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for questions about etale cohomology of schemes, including foundational material and applications.

1 vote
0 answers
423 views

Another stupid question on l-adic sheaves: does a generically zero constructible sheaf vanis...

Suppose that the stalk of a constructible l-adic ($\mathbb{Z}_l$-adic or $\mathbb{Q}_l$-adic) etale sheaf $S$ over a generic (Zariski) point of a variety $V$ is zero. Does this imply that $S$ vanishes …
Mikhail Bondarko's user avatar
1 vote

global complete intersection and independence of $l$

As was pointed above, this statement was proven by Deligne for general smooth projective varieties. Yet in your case you don't require the theory of Deligne's weights, since you have only one cohomolo …
Mikhail Bondarko's user avatar
2 votes
0 answers
275 views

Can one compute the (etale) cohomology with support at a point for a "big" regular $k$-schem...

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 resul …
Mikhail Bondarko's user avatar
7 votes
1 answer
263 views

'Cohomologically approximating' a $\mathbb{Q}[[t]]$-scheme by a one over the henselization o...

For certain matters the henselization $R$ of $\mathbb{Q}[t]$ at $0$ is a 'reasonable approximation' for $\mathbb{Q}[[t]]$ (Artin's approximation theorem and so on). Now, I would like to study certain …
Mikhail Bondarko's user avatar
1 vote
0 answers
114 views

A certain 'coniveau-like' filtration for cohomology: what can one say about the intersection...

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 smoo …
Mikhail Bondarko's user avatar
0 votes
0 answers
211 views

On 'special properties' of various 'sheaf image' functors for a local complete intersection ...

Let $f:X\to Y$ be a local complete intersection morphism (of schemes or varieties) of (relative) dimension $c$ everywhere. Is it true that $f^!\cong f^*[2c]$ (as a functor between the derived categori …
Mikhail Bondarko's user avatar
6 votes
0 answers
305 views

Does one need l to be invertible in S in order to consider the l-adic cohomology of S-scheme...

When Ivorra defines the $l$-adic realization of $S$-motives (i.e. of Voevodsky's motives over a scheme $S$) he demands $l$ to be invertible in $S$. Is this condition really necessary? What happens wit …
Mikhail Bondarko's user avatar
7 votes
1 answer
688 views

Questions on standard (motivic) conjectures

Over an (algebraically closed) characteristic $p$ field, is it known that the cohomological equivalence of cycles relation (with respect to $\mathbb{Q}_l$-adic \'etale cohomology) does not depend on …
Mikhail Bondarko's user avatar
4 votes
2 answers
566 views

If one wants to work with $Q_l$-adic sheaves, should the scheme be of finite type over a 1-d...

In section 6 of his 'Adic Formalism' T. Ekedahl states that $l$-adic sheaves 'behave nicely' for finite type separated schemes over $S$ that is regular of dimension $\le 1$. Is the dimension restricti …
Mikhail Bondarko's user avatar
1 vote
0 answers
236 views

Can etale $X$-schemes be lifted to $Y$, where $X$ is closed in $Y$?

For a closed embedding (of varieties) $X\to Y$ let $U/X$ be etale. Is is true that there necessarily exists an etale $U'/Y$ such that $U'_X=U$? If this is wrong in general, are there any assumptions t …
Mikhail Bondarko's user avatar
1 vote
1 answer
463 views

The cohomology of a $G_m$-bundle

Let $X$ be a smooth variety over an algebraically closed field (whose characteristic could be positive), $Y\to X$ is a $G_m$-bundle ($G_m=\mathbb{A}\setminus \{0\}$). Then I want to have a long exact …
Mikhail Bondarko's user avatar
5 votes
1 answer
2k views

A Kunneth formula for the etale cohomology of the product of ('simple') varieties over not (...

If $X$ and $Y$ are varieties over an algebraically closed field, then in the corresponding derived category of complexes we have $RH_{et}(X\times Y,\mathbb{Z}/l^n\mathbb{Z})\cong RH_{et}(X,\mathbb{Z}/ …
Mikhail Bondarko's user avatar
7 votes
2 answers
2k views

Weights for etale cohomology: why does Deligne's definition work?

For a field $K$ and a variety $X/K$ (whose characteristic could be $0$) I need a 'simple' explanation for the (Deligne's) method of defining weights of the $l$-adic etale cohomology of $\overline{X}$ …
Mikhail Bondarko's user avatar
3 votes
2 answers
264 views

$Pic(X)/l=0$ in terms of $H^*_{et}(X,\mu_{l^n})$?

I would like to calculate Picard groups of certain schemes over fields; I'm mostly interested in the question whether $Pic(X)$ is infinitely $l$-divisible, i.e. whether $Pic(X)/l=0$, $l$ is a prime di …
Mikhail Bondarko's user avatar
0 votes
1 answer
478 views

The fibres of smooth projective families over all geometric points have isomorphic cohomolog...

Let $p:P\to S$ (and $p':P\to S$) be proper smooth morphisms of 'nice' schemes (one may assume that $S$ is a complex variety). It is well-known that the fibres of $p$ (and $p'$) over all geometric poin …
Mikhail Bondarko's user avatar

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