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7
votes
1answer
234 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 ...
12
votes
0answers
218 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 ...
9
votes
0answers
199 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
186 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 ...
6
votes
1answer
378 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 ...
22
votes
0answers
921 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 ...
3
votes
1answer
334 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 $$ ...
6
votes
1answer
350 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 ...
11
votes
1answer
972 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 ...
6
votes
0answers
198 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 ...
6
votes
1answer
371 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
488 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 ...
8
votes
1answer
906 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
619 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 ...
7
votes
1answer
523 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?
16
votes
1answer
1k 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
491 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). ...
19
votes
6answers
5k views

learning crystalline cohomology

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