All Questions
Tagged with cohomology etale-cohomology
19 questions with no upvoted or accepted answers
10
votes
0
answers
479
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
691
views
In Mann's six-functor formalism, do diagrams with the forget-supports map commute?
One of the main goals in formalizing six-functor formalisms is to obtain some sort of "coherence theorem", affirming that "every diagram that should commute, commutes". In these ...
- 711
9
votes
0
answers
703
views
étale vs syntomic vs flat cohomology
Let $\mathscr{A}/X$ be an abelian scheme over $X$ of characterisitic $p$. The étale topology is not fine enough for the Kummer sequence for $\mathscr{A}$ to be (right) exact, but the syntomic and flat ...
user19475
7
votes
0
answers
483
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
6
votes
0
answers
247
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
5
votes
0
answers
213
views
Which pure categories related with Weil cohomology theories are semi-simple?
As far as I understand, the category of pure polarizable Hodge modules is semi-simple, whereas the cohomology of the corresponding schemes is graded polarizable. Is it true that one doesn't have any ...
- 16.9k
4
votes
0
answers
477
views
Cohomology of BG, algebraically
Let $k$ be a field (algebraically closed if you will) and $G$ be a connected reductive group over $k$. I would like to know a purely algebraic computation of the cohomology of $BG$, as the quotient ...
- 805
3
votes
0
answers
69
views
How would you call morphisms of varieties that induce isomorphisms on etale cohomology in low degrees?
In our text we have several statements of the following sort: for a certain morphism $f:X\to Y$ of varieties over an (algebraically closed) field of characteristic $p$ and some $c>0$ the ...
- 16.9k
3
votes
0
answers
292
views
What is the integral cohomology of an Enriques surface over a finite field?
Probably this is well-known, but I could not find it. I would like to understand the integral $2$-adic etale cohomology of an Enriques surface over $\mathbb{F}_q$ in dimension 2: $H_{et}^2(X, \mathbb{...
- 2,809
3
votes
0
answers
217
views
Reference: Relative cohomology of a morphism
Let $f\colon Y \to X$ be a morphism of schemes, the inverse image in $K$-theory always fit into a long exact sequence
$$
\cdots \to K_i(f)\to K_i(X) \xrightarrow {f^*} K_i(Y)\to \cdots
$$
where the ...
- 2,871
3
votes
0
answers
230
views
limit of étale cohomology with supports
I would like to know if the following generalisation of Milne Étale Cohomology, Lemma III.1.16, p. 88 holds: (all cohomology groups with respect to the étale topology)
Let $Y \hookrightarrow X$ quasi-...
user19475
2
votes
0
answers
245
views
Proof of the projection formula (for cohomology of $\mathbf{P}V$)
Let $V\to X$ be a vector bundle (over say a scheme).
Then the cohomology of its projectivisation is
$$\text{H}^*(\mathbf{P}V)\ =\ \text{H}^*(X)[t]/(t^{n+1}+c_1(V)t^n+\cdots+c_n(V))$$
as an algebra, ...
- 5,701
2
votes
0
answers
156
views
Rigid \'etale cohohomology of flag variety minus its rational points e.g $p$-adic Drinfeld half plane
Let $Fl=G/B$ over $\mathbb Q_p$ be the flag variety of a quasi-split reductive group $G$ over $\mathbb Q_p$, then $X=Fl-Fl(Q_p)$ shall exist as a rigid analytic variety over $\mathbb Q_p$, how to ...
- 6,622
2
votes
0
answers
281
views
Can one compute the (etale) cohomology with support at a point for a "big" regular $k$-scheme via limit arguments?
I am trying to understand the coniveau spectral sequence for the cohomology of a "big" regular scheme over a field. This involves cohomology with support at points, and I am getting some strange ...
- 16.9k
2
votes
0
answers
347
views
l-adic cohomology and perverse sheaves
Let consider the map $tr:\mathbb{G}_{m}^{n}\rightarrow\mathbb{A}^{1}_{\mathbb{F}_{q}}$ given by the sum of the coordinates and let $\psi:\mathbb{F}_{q}\rightarrow\mathbb{Q}_{l}^{*}$ a non trivial ...
- 3,472
1
vote
0
answers
116
views
A certain 'coniveau-like' filtration for cohomology: what can one say about the intersection of $Ker H^i(X)\to H^i(Z)$ for $Z$ running through subvarieties of $X$ of dimension $m$?
Let $X$ be a smooth variety (say, a complex one; denote its dimension by $n$). What can one say about the intersection of $Ker (H^i(X)\to H^i(Z))$ for $Z$ running through (closed, not necessarily ...
- 16.9k
1
vote
0
answers
232
views
What can one say about a smooth variety whose lower cohomology is trivial?
Let $X$ be a smooth (quasi-affine) complex variety; suppose that its cohomology (say, with integral coefficients) is trivial in degrees $0 < i\le s$ (for some $s>0$). What can one say about such ...
- 16.9k
1
vote
0
answers
333
views
Does n-multiplication maps of cohomology groups vanish if it vanishes at the 0th cohomology?
In general, we know that a morphism $f=(f ^ {q})$ between universal (cohomological) $\delta$ functors $S=(S ^ {q}),T=(T ^ {q})\ $vanishes if and only if $f ^ {0} \ \colon \ S^{0} \to T^{0}$ vanishes....
- 945
0
votes
0
answers
628
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"Cohomology with compact support isomorphic to homology" theorems
I am collecting theorems throughout different fields which say roughly something of the form "Cohomology with compact support isomorphic to homology".
I'm studying this situation (and its ...
user30211