--- date: 2021-04-21 tags: [ web/quick-notes ] title: "Prismatic cohomology" --- # 2021-04-21 ## 15:05: Bhargav Bhatt (Harvard NT Seminar) #projects/notes/seminars #web/blog #arithmetic-geometry/prisms - One can take [Unsorted/etale cohomology](Unsorted/etale%20cohomology) of varieties, and later refine to [schemes](schemes), and thus take it for the base field even when it's not algebraically closed and extract arithmetically interesting information. - [prismatic cohomology](prismatic%20cohomology.md), meant to relate a number of other cohomology theories - [prism](Unsorted/prismatic%20cohomology.md) : a pair $(A, I)$ where $A$ is a commutative ring with a derived Frobenius lift $\phi:A\to A$, i.e. a $\delta\dash$structure. - $I \normal A$ is an ideal defining a [Cartier divisor](Cartier%20divisor). - $A$ is $(P, I)\dash$complete. - Any ideal generator $d\in I$ satisfies $\phi(d) = d^p + p\cdot u, u\in A\units$. - Fix a scheme and study prisms over it. Need these definitions to have stability under base-change. - Examples: - $A \da \ZZ_p$ and $\phi = \id$ with $I = \gens{ p }$ yields [crystalline cohomology](Unsorted/crystalline%20cohomology.md). - $A \da \ZZ_p\fps{u}, \phi(u) = u^p$. Then $I = \gens{ E(u) }$ is generated by an [Eisenstein polynomial](Eisenstein%20polynomial.md). Here $A/I = \OO_K$ - Prismatic [site](Unsorted/site.md) : fix a base prism $(A, I)$ for $X$ a \(p\dash \)adic [formal scheme](formal%20scheme) over $A/I$. Define \[ (X/A)_\prism = \ts{ (A, I) \to (B, J) \in \Mor(\Prism), \Spf(B/J) \to X \text{ over } A/I } ,\] topologized via the [flat topology](flat%20topology) on $B/J$. - There is a [structure sheaf](structure%20sheaf.md) $\OO_{\prism}$ where $(B, J) \to B$. Take $\RR \Gamma$, which receives a [Frobenius](Frobenius) action, to define a cohomology theory. Why is this a good idea? - Absolute [prismatic](Unsorted/prismatic%20cohomology.md) sites: for $X\in \Sch(\padic)$, define \[ X_\prism \da \ts{ (B, J) \in \Prism,\, \Spf(B/J) \to X } .\] Take [sheaf cohomology](sheaf%20cohomology.md) to obtain $\RR\Gamma_\prism(X) \da \RR \Gamma(X_\prism, \OO_\prism) \selfmap_\phi$. - The category $\Prism$ doesn't have a final object, so has interesting cohomology. Relates to the [algebraic K theory](Unsorted/K-theory.md) of $\ZZ_p$? - Questions: let $X_{/\ZZ_p}$ be a smooth [formal scheme](formal%20scheme). - What is the [cohomological dimension](cohomological%20dimension.md) of $\RR \Gamma_\prism(X)$? - What are the [F-crystals](Unsorted/Hodge%20F-crystal.md) on $X_\prism$? - Produce finite flat $B\dash$modules? - Bhatt and Lurie: found a stacky way to understand the absolute prismatic site of $\ZZ_p$. Drinfeld found independently. - Construction due to Simpson: take $X\in \Var(\Alg)$, define a de Rham presheaf \[ X_{\dR}: \CCalg^{\fp} &\to \Set \\ R &\mapsto X(R_\red) .\] - 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](Cartier-Witt%20stack), a.k.a. the prismatization of $\ZZ_p$ - Define $\WCart$ to be the [formal stack](formal%20stack) on $p\dash$complete rings. - Plug in a $p\dash$nilpotent ring $R$ to extract all (derived) prism structure on $W(R)$. - Prisms aren't [base change](base%20change.md) compatible without the derived part. - This is a [groupoid](groupoid.md). - An explicit presentation: $\WCart_0(R)$ are distinguished [Witt vectors](Archive/AWS2019/Witt%20vectors.md) 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. $\WCart = \WCart_0 / W^*$ is a presentation as a [stack quotient](stack%20quotient). - Receives a natural Frobenius action, which is a [derived Frobenius lift](derived%20Frobenius%20lift). - Start by understanding its points, suffices to evaluate on fields of characteristic $p$. - If $k\in \Field(\Perf)_{\chp}$, $\WCart(k) = \ts{ \pt }$, with the point represented by $(W(k), ?)$. - Yields a (geometric?) point $x_{\FFp}: \spec(\FFp) \to \WCart$. - Analogy to understanding [Hodge-Tate cohomology](Hodge-Tate%20cohomology). Similar easy locus in this stack. - Take 0th component of distinguished [Witt vectors](Archive/AWS2019/Witt%20vectors.md) to get a diagram \begin{tikzcd} \WCart &&& {\hat{\AA^1}/\GG_m} \\ \\ \\ {\WCart^{\mathrm{HT}}} &&& {B\GG_m} \arrow[dashed, from=4-1, to=4-4] \arrow[from=1-1, to=1-4] \arrow[hook, from=4-4, to=1-4] \arrow[hook, from=4-1, to=1-1] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMCwwLCJcXFdDYXJ0Il0sWzMsMCwiXFxoYXR7XFxBQV4xfS9cXEdHX20iXSxbMywzLCJCXFxHR19tIl0sWzAsMywiXFxXQ2FydF57XFxtYXRocm17SFR9fSJdLFszLDIsIiIsMCx7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX1dLFswLDFdLFsyLDEsIiIsMSx7InN0eWxlIjp7InRhaWwiOnsibmFtZSI6Imhvb2siLCJzaWRlIjoidG9wIn19fV0sWzMsMCwiIiwxLHsic3R5bGUiOnsidGFpbCI6eyJuYW1lIjoiaG9vayIsInNpZGUiOiJ0b3AifX19XV0=) - The bottom-left is this [Hodge-Tate stack](Hodge-Tate%20stack) - Now has a better chance of being an [algebraic stack](Unsorted/stacks%20MOC.md) instead of a [formal stack](formal%20stack). Bottom arrow kills the formal direction. - Will be [classifying stack](classifying%20stack.md) of a [group scheme](group%20scheme) : need to produce a point and take automorphisms. - Take the distinguished element $V(?) \in W(\ZZ_p)$. Produces a map \[ \Spf(\ZZ_p) \mapsvia{\pi_\HT} \WCart^\HT .\] - Fact: $\pi_\HT$ is a flat cover and $\Aut(\pi_\HT) = W^*[ F]$. - **Upshot**: $\WCart^\HT = \B W^* [F]$ is a classifying stack. [quasicoherent sheaves](Unsorted/quasicoherent%20sheaf.md) on the left and representations of the (classifying stack of the) group scheme on the right. I.e. $\D_\qc(\WCart^\HT) = \RR (W^*[F])$. - [Teichmüller lift](Teichmüller%20lift) yields a $\ZZ/p$ grading on the LHS. - Something about Deligne-Illusie? [Unsorted/Hodge-to-deRham spectral sequence](Unsorted/Hodge-to-deRham%20spectral%20sequence.md) - Upshot: a [divisor](divisor.md) inside is easy to understand. - Fact: $\D_\qc(\WCart)$ are equivalent to \[ \inverselim_{(A, I)\in \Prism} \D_{(P, I)-?}(A) .\] - Diffracted Hodge cohomology: let $X\in \Schf_{\ZZ_p}$. Get a prismatic structure sheaf using the assignment $(A, I) \to \RR \Gamma_\prism \qty{ (X\tensor A/I) / A}$. - Heuristic: $\spec \ZZ_p$ should be 1-dimensional over something. - Get an absolute comparison: $\cohdim \RR \Gamma_\prism (X) \leq d+1$ where $d = \reldim X_{/\ZZ_p}$. - There is a deRham comparison: \[ X_{\FFp}^* H_\prism(X) \cong \RR \Gamma_\dR (X_{\FFp}) .\] - There is a [Hodge-Tate comparison](Hodge-Tate%20comparison): the object $H_\prism(X)$ restricted to $\WCart^\HT$ has an increasing filtration with $\gr_i = \RR \Gamma(X, \Omega^i_X)[-i]$. - Use representation interpretation, then $\mu_p \actson \gr_i$ by weight $-i$. - Combine these comparisons to get [Deligne-Illusie](Deligne-Illusie) : if $\reldim X < p$, then \[ \RR \Gamma_\dR(X_{\FFp}) \cong \bigoplus_{i} \RR \Gamma(X_{\FFp}, \Omega^i[-i]) .\] Get a lift to characteristic zero, yields Hodge-to-deRham degeneration there. - An $F\dash$crystal on $X_\prism$ is a vector bundle $\bundle{E} \in \Vect(X_\prism, \OO_\prism)$? Plus some extra data. - Infinite tensor product: \[ I_\prism \tensor F^* I_\prism \tensor (F^2)^* I_\prism \tensor \cdots .\] Converges to some object \( \OO_\prism \ts{ 1 } \in \Pic(X_\prism, \OO_\prism ) \), twisted? Yields isomorphism of sheaves after inverting $I_\prism$, \[ F^* \OO_\prism \ts{ 1 } \cong I_\prism\inv \tensor \OO_\prism \ts{ 1 } .\] - Convergence: this is a formal stack, any [thickening](thickening.md) are identified with something... On each finite approximation, most terms are $\OO_X$. - Some analog of [Artin-Schreier](Artin-Schreier.md) here, taking fixed points? - There is a natural functor from $F\dash$crystals on $X$ to local $\ZZ_p$ systems on a geometric fiber $X_?$? - Main theorem: produces [descent](descent.md) data, uses work on Beilinson fiber sequence (Benjamin Antieau, Morrow, others?) - Can say \[ H^i(\Delta_{\ZZ_p}) = \begin{cases} \ZZ_p & i=0 \\ \prod_{\NN} \ZZ_p & i=1. \end{cases} \] Can compute using [HH](Unsorted/HH.md)? $\pi_{-1}( \TP(\ZZ_p) )$ is where the $i=1$ part comes from. - [HH](Unsorted/HH.md) corresponds to [prismatic cohomology](prismatic%20cohomology.md) - [THH](Unsorted/HH.md) corresponds to [Hodge-Tate cohomology](Hodge-Tate%20cohomology). - Prismatic is filtered by things that look like Hodge-Tate - Absolute = arithmetic (take eigenspaces, related to [motivic cohomology](motivic%20cohomology.md), relative = geometric? - Link to [K-theory](Unsorted/K-theory.md) comes from eigenspaces somehow. - Similar to situation in [etale cohomology](Unsorted/etale%20cohomology): need absolute and relative to compute either. ## Why are [triangulated categories](triangulated%20categories.md) and [derived categories](derived%20category.md) important? - Homological algebra lives in the [derived category](derived%20category.md) - In AG, tight link between [birational](Unsorted/birational.md) equivalence (of say smooth [projective varieties](projective%20varieties) and equivalence of $\derivedcat{ \Coh X }$, the derived category of [coherent sheaves](coherent%20sheaves.md) - See the [Bondal-Orlov conjecture](Bondal-Orlov%20conjecture). - [birational](birational.md) is a weakening of isomorphism between varieties - Being derived equivalent is a weakening of having equivalent $\DCoh$ - Both recover actual isomorphisms in the case of smooth projective varieties - Rep theory: having equivalent derived categories is called [Morita equivalence](Morita%20equivalence.md). - [Derived equivalence](Derived%20equivalence) is a weakening of [Morita equivalence](Morita%20equivalence.md) - Can replace an algebra by a much simpler derived-equivalent one - Use to study [blocks](blocks) of [group algebra](group%20algebra.md) - Lots of numerical consequences? ## A Roadmap to [Hill-Hopkins-Ravenel](Hill-Hopkins-Ravenel.md) [Roadmap to HHR](https://mathoverflow.net/questions/257885/roadmap-to-hill-hopkins-ravenel) ## Some [Lurie stuff](Lurie%20stuff) [Lurie's Seminar on Algebraic Topology](http://people.math.harvard.edu/~lurie/915.html) [A bunch of suggested papers](http://people.math.harvard.edu/~lurie/915list.pdf) [Lurie's Topics in Geometric Topology](http://people.math.harvard.edu/~lurie/937.html) ### The Relationship Between [THH](Unsorted/HH.md) and [K-theory](Unsorted/K-theory.md) > Some remarks on $\THoH$ and $K\dash$Theory, no clue what the original source was: - [algebraic K theory](Unsorted/K-theory.md) is hard, using [Topological Hochschild homology](Unsorted/HH.md) somehow makes computations easier. - \(K\dash\)theory says something about [vector bundles](vector%20bundles.md), [topological Hochschild cohomology](Unsorted/HH.md) describes [monodromy](monodromy) of vector bundles around infinitesimal loops - For $X$ a nice [scheme](scheme.md), take $LX$ the derived [free loop space](free%20loop%20space) : the [derived stacks](derived%20stacks.md) $\Map_\DSt(S^1, X)$. - Points of $LX$: infinitesimal loops in $X$ - Identify $\THoH(X) \mapsvia{\sim} \OO(LX)$ (global functions) - Corollary of a result in Ben-Zvi--Francis--Nadler "Integral Transforms and Drinfeld Centers in [derived algebraic geometry](derived%20algebraic%20geometry.md)"? - [Dennis trace](Dennis%20trace) : a comparison $K(X) \to \THoH(X)$, takes $E\in \VectBundle$ 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 $S^1\dash$[equivariant](equivariant.md) because loops! Just equip $K(X)$ with the trivial $S^1$ action. - Take [homotopy fixed points](homotopy%20fixed%20points.md) to get something smaller than $\THoH$: $\THC^-$, topological negative cyclic homology - See Connes' [negative cyclic homology](Unsorted/HH.md) - [Dennis trace](Dennis%20trace) is invariant under *all* covering maps of circles, even multisheeted - Encoded not in a group action by a [cyclotomic structure](cyclotomic%20structure). - Take [homotopy fixed points](homotopy%20fixed%20points.md) of the cyclotomic structure on $\THoH$ to get $\TC$, [Topological cyclic homology](Unsorted/Topological%20cyclic%20homology.md) - There is a map $K\to \TC$ - Theorem of Dundas-Goodwillie-McCarthy: whenever $A\to A'$ is a nilpotent extension of connective [ring spectra](ring%20spectra.md), $$ K(A') \mapsvia{\sim} K(A) \fiberprod{\TC(A)} \TC(A') $$ ### [Eilenberg-MacLane spaces](Eilenberg-MacLane%20spaces.md) - Some good stuff from Akhil Mathew on EM spaces: - [Blog post](https://amathew.wordpress.com/2010/12/06/eilenberg-maclane-spaces/amp/) ### Why Care About [Stacks](Projects/2022%20Algebraic%20Geometry%20Oral%20Exam/500%20Stacks%20and%20Moduli.md)? - Why shouldanyone care about stacks? #why-care ![](attachments/image_2021-04-21-19-17-08.png) ![](attachments/image_2021-04-21-19-17-15.png) - Why should I care about [derived stacks](derived%20stacks.md)? #todo/questions - Note from Arun: one can get [TMF](Topological%20modular%20forms) and [tmf](Topological%20modular%20forms) along with their ring structures without doing [Unsorted/obstruction theory](Unsorted/obstruction%20theory.md) ## Homotopy Theory is Connected to [Lie algebra cohomology](Lie%20algebra%20cohomology) ![](attachments/image_2021-04-21-20-24-13.png) ## [schemes](schemes) and [class field theory](class%20field%20theory.md) - Definitions of schemes and scheme-y curves ![](attachments/image_2021-04-21-20-29-40.png) - Definitions of schemes and scheme-y curves ![](attachments/image_2021-04-21-20-29-40.png) - Grothendieck's fundamental group ![](attachments/image_2021-04-21-20-31-13.png) - Statement of class field theory in terms of fundamental groups ![](attachments/image_2021-04-21-20-32-14.png) - See [Arithmetic schemes](Arithmetic%20schemes) - [Idele group](Idele%20group) for [arithmetic schemes](arithmetic%20schemes) ![](attachments/image_2021-04-21-20-33-57.png) -Actual class group for schemes ![](attachments/image_2021-04-21-20-34-25.png) - [Wiesend’s finiteness theorem](Wiesend’s%20finiteness%20theorem) is one of the strongest and most beautiful results in higher [Global class field theory](Global%20class%20field%20theory)? - The main aim of higher [global class field theory](Unsorted/class%20field%20theory.md) is to determine the abelian fundamental group $\pi_1^{\ab}(X)$ of a regular [arithmetic scheme](arithmetic%20scheme) $X$, i.e. of a connected [regular scheme](regular%20scheme.md) [separated scheme](separated%20scheme) scheme [flat morphism](Unsorted/faithfully%20flat.md) and of [finite type](finite%20type.md) over $\ZZ$, in terms of an arithmetically defined [class groups](class%20groups) - $C(X)$. - Fundamental theorem of class field theory? ![](attachments/image_2021-04-21-20-38-59.png)