Prismatic cohomology

15:05: Bhargav Bhatt (Harvard NT Seminar)

• One can take Unsorted/etale cohomology of varieties, and later refine to schemes, and thus take it for the base field even when it’s not algebraically closed and extract arithmetically interesting information.

• prismatic cohomology, meant to relate a number of other cohomology theories

• prism : a pair $$(A, I)$$ where $$A$$ is a commutative ring with a derived Frobenius lift $$\phi:A\to A$$, i.e. a $$\delta{\hbox{-}}$$structure.

• $$I {~\trianglelefteq~}A$$ is an ideal defining a Cartier divisor.
• $$A$$ is $$(P, I){\hbox{-}}$$complete.
• Any ideal generator $$d\in I$$ satisfies $$\phi(d) = d^p + p\cdot u, u\in A^{\times}$$.
• Fix a scheme and study prisms over it. Need these definitions to have stability under base-change.

• Examples:

• $$A \coloneqq{\mathbf{Z}}_p$$ and $$\phi = \operatorname{id}$$ with $$I = \left\langle{ p }\right\rangle$$ yields crystalline cohomology.
• $$A \coloneqq{\mathbf{Z}}_p{\left[\left[ u \right]\right] }, \phi(u) = u^p$$. Then $$I = \left\langle{ E(u) }\right\rangle$$ is generated by an Eisenstein polynomial. Here $$A/I = {\mathcal{O}}_K$$
• Prismatic site : fix a base prism $$(A, I)$$ for $$X$$ a $$p{\hbox{-}}$$adic formal scheme over $$A/I$$. Define \begin{align*} (X/A)_\prism = \left\{{ (A, I) \to (B, J) \in \mathop{\mathrm{Mor}}(\mathsf{Prism}), \operatorname{Spf}(B/J) \to X \text{ over } A/I }\right\} ,\end{align*} topologized via the flat topology on $$B/J$$.

• There is a structure sheaf $${\mathcal{O}}_{\prism}$$ where $$(B, J) \to B$$. Take $${\mathbf{R}}\Gamma$$, which receives a Frobenius action, to define a cohomology theory. Why is this a good idea?

• Absolute prismatic sites: for $$X\in {\mathsf{Sch}}(p{\hbox{-}}\text{adic})$$, define \begin{align*} X_\prism \coloneqq\left\{{ (B, J) \in \mathsf{Prism},\, \operatorname{Spf}(B/J) \to X }\right\} .\end{align*} Take sheaf cohomology to obtain $${\mathbf{R}}\Gamma_\prism(X) \coloneqq{\mathbf{R}}\Gamma(X_\prism, {\mathcal{O}}_\prism) {\circlearrowleft}_\phi$$.

• The category $$\mathsf{Prism}$$ doesn’t have a final object, so has interesting cohomology. Relates to the algebraic K theory of $${\mathbf{Z}}_p$$?

• Questions: let $$X_{/{\mathbf{Z}}_p}$$ be a smooth formal scheme.

• What is the cohomological dimension of $${\mathbf{R}}\Gamma_\prism(X)$$?
• What are the F-crystals on $$X_\prism$$?
• Produce finite flat $$B{\hbox{-}}$$modules?
• Bhatt and Lurie: found a stacky way to understand the absolute prismatic site of $${\mathbf{Z}}_p$$. Drinfeld found independently.

• Construction due to Simpson: take $$X\in {\mathsf{Var}}( \mathsf{Alg} )$$, define a de Rham presheaf \begin{align*} X_{\mathrm{dR}}: {}_{\\CC} \mathsf{Alg} ^{{ \operatorname{fp} }} &\to {\mathsf{Set}}\\ R &\mapsto X(R_{ \text{red} }) .\end{align*}

• Translates other cohomology theories into something about coherent sheaves..?
• Can reduce to studying e.g. a vector bundle on a more complicated object.
• Def: Cartier-Witt stack, a.k.a. the prismatization of $${\mathbf{Z}}_p$$

• Define $$\mathsf{WCart}$$ to be the formal stack on $$p{\hbox{-}}$$complete rings.
• Plug in a $$p{\hbox{-}}$$nilpotent ring $$R$$ to extract all (derived) prism structure on $$W(R)$$.

• Prisms aren’t base change compatible without the derived part.

• This is a groupoid.

• An explicit presentation: $$\mathsf{WCart}_0(R)$$ are distinguished Witt vectors in $$W(R)$$. Given by $$[a_0, a_1, \cdots ]$$ where $$a_0$$ is nilpotent and $$a_1$$ is a unit. This is a formal affine scheme. $$\mathsf{WCart}= \mathsf{WCart}_0 / W^*$$ is a presentation as a stack quotient.

• Receives a natural Frobenius action, which is a derived Frobenius lift.
• Start by understanding its points, suffices to evaluate on fields of characteristic $$p$$.

• If $$k\in \mathsf{Field}(\mathsf{Perf})_{\operatorname{ch. p}}$$, $$\mathsf{WCart}(k) = \left\{{ {\operatorname{pt}}}\right\}$$, with the point represented by $$(W(k), ?)$$.

• Yields a (geometric?) point $$x_{{{ \mathbf{F} }_p}}: \operatorname{Spec}({{ \mathbf{F} }_p}) \to \mathsf{WCart}$$.
• Analogy to understanding Hodge-Tate cohomology. Similar easy locus in this stack.

• Take 0th component of distinguished Witt vectors to get a diagram

• The bottom-left is this Hodge-Tate stack

• Now has a better chance of being an algebraic stack instead of a formal stack. Bottom arrow kills the formal direction.

• Will be classifying stack of a group scheme : need to produce a point and take automorphisms.

• Take the distinguished element $$V(?) \in W({\mathbf{Z}}_p)$$. Produces a map \begin{align*} \operatorname{Spf}({\mathbf{Z}}_p) \xrightarrow{\pi_{\operatorname{HT}}} \mathsf{WCart}^{\operatorname{HT}} .\end{align*}

• Fact: $$\pi_{\operatorname{HT}}$$ is a flat cover and $$\mathop{\mathrm{Aut}}(\pi_{\operatorname{HT}}) = W^*[ F]$$.
• Upshot: $$\mathsf{WCart}^{\operatorname{HT}}= {\mathbf{B}}W^* [F]$$ is a classifying stack. quasicoherent sheaves on the left and representations of the (classifying stack of the) group scheme on the right. I.e. $${ \mathsf{D} }_{\operatorname{qc}}(\mathsf{WCart}^{\operatorname{HT}}) = {\mathbf{R}}(W^*[F])$$.

• Fact: $${ \mathsf{D} }_{\operatorname{qc}}(\mathsf{WCart})$$ are equivalent to \begin{align*} \varprojlim_{(A, I)\in \mathsf{Prism}} { \mathsf{D} }_{(P, I)-?}(A) .\end{align*}

• Diffracted Hodge cohomology: let $$X\in {\mathsf{Schf}}_{{\mathbf{Z}}_p}$$. Get a prismatic structure sheaf using the assignment $$(A, I) \to {\mathbf{R}}\Gamma_\prism \qty{ (X\otimes A/I) / A}$$.

• Heuristic: $$\operatorname{Spec}{\mathbf{Z}}_p$$ should be 1-dimensional over something.

• Get an absolute comparison: $$\operatorname{cohdim}{\mathbf{R}}\Gamma_\prism (X) \leq d+1$$ where $$d = \operatorname{reldim}X_{/{\mathbf{Z}}_p}$$.

• There is a deRham comparison: \begin{align*} X_{{{ \mathbf{F} }_p}}^* H_\prism(X) \cong {\mathbf{R}}\Gamma_\mathrm{dR}(X_{{{ \mathbf{F} }_p}}) .\end{align*}

• There is a Hodge-Tate comparison: the object $$H_\prism(X)$$ restricted to $$\mathsf{WCart}^{\operatorname{HT}}$$ has an increasing filtration with $${\mathsf{gr}\,}_i = {\mathbf{R}}\Gamma(X, \Omega^i_X)[-i]$$.

• Use representation interpretation, then $$\mu_p \curvearrowright{\mathsf{gr}\,}_i$$ by weight $$-i$$.
• Combine these comparisons to get Deligne-Illusie : if $$\operatorname{reldim}X < p$$, then \begin{align*} {\mathbf{R}}\Gamma_\mathrm{dR}(X_{{{ \mathbf{F} }_p}}) \cong \bigoplus_{i} {\mathbf{R}}\Gamma(X_{{{ \mathbf{F} }_p}}, \Omega^i[-i]) .\end{align*} Get a lift to characteristic zero, yields Hodge-to-deRham degeneration there.

• An $$F{\hbox{-}}$$crystal on $$X_\prism$$ is a vector bundle $$\mathcal{E} \in { \mathsf{Vect} }(X_\prism, {\mathcal{O}}_\prism)$$? Plus some extra data.

• Infinite tensor product: \begin{align*} I_\prism \otimes F^* I_\prism \otimes(F^2)^* I_\prism \otimes\cdots .\end{align*} Converges to some object $${\mathcal{O}}_\prism \left\{{ 1 }\right\} \in \operatorname{Pic}(X_\prism, {\mathcal{O}}_\prism )$$, twisted? Yields isomorphism of sheaves after inverting $$I_\prism$$, \begin{align*} F^* {\mathcal{O}}_\prism \left\{{ 1 }\right\} \cong I_\prism^{-1}\otimes{\mathcal{O}}_\prism \left\{{ 1 }\right\} .\end{align*}

• Convergence: this is a formal stack, any thickening are identified with something… On each finite approximation, most terms are $${\mathcal{O}}_X$$.
• Some analog of Artin-Schreier here, taking fixed points?

• There is a natural functor from $$F{\hbox{-}}$$crystals on $$X$$ to local $${\mathbf{Z}}_p$$ systems on a geometric fiber $$X_?$$?

• Main theorem: produces descent data, uses work on Beilinson fiber sequence (Benjamin Antieau, Morrow, others?)

• Can say \begin{align*} H^i(\Delta_{{\mathbf{Z}}_p}) = \begin{cases} {\mathbf{Z}}_p & i=0 \\ \prod_{{\mathbb{N}}} {\mathbf{Z}}_p & i=1. \end{cases} \end{align*} Can compute using HH? $$\pi_{-1}( {\operatorname{TP}}({\mathbf{Z}}_p) )$$ is where the $$i=1$$ part comes from.

• HH corresponds to prismatic cohomology
• THH corresponds to Hodge-Tate cohomology.
• Prismatic is filtered by things that look like Hodge-Tate

• Absolute = arithmetic (take eigenspaces, related to motivic cohomology, relative = geometric?

• Link to K-theory comes from eigenspaces somehow.
• Similar to situation in etale cohomology: need absolute and relative to compute either.

Why are triangulated categories and derived categories important?

• Homological algebra lives in the derived category
• In AG, tight link between birational equivalence (of say smooth projective varieties and equivalence of $\mathbf{D} { {\mathsf{Coh}}X }$, the derived category of coherent sheaves
• See the Bondal-Orlov conjecture.
• birational is a weakening of isomorphism between varieties
• Being derived equivalent is a weakening of having equivalent $${\mathsf{DCoh}}$$
• Both recover actual isomorphisms in the case of smooth projective varieties
• Rep theory: having equivalent derived categories is called Morita equivalence.
• Derived equivalence is a weakening of Morita equivalence
• Can replace an algebra by a much simpler derived-equivalent one
• Use to study blocks of group algebra
• Lots of numerical consequences?

Some Lurie stuff

The Relationship Between THH and K-theory

Some remarks on $${\operatorname{THH}}$$ and $$K{\hbox{-}}$$Theory, no clue what the original source was:

• algebraic K theory is hard, using Topological Hochschild homology somehow makes computations easier.

• $$K{\hbox{-}}$$theory says something about vector bundles, topological Hochschild cohomology describes monodromy of vector bundles around infinitesimal loops

• For $$X$$ a nice scheme, take $$LX$$ the derived free loop space : the derived stacks $$\mathop{\mathrm{Maps}}_{ \operatorname{DSt}}(S^1, X)$$.

• Points of $$LX$$: infinitesimal loops in $$X$$
• Identify $${\operatorname{THH}}(X) \xrightarrow{\sim} {\mathcal{O}}(LX)$$ (global functions)

• Corollary of a result in Ben-Zvi–Francis–Nadler “Integral Transforms and Drinfeld Centers in derived algebraic geometry”?
• Dennis trace : a comparison $$K(X) \to {\operatorname{THH}}(X)$$, takes $$E\in { {\mathsf{Bun}}\qty{\operatorname{GL}_r} }$$ to the canonical monodromy automorphism of the pullback of $$E$$ to $$LX$$

• Use the map $$LX\to X$$ sending a loop to its basepoint
• Traces are $$S^1{\hbox{-}}$$equivariant because loops! Just equip $$K(X)$$ with the trivial $$S^1$$ action.

• Take homotopy fixed points to get something smaller than $${\operatorname{THH}}$$: $${\operatorname{THC}}^-$$, topological negative cyclic homology

• Dennis trace is invariant under all covering maps of circles, even multisheeted

• Encoded not in a group action by a cyclotomic structure.
• Take homotopy fixed points of the cyclotomic structure on $${\operatorname{THH}}$$ to get $${\operatorname{TC}}$$, Topological cyclic homology
• There is a map $$K\to {\operatorname{TC}}$$
• Theorem of Dundas-Goodwillie-McCarthy: whenever $$A\to A'$$ is a nilpotent extension of connective ring spectra, \begin{align*} K(A') \xrightarrow{\sim} K(A) \underset{\scriptscriptstyle {{\operatorname{TC}}(A)} }{\times} {\operatorname{TC}}(A') \end{align*}

Eilenberg-MacLane spaces

• Some good stuff from Akhil Mathew on EM spaces:

• Why shouldanyone care about stacks? #why-care

schemes and class field theory

• Definitions of schemes and scheme-y curves

• Definitions of schemes and scheme-y curves

• Grothendieck’s fundamental group

• Statement of class field theory in terms of fundamental groups

• See Arithmetic schemes

• Idele group for arithmetic schemes

-Actual class group for schemes

• Wiesend’s finiteness theorem is one of the strongest and most beautiful results in higher Global class field theory?

• The main aim of higher global class field theory is to determine the abelian fundamental group $$\pi_1^{{\operatorname{ab}}}(X)$$ of a regular arithmetic scheme $$X$$, i.e. of a connected regular scheme separated scheme scheme flat morphism and of finite type over $${\mathbf{Z}}$$, in terms of an arithmetically defined class groups

• $$C(X)$$.

• Fundamental theorem of class field theory?