group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
algebraic topology – application of higher algebra and higher category theory to the study of (stable) homotopy theory
(2,1)-quasitopos?
structures in a cohesive (∞,1)-topos
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.
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 $p \in I + \phi(I)A$.
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.
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 $\Delta_{R/A}:=R\Gamma((R/A)_{\Delta},\mathcal{O}_{\Delta})$.
We have the following comparison theorems relating prismatic cohomology to the crystalline and étale cohomology:
Let $A,(p)$ be a bounded prism and let $R$ be a smooth $A/p$-algebra. There exists a canonical isomorphism
in $D(A)$, compatible with the action of Frobenius on both sides.
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
where $d$ is a generator of $I$.
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.
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
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).
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 $\mathcal{O}_{\mathbb{C}_{p}}$):
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 $\mathbb{Z}/p^{n}$.
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.
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:
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
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]
Last revised on October 4, 2024 at 21:36:11. See the history of this page for a list of all contributions to it.