# Questions tagged [crystalline-cohomology]

The crystalline-cohomology tag has no usage guidance.

61
questions

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### About the filtration of crystalline cohomology

Suppose $K$ is an finite unramified extension of $\mathbb Q_p$ with residue field $k$, and let $Y$ be an proper smooth variety defined over $k$. We know if $Y$ admits a proper smooth lifting $X/W(k)$ ...

1
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1
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160
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### Crystalline fibre of a morphism of Galois cohomology stacks

Let $K = \mathbb{Q}_p$, $G = G_K$ its absolute Galois group. Let
$$1\longrightarrow A\longrightarrow B\longrightarrow C\longrightarrow 1$$
be a split exact sequence of (not necessarily abelian) group ...

3
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2
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### (crystalline cohomology version's) Tate's conjecture for K3 surfaces

Let $X$ be a K3 over $\overline{\mathbb{F}_p}$. The (crystalline version's) Tate conjecture predicts:
$c_1: Pic(X)\otimes\mathbb{Q}_p\rightarrow H^2_{crys}(X/W)^{\Phi=p}\otimes\mathbb{Q}_p$
is an ...

2
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0
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### About an argument in absolute prismatic cohomology

In Bhatt-Lurie Absolute prismatic cohomology, proof of Corollary 4.1.15, it asserts that extension of scalars along the quotient map is conservative and preserves small limits:
I think the ...

7
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### Is there a cohomology theory wider than crystalline?

We know crystalline cohomology is calculated by taking an inverse limit:
$$H_{cris}^i:=\varprojlim_nH_{cris}^i(X/W_n(k))$$
provided $X$ projective smooth over a perfect field $k$ of char $p$.
I want ...

3
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0
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197
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### Nygaard filtration on Fontaine's period ring

Let $K$ be a discretely valued extension of $\mathbb{Q}_p$ with perfect residue field $k$, and $\mathcal{C}$ a completed algebraic closure of $K$ with the ring of integers $\mathcal{O}_{\mathcal{C}}$. ...

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188
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### Compute de Rham-Witt sheaves

I am really new to this, but I am having a hard time understanding all the de Rham-Witt construction.
It seems to be really difficult to compute anything with those beasts: like I cannot find any ...

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1
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446
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### Pairing of cotangent and tangent bundles

I am reading the survey paper: "The de-Rham Witt complex and Crystalline cohomology" by Luc Illusie.
In math line (2.1.12), Illusie considers the pairing $\langle-,-\rangle:\Omega_{X/S}^1\...

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249
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### de Rham Witt complex vs. de Rham complex of the Witt ring

I am reading the paper "Revisiting the de Rham-Witt complex" by Bhatt-Lurie-Mathew and I am a bit confused about the difference between $W\Omega_R^*$ and $\hat{\Omega}^*_{W(R)}$.
Let $\...

4
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1
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544
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### A comparison theorem between crystalline cohomology and étale cohomology

Suppose $X/\mathbb F_q$ is a smooth projective variety. Katz-Messing (eudml) shows that the characteristic polynomial of the Frobenius on $H^i_{et}(\overline{X},\mathbb Q_\ell)$ and $H^i_{crys}(X)$ ...

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### D-modules as ind-coherent sheaves over positive characteristics?

There is an interpretation of D-modules over "sufficiently nice" prestacks $X$ (read: various finiteness conditions apply, perhaps even smoothness) by Gaitsgory and Rozenbylum (see chapter I....

3
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428
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### The cycle class map with values in crystalline cohomology

Let $ k = \mathbb{F}_q $ be a finite field of characteristic $ p > 0 $.
Let $ X $ be a smooth proper scheme of dimension $ d $ over $ k $.
Consider the associated $ K $ - linear cycle class map ...

7
votes

1
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240
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### Choice of topology in the (log) crystalline site

Let $X$ be a scheme or fs log scheme over a finite field. There seem to be several slightly different definitions of the (log) crystalline site of $X/S$ available in the literature, depending on ...

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0
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168
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### How to construct a sheaf on the infinitesimal site from a stratified module

Let $X\to S$ be a morphism of schemes.
Proposition 2.11 of the book "Notes on crystalline cohomology" by Berthelot and Ogus states that a stratified $\mathcal{O}_X$-module $(E,\{\...

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359
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### About derived divided power envelope

Assume $A$ is a $\mathbb{Z}_{(p)}$-algebra with ideal $I$ and $A,A/I$ are $p$-torsionfree.
In this survey, Akhil Mathew defines the derived divided power envelope $LD_I(A)$ in Construction 7.15, after ...

5
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### Pushforward of crystals in mixed/positive characteristic

Is there a good reference for pushforward of crystals along smooth maps in mixed/positive characteristic with respect to the crystalline site? Intuitively I'm confused on what the pushforward looks ...

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### Verifying the Lefschetz Conditions for crystalline cohomology

For context, I am rather new to the whole business of abstract Weil cohomology theories and motives in general, so if I am not making sense somewhere, do let me know!
In many of the literature that I ...

4
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0
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### Looking for a crystalline analogue of , $\mathcal {Z}_{\sim}^* (X)_F \simeq \mathcal{Z}_{\sim}^* (X_{k ' })_F^{\mathrm{Gal} ( k ' / k )} $

Is there a crystalline analogue for the following formula, using the crystalline Frobenius $ F_v $ instead of the absolute Galois group $ \mathrm{Gal} (\overline{k} / k) $ ?
Here is the formula, which ...

2
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0
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265
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### Looking for the exact and the precise statement of Ogus conjecture

I have been looking for several weeks for the exact and the precise statement of Ogus conjecture, but, I cannot find it.
The only book which made me discover the statement of this conjecture is that ...

1
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0
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250
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### Berthelot-Ogus comparison isomorphism

On the link, page, $ 2 $, the Berthlot-Ogus isomorphism theorem is stated as follows,
We have a canonical isomorphism, $$ \rho_{\mathrm{cris}} \ : \ H_{\mathrm{cris}}^{i} (X) \otimes_{K_ {0}} K \to H_{...

4
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1
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### Frobenius actions on de Rham cohomology of ordinary elliptic curves

In appendix 2 of Katz's "p-Adic properties of modular schemes and modular forms", he describes a certain "Frobenius" endomorphism on the de Rham cohomology of ordinary elliptic ...

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### $F$-isocrystals defined via a lift of a scheme

Let $X$ be a smooth affine scheme over a finite field $k$. Then there exists a smooth affine formal scheme $\mathfrak{X}$ over $W(k)$ with a lift $\sigma$ of the Frobenius. A convergent $F$-isocrystal ...

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### Isocrystals on simply connected varieties

Esnault and Shiho - Convergent isocrystals on simply connected varieties proves that there are no non-trivial convergent isocrystals on simply connected varieties. There is another similar result in ...

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### Pullback in crystalline cohomology

Let $f:X\to Y$ be a morphism of schemes over a perfect field $k$. If $f$ is flat, is the pullback $f^*$ from quasi-coherent crystals on $Y$ to quasi-coherent crystals on $X$ exact? This is pullback of ...

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### On a series of lectures of Deligne on crystalline cohomology in characteristic $0$

In the introduction of Berthelot's book on crystalline cohomology [Ber74], one finds, on page 11, the following passage:
i) des travaux de P. Deligne ([14], non publiés) prouvant en particulier le ...

12
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2
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893
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### Is there an $\mathbb{R}$-valued cohomology theory for varieties over $\mathbb{F}_p$?

If $E$ is a supersingular elliptic curve over $\mathbb{F}_{p^m}$ with $m\geq 2$ its endomorphism ring is a maximal order in a quaternion algebra ramified at $p$ and $\infty$ so there can't be a Weil ...

16
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1
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964
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### Reconstruct a variety from its crystalline topos

Let $k$ be a perfect field of positive characteristic. Let $X$ be a smooth projective geometrically connected $k$-scheme with a $k$-point.
Can we reconstruct $X$ from its small crystalline topos $((X/...

4
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1
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363
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### Can Hodge symmetry fail if there is a lift to $W_2$ and the crystalline cohomology is torsion-free?

Let $f:X\to \mathrm{Spec}\:\mathbb{F}_p$ be a smooth proper morphism with $p>\mathrm{dim}\:X$. Assume that $H^i_{\mathrm{crys}}(X/\mathbb{Z}_p)$ is torsion-free for all $i\geq 0$ and that there is ...

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1
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### Can $h^{1, 0}$ and $h^{1, 1}$ jump for smooth projective surfaces over $\mathbb{Z}[1/N]$?

Let $N$ be a positive integer. Let $f:X\to S=\mathrm{Spec}\:\mathbb{Z}[1/N]$ be a smooth projective morphism of relative dimension 2 such that $R^1f_*\mathcal{O}_X$ and $R^2f_*\mathcal{O}_X$ are both ...

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### Frobenius action on de Rham cohomology

Let $X$ be a smooth projective $k$-scheme, where $k=\mathbb{F}_p$ and $p$ is prime. We have an identification of the de Rham cohomology of $X$ with $H^*_{crys}(X/k)$: $H^*_{DR}(X/k)\cong H^*_{crys}(X/...

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### Non-abelian Berthelot comparison?

Berthelot's comparison theorem connects the algebraic de Rham cohomology of a $\mathbb{Z}_p$-scheme and the crystalline cohomology of its special fiber. Is there a statement on the level of homotopy ...

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### Explicit computations with crystalline cohomology

I am currently studying crystalline cohomology and while all the talk about crystalline topoi is nice, I would like to see some explicit computations. What are some references on this subject which ...

4
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### When does a morphism of schemes induce a morphism of crystalline sites (not topoi)?

Here it is stated that the crystalline site of a scheme is not functorial in general. Is there a non-tautological characterization of morphisms of schemes which in fact do induce morphisms of ...

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### Reference request: Newton above Hodge

Let $K$ be a p-adic field, and let $\mathcal{O}$ be the ring of integers inside $K$ with residue field $k$. Let $\mathcal{X}$ be a smooth proper formal scheme over $\mathcal{O}$ (with topology given ...

6
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### Frobenius automorphisms of cohomology of a variety

Suppose $X$ is a smooth variety defined over $\mathbb{Q}$. There are (at least) two automorphisms of cohomology groups of $X$ that are called "Frobenius", and I would like to understand how they are ...

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### flat/crystalline cohomology of abelian variety

Let $A/k$ be an abelian variety over an algebraically closed field and $\ell \neq \mathrm{char}\,k$.
In http://jmilne.org/math/articles/1986b.pdf, Theorem 15.1(b) it is proved that $$H^r_{et}(A, R) = \...

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### Universal property of $A_{\mathrm{cris}}/p^n$

It is well known that the ring $A_{\mathrm{cris}}$ of Fontaine is the universal $p$-adically complete divided power thickening of $\mathcal{O}_{\mathbb{C}_p}$ over $\mathbb{Z}_p$; in fact, this is one ...

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### Applications of Crystalline Cohomology for Physics

I know this is a very vague question, so I may restrict it to quantum theories for their more category theoretic setting. Even if the concept of crystalline cohomology is very abstract, it has made me ...

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### Torsionfree crystalline cohomology implies torsionfree etale cohomology?

Let $X$ be a smooth projective variety over an algebraically closed field $k$ of characteristic $p>0$. Let $W=W(k)$ be the ring of Witt vectors of $k$.
Assume that the crystalline cohomology $H^2_{...

9
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### Grothendieck's motivation of crystalline cohomology

Here Illusie mentions Grothendieck's observation that using Gauss-Manin connection one can give a non-canonical isomorphism between de Rham cohomology of smooth schemes over $W(k)$ with isomorphic ...

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### Interpretation of the formal groups arising from the DeRham-Witt complex

In the accepted answer to this question, it is shown that for a proper algebraic variety $X$ we have that $H^{r-i}(X, W\Omega^i)[1/p]$ has slopes from the interval $[i, i+1[$, so namely is isomorphic ...

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### Are the Eigenvalues of the Frobenius on Crystalline cohomology bounded by degree?

Excuse me if this question is trivial or trivially false, or not at this sites level. Lets work over an algebraically closed field of characteristic $p>0$, say $k$. The action of the Frobenius on $...

5
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### A crystalline version of an isomorphism of Beauville and Donagi

Let $k$ be an algebraically closed field of characteristic $p>0$ and write $W:=W(k)$ for its ring of Witt vectors. Consider a smooth cubic fourfold $X_{0}\subset\mathbb{P}^{5}_{k}$ and let $F(X_{0})...

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1
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### Morphisms for good reduction are maps respecting filtration

Please see edits below!
So, let $A,A'/K$ be abelian varieties where $K$ is a $p$-adic local field with residue field $k$. Suppose further that they have good reduction with models $\mathscr{A},\...

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### Is the Dieudonne module actually a cohomology group?

One often times thinks of the Dieudonne module $M(X)$ of a $p$-divisible group (say over $k$, a perfect characteristic $p$ field) as being some sort of cohomology theory
$$M:\left\{p\text{- divisible ...

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### Extension of Messing-Mazur-Oda to general groups

The following may be well-known (or obviously false), but I can't find a counterexample or a reference.
Suppose that $k$ is some perfect field (one can assume algebraically closed, if that makes you ...

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363
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### Are there good properties of the divided power completion map?

Let $Y \to X$ be a closed immersion of smooth schemes over, say, the ${\rm Spec}(\mathbb{Z}_p)$. The completion map $$X_{/Y}\to X$$ is an ind-closed immersion (sometimes called pseudo-closed immersion)...

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### What is the current state of the crystalline analogue of the Weil conjectures?

In "F-isocrystals on open varieties results and conjectures" Faltings says:
"Finally, we extend the theory of weights and show as much as possible of the crystalline analogue of the Weil ...

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1k
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### The most important facts, modern surveys, and readable introductions to p-adic cohomology theories (crystalline cohomology and the mysterious functor)

I would like to organize a seminar on crystalline cohomology; I dream of understanding the Beilinson's recent paper on the mysterious functor (http://www.ams.org/journals/jams/2012-25-03/S0894-0347-...

4
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798
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### Base change in crystalline cohomology?

Does one have a base change theorem in crystalline cohomology like in étale cohomology?
Suppose one has the following cartesian diagram
$$
\newcommand{\ra}[1]{\!\!\!\!\!\!\!\xrightarrow{\quad#1\quad}...