Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [crystalline-cohomology]

The tag has no usage guidance.

2
votes
1answer
103 views

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 ...
8
votes
0answers
121 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 ...
5
votes
0answers
76 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 ...
11
votes
0answers
363 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 ...
6
votes
1answer
274 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 ...
8
votes
1answer
450 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) = ...
6
votes
0answers
201 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 ...
8
votes
1answer
385 views

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 ...
5
votes
0answers
325 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_{...
7
votes
0answers
559 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 ...
7
votes
0answers
212 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 ...
5
votes
2answers
795 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 $...
5
votes
0answers
246 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})...
8
votes
1answer
467 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},\...
18
votes
0answers
746 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 ...
10
votes
0answers
293 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 ...
2
votes
0answers
251 views

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)...
8
votes
1answer
524 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 ...
24
votes
0answers
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-...
4
votes
1answer
556 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}...
6
votes
1answer
445 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 ...
17
votes
2answers
2k 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 ...
7
votes
0answers
257 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 ...
7
votes
1answer
508 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 ...
3
votes
0answers
638 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 ...
12
votes
1answer
1k 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 ...
9
votes
1answer
702 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 ...
8
votes
1answer
652 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?
17
votes
1answer
2k 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 $Y$ ...
2
votes
0answers
539 views

a counterexample of Serre vs. motivic cohomology

There is a counterexample of Serre showing that there is no Weil cohomology theory with coefficients in $\mathbf{Q}, \mathbf{Q}_p, \mathbf{R}$ over $\mathbf{F}_{p^2}$ (a supersingular elliptic curve). ...
26
votes
6answers
7k views

learning crystalline cohomology

From which sources would you learn about crystalline cohomology and the de-Rham-Witt complex?