n)(Δ R/A[1/d]/p n) ϕ=1R\Gamma_{\et}(\mathrm{Spec}(R[1/p]),\mathbb{Z}/p^{n})\cong (\Delta_{R/A}[1/d]/p^{n})^{\phi=1}

where dd is a generator of II.

Absolute Prismatic Cohomology

By forgetting the choice of base prism, one obtains the absolute prismatic cohomology. Given a p-adic formal scheme XX, its absolute prismatic site X Δ{X}_{\Delta} is the category of all bounded prisms (B,J)(B,J) equipped with a map Spf(B/J)X\mathrm{Spf}(B/J)\to X, topologized with the flat topology. We have sheaves 𝒪 Δ\mathcal{O}_{\Delta}, I ΔI_{\Delta}, and 𝒪¯ Δ\overline{\mathcal{O}}_{\Delta} obtained by remembering BB, JJ, and B/JB/J respectively.

Prismatic Crystals

Let XX be a p-adic formal scheme and let X ΔX_{\Delta} be its absolute prismatic site as above. A prismatic crystal is an assignment

(B,J)X Δ(B)Vect B(B,J)\in X_{\Delta}\mapsto \mathcal{E}(B)\in\mathrm{Vect}_{B}

where Vect B\mathrm{Vect}_{B} is the set of finite projective BB-modules.

A stacky approach to the study of prismatic crystals has been developed independently by Drinfeld (Drinfeld20) and Bhatt-Lurie (BhattLurie22).

Given p-adic formal scheme XX, one can attach a stack, called the Cartier-Witt stack and denoted WCart X\mathrm{WCart}_{X} by Bhatt-Lurie, and called the prismatization of XX and denoted X ΔX^{\Delta} by Drinfeld. When XX is quasi-syntomic, one can identify the \infty-category of crystals of (p,I Δ)(p,I_{\Delta})-complete complexes on (X Δ,𝒪 Δ)(X_{\Delta},\mathcal{O}_{\Delta}) with the derived \infty-category of quasi-coherent sheaves on WCart X\mathrm{WCart}_{X} (Bhatt21, Remark 2.6).

Applications

Prismatic cohomology can be used to obtain the following inequality for XX a proper smooth formal scheme over p\mathbb{C}_{p} (here kk is the residue field of 𝒪 p\mathcal{O}_{\mathbb{C}_{p}}):

dim 𝔽 pH et i(X p,𝔽 p)dim kH dR i(X k)dim_{\mathbb{F}_{p}}H_{et}^{i}(X_{\mathbb{C}_{p}},\mathbb{F}_{p})\leq dim_{k}H_{dR}^{i}(X_{k})

Prismatic cohomology has also been used in Bhatt20 to prove that modulo a prime power the absolute integral closure of an excellent Noetherian domain is Cohen-Macaulay. The proof also uses a p-adic version of the Riemann-Hilbert correspondence being developed in yet-unpublished work of Bhargav Bhatt and Jacob Lurie.

Prismatic cohomology has also been used to construct a version of syntomic cohomology, which describes the graded pieces of a “motivic” filtration on p-adic etale K-theory. This is analogous to the filtration on topological K-theory whose graded pieces are described by the shifted singular cohomology complex, and is also analogous to the filtration on algebraic K-theory by motivic cohomology.

Historical Precursors

Prismatic cohomology was developed by Bhargav Bhatt and Peter Scholze following earlier work on developing an integral version of p-adic Hodge theory. Some of this earlier work includes A infA_inf-cohomology in BhattMorrowScholze16 and integral p-adic Hodge theory via topological Hochschild homology in BhattMorrowScholze16, both of which were developed together with Matthew Morrow.

References

Prismatic cohomology was introduced in

A survey of recent developments is given in

Lecture notes include

Lecture notes taken by Chao Li, from the same lectures (contains some material not included in the above)

Some other lectures on prismatic cohomology:

For some introductory comments see

Recent developments include

Historical precursors of prismatic cohomology are

Last revised on November 29, 2022 at 14:35:12. See the history of this page for a list of all contributions to it.