Questions tagged [crystalline-cohomology]
The crystalline-cohomology tag has no usage guidance.
62
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learning crystalline cohomology
From which sources would you learn about crystalline cohomology and the de-Rham-Witt complex?
25
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0
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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-...
20
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1
answer
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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 $Y$ ...
19
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2
answers
3k
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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 ...
19
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0
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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 ...
16
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1
answer
883
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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 ...
16
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1
answer
970
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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/...
14
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1
answer
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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 ...
13
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2
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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 ...
11
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1
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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 ...
10
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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 ...
10
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638
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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 ...
10
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1
answer
833
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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 ...
9
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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},\...
9
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0
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850
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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 ...
9
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0
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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 ...
8
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1
answer
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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 ...
8
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1
answer
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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?
8
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1
answer
471
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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....
8
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1
answer
712
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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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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 ...
8
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0
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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
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1
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680
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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 ...
7
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1
answer
247
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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 ...
7
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0
answers
558
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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 ...
7
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0
answers
355
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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 ...
7
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0
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268
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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 ...
6
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1
answer
392
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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 ...
6
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1
answer
560
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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 ...
6
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0
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317
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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 ...
6
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0
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257
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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 ...
5
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2
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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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1
answer
817
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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}...
5
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0
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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 ...
5
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0
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428
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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_{...
5
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0
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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})...
4
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1
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477
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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\...
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 ...
4
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390
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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 ...
4
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1
answer
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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 ...
4
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1
answer
580
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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 $H^i_{crys}(X)$ ...
4
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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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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 ...
4
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0
answers
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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 ...
3
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1
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369
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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_*(\...
3
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2
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468
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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 ...
3
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0
answers
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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 $\mathcal{O}_{\mathcal{C}}$. ...
3
votes
0
answers
461
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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 ...
3
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0
answers
386
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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/...
3
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0
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721
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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 ...