Questions tagged [crystalline-cohomology]

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6 answers
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learning crystalline cohomology

From which sources would you learn about crystalline cohomology and the de-Rham-Witt complex?
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25 votes
0 answers
1k views

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-...
Mikhail Bondarko's user avatar
20 votes
1 answer
3k views

Crystalline cohomology via the syntomic site

Hello, Let $k$ be a field of characteristic $p > 0$, and let $Y$ be a $k$-scheme. Consider the sites $Y_{syn}$ and $(Y/W_n)_{cris}$ (where $W_n$ are the Witt vectors of $k$ of length $n$), of $Y$ ...
Nicolás's user avatar
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19 votes
2 answers
3k views

Letter from Grothendieck to Tate on "crystals"

I have downloaded from this link a quite poor quality scan of the letter dating May 1966 that Grothendieck sent to Tate mentioning his ideas about generalizing Monsky-Washnitzer cohomology. I am ...
Filippo Alberto Edoardo's user avatar
19 votes
0 answers
1k views

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 ...
Alex Youcis's user avatar
16 votes
1 answer
883 views

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 ...
Emily's user avatar
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16 votes
1 answer
970 views

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/...
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14 votes
1 answer
2k views

What are p-adic period rings?

I'm reading Illusie's survey on Crystalline cohomology, and I found him talking about those $p$-adic period rings like $B_{\text{dR}}, B_{\text{cris}}$. Can anybody explain what they are and give some ...
Yuhao Huang's user avatar
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13 votes
2 answers
922 views

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 ...
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11 votes
1 answer
1k views

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 ...
S. Li's user avatar
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10 votes
1 answer
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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 ...
Tatu's user avatar
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10 votes
1 answer
638 views

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 ...
M. Carmona's user avatar
10 votes
1 answer
833 views

Crystalline analogue of perverse sheaves

Consider a variety $X$ over a field $k$ and let $\ell$ be a prime different from the characteristic of $k$. One has the derived category $D(X, Q_{\ell})$ of $\ell$-adic sheaves. There are very ...
SGP's user avatar
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9 votes
1 answer
536 views

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},\...
Alex Youcis's user avatar
9 votes
0 answers
850 views

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 ...
SashaP's user avatar
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9 votes
0 answers
401 views

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 ...
Alex Youcis's user avatar
8 votes
1 answer
655 views

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 ...
Dat Minh Ha's user avatar
  • 1,472
8 votes
1 answer
795 views

The Galois representation of a p-divisible group is crystalline

Can someone explain (or give a reference) why the Galois representation attached to a p-divisible group over the ring of integers of a p-adic ring is Crystalline?
hadimath's user avatar
  • 137
8 votes
1 answer
471 views

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....
Dat Minh Ha's user avatar
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8 votes
1 answer
712 views

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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8 votes
0 answers
286 views

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 ...
user avatar
8 votes
0 answers
291 views

simple proof of relation between H^1 crystalline and Dieudonne module?

Hi, Let $k$ be a perfect field of characteristic $p > 0$. Let $A/k$ be an abelian variety. Then the first crystalline cohomology group of $A$ with respect to $W(k)$ (= Witt vectors) is canonically ...
Nicolás's user avatar
  • 2,802
7 votes
1 answer
680 views

How to calculate zeroth crystalline cohomology

I am just learning crystalline cohomology, so I understand the basic set-ups. But I can't really do any calculations. For example, let's choose the base $S=W(k)/p^n$, and let $X$ be an affine scheme ...
natura's user avatar
  • 1,503
7 votes
1 answer
247 views

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 ...
Alexander Betts's user avatar
7 votes
0 answers
558 views

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 ...
Richard's user avatar
  • 481
7 votes
0 answers
355 views

$F$-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 ...
curious math guy's user avatar
7 votes
0 answers
268 views

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 ...
Pax's user avatar
  • 821
6 votes
1 answer
392 views

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 ...
Julian Rosen's user avatar
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6 votes
1 answer
560 views

Vanishing cohomology of de-Rham Witt complex

Let $X$ be a smooth scheme over $\mathbb{F}_{p}$ for a prime number $p$. As far as I understand, there is a surjective morphism from $\Omega^\bullet_{W\mathcal{O}_X} \to W \Omega_{X}^\bullet$ which ...
Jana's user avatar
  • 2,022
6 votes
0 answers
317 views

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 ...
J.P. Gimori's user avatar
6 votes
0 answers
257 views

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 ...
O-Ren Ishii's user avatar
5 votes
2 answers
1k views

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 $...
Pax's user avatar
  • 821
5 votes
1 answer
817 views

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}...
Lan's user avatar
  • 699
5 votes
0 answers
104 views

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 ...
davik's user avatar
  • 2,035
5 votes
0 answers
428 views

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_{...
Monsie's user avatar
  • 81
5 votes
0 answers
303 views

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})...
Oli Gregory's user avatar
  • 1,259
4 votes
1 answer
477 views

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\...
kindasorta's user avatar
  • 1,473
4 votes
1 answer
397 views

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 ...
Leray Jenkins's user avatar
4 votes
1 answer
390 views

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 ...
user avatar
4 votes
1 answer
200 views

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 ...
user197402's user avatar
4 votes
1 answer
580 views

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 $H^i_{crys}(X)$ ...
Asvin's user avatar
  • 7,648
4 votes
0 answers
257 views

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 $\...
Jun Koizumi's user avatar
4 votes
0 answers
119 views

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 ...
Bradley04's user avatar
  • 487
4 votes
0 answers
102 views

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 ...
user avatar
3 votes
1 answer
369 views

F-crystals from crystalline cohomology

In Section 7 of Katz' paper: https://web.math.princeton.edu/~nmk/old/travdwork.pdf He asserts "Crystalline cohomology tells us that for each integer $i \geq 0$, the de Rham cohomology $H^i=Rf_*(\...
onefishtwofish's user avatar
3 votes
2 answers
468 views

(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 ...
Yuan Yang's user avatar
  • 537
3 votes
0 answers
225 views

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 $\mathcal{O}_{\mathcal{C}}$. ...
user145752's user avatar
3 votes
0 answers
461 views

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 ...
Bradley04's user avatar
  • 487
3 votes
0 answers
386 views

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/...
Vitay's user avatar
  • 91
3 votes
0 answers
721 views

Explicit description of O^{cris}_n in Fontaine/Messing

Let $k$ be a perfect field of characteristic $p$, $W(k)$ the Witt ring and $K$ its quotient field. In their article "$p$-adic periods and $p$-adic etale cohomology" Fontaine and Messing give in II.1.4 ...
Matthias Kümmerer's user avatar