17
votes
Is there an $\mathbb{R}$-valued cohomology theory for varieties over $\mathbb{F}_p$?
Let me advertise a conjectural positive answer to a slightly different question. This actually works not only over $\mathbb F_p$ but over its algebraic closure $\overline{\mathbb F}_p$.
Consider the ...
- 21.3k
12
votes
Applications of Crystalline Cohomology for Physics
A talk that explores the physics connection to crystalline cohomology in the context of string theory is Motives and Strings by Jan Stienstra, with a challenge for each community:
Why look for a ...
- 188k
12
votes
Accepted
Is there an $\mathbb{R}$-valued cohomology theory for varieties over $\mathbb{F}_p$?
The answer is no. If $A$ is a simple abelian variety over $\mathbb{F}_p$, then $End(A)\otimes\mathbb{R}$ cannot act on a real vector space of dimension $2dim(A)$ if the center of the endomorphism ...
- 136
11
votes
Accepted
Verifying the Lefschetz Conditions for crystalline cohomology
The Hard Lefschetz theorem can certainly not be deduced formally from the axioms of a Weil cohomology theory given in the Stacks Project. The reason it is called "hard" Lefschetz is that it ...
- 40.2k
9
votes
Accepted
Pairing of cotangent and tangent bundles
For (1), recall that if $R$ is a ring, then a derivation $D: R \to R$ satisfies the Leibniz rule, which by induction on $n$ implies that if $D^n$ denotes the $n$-fold iterate of $D$, then
$$D^n(fg) = \...
- 5,760
8
votes
Accepted
Non-existence of power divided structure on a maximal ideal of truncated polynomial rings (example from Koblitz)
Suppose that $I$ admits a divided power structure. On the one hand, $\gamma_p(x_1x_2+x_3x_4+x_5x_6)$ has to be equal to zero because the element $x_1x_2+x_3x_4+x_5x_6$ is zero in our ring, but let's ...
- 7,377
5
votes
Accepted
F-crystals from crystalline cohomology
I shall try to stick to the notation in Katz's paper. Let $k$ be a perfect field of characteristic $p>0$. Let $S_{\infty}$ be a $p$-adically complete and separated smooth formal $W(k)$-scheme and $...
- 1,404
5
votes
Accepted
Can Hodge symmetry fail if there is a lift to $W_2$ and the crystalline cohomology is torsion-free?
There are counterexamples (at least for some $p$) even if we assume that $X$ lifts all the way to a (non-algebraizable) formal scheme over $\mathbb{Z}_p$. See e.g. Theorem 4.1 in https://arxiv.org/pdf/...
- 7,377
5
votes
Accepted
Frobenius automorphisms of cohomology of a variety
I think you are intending to take everything with smooth proper varieties (or else there is no good notion of a prime of good reduction).
In this case, the two Frobenius elements have the same ...
- 148k
5
votes
Accepted
D-modules as ind-coherent sheaves over positive characteristics?
The following link to a set of notes seems to contain I was looking for. It's about so-called crystalline spaces associated to proper and separated smooth schemes (which are very similar to de Rham ...
- 1,516
5
votes
Frobenius actions on de Rham cohomology of ordinary elliptic curves
It's important to be clear that this map on $H^1_{\mathrm{dR}}$ overlies a highly non-trivial map on the base-ring $R$. You can imagine a case where $R$ is something like $\mathbf{Z}_p\langle X \...
- 37k
5
votes
Accepted
On a series of lectures of Deligne on crystalline cohomology in characteristic $0$
After asking Mr Le Stum for the scan which Emily talked about in the comments to the original post, he kindly sent it to me, with permission to share it publicly. It is, according to him, a "bad ...
- 584
4
votes
Reconstruct a variety from its crystalline topos
Edit: This answer is probably wrong, sorry. The issue is indicated [in bold] below.
Yes, we can reconstruct $X$ (as a $k$-scheme).
It would be kind of trivial if you had asked for the small Zariski ...
- 1,110
4
votes
Interpretation of the formal groups arising from the DeRham-Witt complex
This is an old question but since it hasn't received much attention, let me just point out "the next" example beyond that given in the question:
Let $k$ be a perfect field of characteristic $...
- 1,404
3
votes
Compute de Rham-Witt sheaves
This is explained in both:
Corollaire 2.15, p. 561 in Luc Illusie "Complexe de de Rham-Witt et cohomologie cristalline", Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 12 ...
3
votes
Accepted
(crystalline cohomology version's) Tate's conjecture for K3 surfaces
This is related to a phenomenon called 'hypersymmetry' (or the lack of it). The term comes from Chai and Oort, who explored this for abelian varieties. The essential point is that the category of F-...
- 3,032
3
votes
Accepted
Non-abelian Berthelot comparison?
Yes. A Google search will immediately give you lots of articles in varying generality (see e.g. work of Shiho), but one article I'm particularly fond of is Kim and Hain's "A de Rham-Witt approach to ...
- 1,404
3
votes
Accepted
Reference request: Newton above Hodge
Frobenius and the Hodge Filtration of the generic fiber has an alternative proof of "Newton over Hodge".
- 328
2
votes
Accepted
Can $h^{1, 0}$ and $h^{1, 1}$ jump for smooth projective surfaces over $\mathbb{Z}[1/N]$?
Yes. An Enriques surface with classical reduction at $p=2$ gives such an example. See Illusie "Complexe de de Rham-Witt et cohomologie cristalline" Prop. II 7.3.8(b), p. 658.
- 16.1k
2
votes
Accepted
About the filtration of crystalline cohomology
Let me upgrade my comment to an answer:
The answer is no, the filtration is not independent of the lifting. In fact the relationship between liftings of $Y$ and filtrations lifting the Hodge ...
- 1,404
1
vote
(crystalline cohomology version's) Tate's conjecture for K3 surfaces
I see where is the problem.
First of all, the right statement of Tate conjecture is for $X$ over $\mathbb{F}_q$ (not over $\overline{\mathbb{F}_q}$!), and Tate conjecture predicts:
$c_1: Pic(X)\otimes\...
- 547
1
vote
Accepted
Choice of topology in the (log) crystalline site
It seems that the categories of log-crystals in the strict etale and Kummer etale topologies are not actually equivalent to one another, contrary to what I expected.
Here's a sketch of a proof. ...
- 1,790
Only top scored, non community-wiki answers of a minimum length are eligible