Prismatic cohomology

15:05: Bhargav Bhatt (Harvard NT Seminar)

#projects/notes/seminars #web/blog #subjects/arithmetic-geometry/prisms

  • 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:

    • \(A \coloneqq{\mathbb{Z}}_p\) and \(\phi = \operatorname{id}\) with \(I = \left\langle{ p }\right\rangle\) yields crystalline cohomology.
    • \(A \coloneqq{\mathbb{Z}}_p{\left[\left[ u \right]\right] }, \phi(u) = u^p\). 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 \({\mathbb{R}}\Gamma\), 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 \({\mathbb{R}}\Gamma_\prism(X) \coloneqq{\mathbb{R}}\Gamma(X_\prism, {\mathcal{O}}_\prism) {\circlearrowleft}_\phi\).

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

  • Questions: let \(X_{/{\mathbb{Z}}_p}\) be a smooth formal scheme.

    • What is the cohomological dimension of \({\mathbb{R}}\Gamma_\prism(X)\)?
    • 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 \({\mathbb{Z}}_p\). Drinfeld found independently.

  • Construction due to Simpson: take \(X\in {\mathsf{Var}}({\mathsf{Alg}})\), define a de Rham presheaf \begin{align*} X_{\mathrm{dR}}: {\mathsf{Alg}_{\mathbb{C}} }^{{ \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 \({\mathbb{Z}}_p\)

    • 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 \(x_{{\mathbb{F}_p}}: \operatorname{Spec}({\mathbb{F}_p}) \to \mathsf{WCart}\).
  • Analogy to understanding Hodge-Tate cohomology. Similar easy locus in this stack.

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

Link to 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 \(V(?) \in W({\mathbb{Z}}_p)\). Produces a map \begin{align*} \operatorname{Spf}({\mathbb{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. \({ \mathsf{D} }_{\operatorname{qc}}(\mathsf{WCart}^{\operatorname{HT}}) = {\mathbb{R}}(W^*[F])\).

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

  • Heuristic: \(\operatorname{Spec}{\mathbb{Z}}_p\) should be 1-dimensional over something.

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

  • There is a deRham comparison: \begin{align*} X_{{\mathbb{F}_p}}^* H_\prism(X) \cong {\mathbb{R}}\Gamma_\mathrm{dR}(X_{{\mathbb{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 \({\mathsf{gr}\,}_i = {\mathbb{R}}\Gamma(X, \Omega^i_X)[-i]\).

    • 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*} {\mathbb{R}}\Gamma_\mathrm{dR}(X_{{\mathbb{F}_p}}) \cong \bigoplus_{i} {\mathbb{R}}\Gamma(X_{{\mathbb{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 \({\mathbb{Z}}_p\) 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_{{\mathbb{Z}}_p}) = \begin{cases} {\mathbb{Z}}_p & i=0 \\ \prod_{{\mathbb{N}}} {\mathbb{Z}}_p & i=1. \end{cases} \end{align*} Can compute using HH? \(\pi_{-1}( {\operatorname{TP}}({\mathbb{Z}}_p) )\) 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?

A Roadmap to Hill-Hopkins-Ravenel

Roadmap to HHR

Some Lurie stuff

Lurie’s Seminar on Algebraic Topology

A bunch of suggested papers

Lurie’s Topics in Geometric Topology

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 Care About Stacks?

  • Why shouldanyone care about stacks? #why-care

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Homotopy Theory is Connected to Lie algebra cohomology

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schemes and class field theory

  • Definitions of schemes and scheme-y curves

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  • Definitions of schemes and scheme-y curves

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  • Grothendieck’s fundamental group

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  • Statement of class field theory in terms of fundamental groups

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  • See Arithmetic schemes

  • Idele group for arithmetic schemes

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-Actual class group for schemes

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  • 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 \({\mathbb{Z}}\), in terms of an arithmetically defined class groups

  • \(C(X)\).

  • Fundamental theorem of class field theory?

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