All Questions
23
questions
2
votes
1
answer
276
views
Geometric Interpretation of absolute Hodge cohomology
$\quad$Let $\mathcal{Sch}/\mathbb C$ denote the category of schemes over $\mathbb C$. For an arbitrary $X\in\mathcal{Ob}(\mathcal{Sch}/\mathbb C)$, Deligne in his Article defined a polarizable Hodge ...
user508433
12
votes
1
answer
876
views
Comparing singular cohomology with algebraic de Rham cohomology
Let $X$ be a smooth projective variety over a number field $K$. Then there are two cohomology groups we can attach to $X$: the algebraic de Rham cohomology group
$H^k_{\text{dR}}(X/K), $
which is a ...
- 1,889
2
votes
1
answer
120
views
Checking formality of a perfect complex Zariski-locally
Let $X$ be a locally Noetherian scheme and $K^{\bullet}$ a perfect complex of $\mathcal{O}_X$-modules. We say $K^{\bullet}$ is "formal" if it is quasi-isomorphic to the complex $\bigoplus_{n}...
user178246
10
votes
0
answers
465
views
How do I produce a basis of cohomology?
Suppose I am discussing a smooth projective variety over an algebraically closed field with my friend on the phone and I want to make a statement about its $l$-adic cohomology (integral or rational). ...
user158636
9
votes
0
answers
350
views
Would full resolution of singularities have cohomological implications beyond the alteration theory?
De Jong's result on alterations allows one to show the potential semistability of certain Galois representations arising from cohomology of varieties (among other things). If we knew the existence of ...
user158636
13
votes
2
answers
932
views
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 ...
user158636
3
votes
0
answers
283
views
Kummer theory if $\ell = p$
Background. Let $k$ be a field and let $\ell$ be an integer which is divisible in $k$. Then one has a short exact sequence of abelian étale sheaves
$$ 0 \to \mu_\ell \to \mathbb{G}_m \xrightarrow{(\,\...
- 317
18
votes
1
answer
3k
views
Conjectures of Peter Scholze about q-de Rham complex: examples
Peter Scholze formulated several conjectures about $q$-de Rham complex in the paper
Canonical $q$-deformations in arithmetic geometry, Ann. Fac. Sci. Toulouse Math. (6) 26 (2017), no. 5, pp 1163–...
- 4,316
56
votes
2
answers
10k
views
What is prismatic cohomology?
Prismatic cohomology is a new theory developed by Bhatt and Scholze; see, for instance, these course notes. For the sake of the community, it would be great if the following question is discussed in ...
- 2,711
16
votes
0
answers
499
views
Are there smooth and proper schemes over $\mathbb Z$ whose cohomology is not of Tate type
Is there an example of smooth and proper scheme $X \to \mathrm{Spec}(\mathbb Z)$, and an integer $i$ such that $H^i(X, \mathbb Q)$ is not a Hodge structure of Tate type?
Alternatively: such that $H^...
- 171
6
votes
0
answers
244
views
Torsors for discrete groups in the etale topology
Let $S$ be a smooth variety over $\mathbb C$ or a smooth quasi-projective integral scheme over Spec $\mathbb{Z}$.
Let $G$ be an (abstract) discrete group. For instance, $G =\mathbb{Z}^n$ or $G$ a ...
- 61
15
votes
2
answers
2k
views
Meaning of the determinant of cohomology
The Arakelov intersection number on arithmetic surfaces is defined as an "extension" of the classical intersection number on algebraic surfaces. It was introduced to get a nice intersection theory ...
- 319
7
votes
0
answers
466
views
independence of $\ell$ for $p$-adic cohomology of varieties over finite fields
Let $X/k$ be a smooth projective geometrically integral variety ($X = A$ an Abelian variety suffices) over $k = \mathbf{F}_q$ with absolute Galois group $\Gamma$, $\bar{X} = X \times_k \bar{k}$, $q = ...
user19475
0
votes
0
answers
79
views
Good cohomological setting for binary operations on arithmetical functions
Is there currently a good abstract theory (derived from algebraic geometry and cohomological theories) to study binary operations on arithmetical functions like the Dirichlet convolution $$f\star g = \...
- 1,007
4
votes
1
answer
607
views
finitness of syntomic/fppf cohomology with coefficients in a finite flat group scheme
Let $X/k$ be a smooth projective variety over a finite field of characteristic $p$ and $\mathscr{A}/X$ be an Abelian scheme.
Is then $H^1_\mathrm{SYN}(X,\mathscr{A}[p]) = H^1_\mathrm{fppf}(X,\mathscr{...
user19475