nLab prismatic cohomology

Idea

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

Prismatic cohomology relies heavily on the idea of a $\delta$-ring, introduced by Andre Joyal to formalize how a ring equipped with a Frobenius lift is analogous to ring equipped with a derivation.

Prisms

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

Examples of Prisms

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

Relation to Perfectoid Rings

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

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

Comparison theorems

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

Theorem

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

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

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

Theorem

Let $(A,I)$ be a perfect prism and let $R$ be a $p$-complete $A/I$-algebra. For all $n\geq 1$, there exists a canonical isomorphism

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

where $d$ is a generator of $I$.

Absolute Prismatic Cohomology

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

The absolute prismatic cohomology $R\Gamma(X_{\Delta},\mathcal{O}_{\Delta})$ is equipped with a filtration called the Nygaard filtration, playing a role analogous to that of the Hodge filtration for de Rham cohomology.

Prismatic Crystals

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

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

where $\mathrm{Vect}_{B}$ is the set of finite projective $B$-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 $X$, one can attach a stack, called the Cartier-Witt stack and denoted $\mathrm{WCart}_{X}$ by Bhatt-Lurie, and called the prismatization of $X$ and denoted $X^{\Delta}$ by Drinfeld. When $X$ is quasi-syntomic, one can identify the $\infty$-category of crystals of $(p,I_{\Delta})$-complete complexes on $(X_{\Delta},\mathcal{O}_{\Delta})$ with the derived $\infty$-category of quasi-coherent sheaves on $\mathrm{WCart}_{X}$ (Bhatt21, Remark 2.6).

Applications

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

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

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. This is used by Antieau, Krause and Nikolaus in AKN22 to determine the K-groups of $\mathbb{Z}/p^{n}$.

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 $A_inf$-cohomology in BhattMorrowScholze16 and integral p-adic Hodge theory via topological Hochschild homology in BhattMorrowScholze16, both of which were developed together with Matthew Morrow.

Related entries

• p-adic Hodge theory
• perfectoid spaces
• topological Hochschild homology
• arithmetic differential geometry
• Borger's absolute geometry

References

Prismatic cohomology was introduced in

• Bhargav Bhatt, Peter Scholze, Prisms and Prismatic Cohomology, preprint (2019) arXiv:1905.08229

A survey of recent developments is given in

• Bhargav Bhatt, Algebraic Geometry in Mixed Characteristic, preprint (2021) arXiv:2112.12010

Lecture notes include

• Bhargav Bhatt, Prismatic cohomology, course at Columbia University, 2018 lecture notes

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

• Bhargav Bhatt, Prismatic cohomology, course at Columbia University, 2018 (lecture notes taken by Chao Li) lecture notes

Some other lectures on prismatic cohomology:

• Kiran Kedlaya, Notes on prismatic cohomology, 2020 lecture notes

• Matthew Emerton, Prismatic cohomology, lecture notes (2020) [webpage]

For some introductory comments see

• Terence Tao, Prismatic cohomology, (blog post)

• Riccardo Pengo (MO answer)

Recent developments include

• Vladimir Drinfeld, Prismatization (2020) arXiv:2005.04746

• Bhargav Bhatt, Jacob Lurie, Absolute Prismatic Cohomology, preprint (2022) arXiv:2022.06120

• Bhargav Bhatt, Jacob Lurie, The Prismatization of $p$-adic Formal Schemes, preprint (2022) arXiv:2022.06124

• Bhargav Bhatt, Cohen-Macaulayness of Absolute Integral Closures, preprint (2020) arXiv:2008.08070

• Jeremy Hahn, Arpon Raksit, Dylan Wilson, A motivic filtration on the topological cyclic homology of commutative ring spectra, preprint (2022) arXiv:2206.11208

An application to algebraic K-theory is

• Benjamin Antieau, Achim Krause, Thomas Nikolaus, On the K-theory of $\mathbb{Z}/p^{n}$ – Announcement [arXiv:2204.03420]

Historical precursors of prismatic cohomology are

• Bhargav Bhatt, Matthew Morrow, Peter Scholze Integral $p$-adic Hodge Theory [arXiv:1602.03148]

• Bhargav Bhatt, Matthew Morrow, Peter Scholze, Topological Hochschild Homology and Integral $p$-adic Hodge Theory [arXiv:1802.03261]