# Questions tagged [crystalline-cohomology]

The tag has no usage guidance.

61 questions
Filter by
Sorted by
Tagged with
172 views

### About the filtration of crystalline cohomology

Suppose $K$ is an finite unramified extension of $\mathbb Q_p$ with residue field $k$, and let $Y$ be an proper smooth variety defined over $k$. We know if $Y$ admits a proper smooth lifting $X/W(k)$ ...
1 vote
160 views

### Crystalline fibre of a morphism of Galois cohomology stacks

Let $K = \mathbb{Q}_p$, $G = G_K$ its absolute Galois group. Let $$1\longrightarrow A\longrightarrow B\longrightarrow C\longrightarrow 1$$ be a split exact sequence of (not necessarily abelian) group ...
437 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 ...
278 views

### About an argument in absolute prismatic cohomology

In Bhatt-Lurie Absolute prismatic cohomology, proof of Corollary 4.1.15, it asserts that extension of scalars along the quotient map is conservative and preserves small limits: I think the ...
548 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 ...
197 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 $\mathcal{O}_{\mathcal{C}}$. ...
188 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 ...
446 views

544 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 $H^i_{crys}(X)$ ...
422 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....
428 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 ...
240 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 ...
1 vote
168 views

829 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 ...
261 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 ...