Questions tagged [etale-cohomology]
for questions about etale cohomology of schemes, including foundational material and applications.
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General cohomology groups and motives
Let $X$ be a variety over $\mathbb{Q}$. Let $\mathcal{F}$ be a sheaf on $X$. Then we have an action of $Gal(\mathbb{Q})$ on $H_{et}^i(X,\mathcal{F})$. In certain cases we can say a lot about this ...
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vanishing of étale cohomology groups with small support with values in an abelian scheme
Let $S/k$ be a smooth variety and $A/S$ be an abelian scheme. Let $Z \hookrightarrow S$ be a reduced closed subscheme of codimension $\geq 2$.
I want to show that in this situation, $H^i_Z(S, A) = 0$ ...
user19475
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Etale cohomology in the $p$-adic setting
Can we hope for application of Etale cohomology techniques in proving results concerning semialgebraic subsets of $\mathbb{Q}_p^n$?
Recall that semialgebraic subsets are obtained from $p$-adic ...
user16974
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is the presheaf of automorphisms a sheaf?
In Chapter III,$\S 4$ of Milne's Etale cohomology a correspondence between twisted forms and Cech cohomology cocycles is described.
Fix some Grothendieck topology, say, etale, and let $A$ be a ...
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Textbook for Etale Cohomology
What is the best textbook (or book) for studying Etale cohomology?
user16974
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Galois cohomology of generic points of formal completions (of components of a hypercovering of the subvariety): examples or general statements?
Let $Y$ be a closed smooth subvariety in a (smooth) affine variety $X$. What can one say about the etale cohomology of the generic points of the formal completion of $X$ along $Y$ i.e. about the ...
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Is the Gelfand-Graev character isomorphic to a cohomology group for some sheaf on a Deligne-Lusztig variety?
Deligne-Lusztig theory
is awesome. You take a maximal torus $T$, you take a character $\theta$, construct a variety $X_T$$^*$, take etale cohomology, get a virtual character $R_T^\theta$, maybe it's ...
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Why does a group action on a scheme induce a group action on cohomology?
This is probably totally obvious but I have no clue how this is done: Say you have an endomorphism $f:X \rightarrow X$ of schemes. Why (if true, perhaps some additional assumptions are necessary!) do ...
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The etale fundamental group and etale cohomology with compact support
Before me, the following was asked:
etale fundamental group and etale cohomology of curves
However, that question dealt only with projective curves.
Question
Let $X$ be any scheme (or if you prefer ...
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Basic properties of Nisnevich cohomology; $l'$-topology?
I would like to know more about Nisnevich cohomology (especially, on its properties that could be easily formulated). In particular, I would like to know which of the following statements are true, ...
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reading off invariants of a scheme $X$ from $D^b_c(X, \bar{\mathbf{Q}}_\ell)$
Which invariants of a scheme $X$ can be read off from $D^b_c(X, \bar{\mathbf{Q}}_\ell)$ (the bounded derived category of $\bar{\mathbf{Q}}_\ell$-sheaves on $X$, see e.g. [Kiehl-Weissauer])?
user19475
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The restriction of the Gersten resolution (for etale cohomology) onto a closed subvariety.
There is a very important result of Bloch and Ogus: for any smooth variety $X$ and fixed $r\in \mathbb{Z}$, $r\ge 0$, $l$ is prime to the residue field characteristic, the Zariski sheafification of ...
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Ordered Cech(-like) complexes that compute etale cohomology (of fields!)
It is well known (cf. Equivalence of ordered and unordered cech cohomology. ) that for 'usual' topologies one can compute the cohomology of sheaves either using unordered Cech complexes or ordered ...
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On inverse images with respect to Zariski-etale topology.
For a variety $X$ I define its Zariski-etale site as follows: the category is the category of etale $X$-schemes, and the coverings are Zariski ones. Note that this topology is more coarse than the ...
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What are the easiest cases of base change (for sheaves on sites)?
I have a closed embedding of schemes $i:X'\to X$, and for each of them I consider three Grothendieck topologies for the category of the corresponding (relatively) \'etale schemes: the \'etale one, the ...
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Do inverse images respect flabby sheaves?
Let $i:Y\to X$ be a closed embedding of varieties, and let $S$ be a flabby \'etale (or Nisnevich) sheaf of abelian groups on $X$. Is $i^*S$ flabby also? I am mostly interested in the case when $S=i_{x*...
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Comparison of etale and formal etale cohomologies for l=p
Let $K$ be a finite extension of $\mathbb{Q} _p$ with a field of integers $\mathcal{O} _K$. Let $X$ be a semistable proper scheme over $\mathcal{O} _K$, and $\mathcal{X}$ the associated p-adic formal ...
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Functoriality properties of the perverse $t$-structure for torsion (constructible complexes of) sheaves
I would like to apply the usual 'functoriality properties' of the perverse $t$-structure to torsion (constructible complexes of) sheaves (I am in the algebraic setting, so these are etale sheaves, ...
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On 'special properties' of various 'sheaf image' functors for a local complete intersection morphism
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 ...
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Finiteness of étale Cohomology Groups
Mr. Milne, in "Étale Cohomology", gives the following proposition (p.224, Corollary VI.2.8):
Proposition: Let $F$ a constructible sheaf on $X_{et}$, the small étale site of $X$, $X$ proper over a ...
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On (the cohomology of) Hensel pairs
I would like to study the cohomology of the Henselization $H_X(Z)$ of a closed subvariety $Z$ of a variety $X$.
I would like the following facts to be true (and to make sense!:)).
a.) The motivic ...
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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 ...
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Does one need to sheafify when defining the inverse image of a sheaf with respect to an embedding?
This seems to be a rather simple (stupid?:)) question; yet I was not able to find an answer quickly.
For a morphism $f:X\to Y$ of schemes (or topological spaces) and an (etale or topological) sheaf $...
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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....
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Principal bundles in the etale and Zariski topology and extensions of the structure group
Say $G$ is a reductive group over a field $k$. I usually take $k = \mathbb{C}$ so assume what you want about the field except maybe that its finite. If $X$ is a scheme over $k$ then a principal $G$ ...
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Cohomologie Etale
Is there an English translation available for Deligne's Cohomologie Etale (Arcata) that is now part of the SGA 4 1/2 ?? Atleast for the first two sections - Grothendieck Topologies and Etale Topology.
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locally constant constructible sheaves and finite etale coverings
Maybe it is well known to experts or maybe it is just a stupid idea, but I will ask any way.
We know that if $X$ is a topological space, then there is an equivalence of categories between the ...
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The category of l-adic sheaves
I'm currently trying to understand the construction of the category of l-adic constructible sheaves as in SGA5, and it seems that quite a lot of machinery (the MLAR condition, localization of the ...
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Galois action on the derived category of $l$-adic sheaves
Hi,
Suppose that $G$ is a group acting on a scheme $X$, and $F$ is an $l$-adic sheaf on $X$ EDIT: with an action of $G$ (thanks Torsten).
Is it true that $R\Gamma(X,F)$ is well-defined as an object ...
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global complete intersection and independence of $l$
Hello,
I remember reading that if $X/\mathbf F_q$ is a projective smooth global complete intersection, then the characteristic polynomial of the $\mathbf F_q$-linear Frobenius of $X$
on $H^i_{et}(X\...
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Additive form of Hilbert 90 for schemes?
First, I am by no means well-versed on cohomology so I apologize if this is too elementary.
I have been going through some basics of etale cohomology, with my ultimate goal being an understanding of ...
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Can proper-smooth base change be used to show that varieties cannot be lifted to characteristic zero?
Recall the following corollary to the proper and smooth base change theorems:
Let $\pi: X \to S$ be a proper, smooth morphism. Then the direct images $R^i \pi_* \mathcal{F}$ are locally constant ...
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Serre and Tate's conjectures on étale cohomology
In the appendix of Serre and Tate "Good Reduction of Abelian Varieties" [Annals of Mathematics 88 (1968), 492-517], the authors make the following conjectures.
Suppose that $X$ is a smooth proper ...
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Special case of Leray spectral sequence
I am looking for a reference for what is stated in Srinivasan's book "Representations of Finite Chevalley Groups", which is apparently a special case of Leray spectral sequence. I'll quote the ...
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For an l-adic sheaf (F_n), why is the complex F_n of finite Tor dimension?
Let $X$ be a variety and let $\mathcal{F}=(\mathcal{F}_n)_{n\geq 0}$ be a (constructible) $\ell$-adic sheaf on $X$. Let $K_n$ be the object in the derived category $D(X,\mathbf{Z}/\ell^{n+1})$ of ...
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structure of $T_\ell A$ for $A/\mathbf{F}_q$ an abelian variety
Can someone give me references for the structure of the $G_{\mathbf{F}_q}$-module $T_\ell A$, $A/\mathbf{F}_q$ an abelian variety?
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Galois descent for K-groups (or for étale cohomology groups)
Let $F/K$ be a Galois extension of number fields with Galois group $G$. Let $\mathcal{O}_F$ and $\mathcal{O}_K$ be the associated rings of integers, and let $n\geq 1$.
When is
$$
K_{2n-1}(\...
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How is etale cohomology of integer rings related to Galois cohomology?
In the paper of Bloch and Kato in the Grothendieck Festschrift, and some other papers relating to the Bloch-Kato conjecture and the ETNC, the cohomology groups
$H^i_{\mathrm{et}}(\operatorname{Spec} ...
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Twisted forms and $\check{H}^1$
I am reading Milne's Étale cohomology, III.4.
A twisted form of an object $Y$ (a scheme, a sheaf of modules, of algebras...) over a scheme $X$ is an object $Y'$ such that there exists a covering in ...
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how to find the varieties whose cohomology realizes certain representations?
The cohomology of Shimura varieties and Drinfeld shtukas is conjectured to realize the representations sought for in the Langlands programme/conjectures, the cohomology of Deligne-Lusztig varieties ...
user19475
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Definition of etale neighbourhood
In Milne's EC, he defines an etale neighbourhood of a point $x \in X$ by a pair $(Y,y)$ with an etale morphism $f: Y \to X$ where $f(y) = x$, such that $k(x) = k(y)$.
What I don't understand is this ...
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Galois cohomology groups given by étale cohomology
What are cases when Galois cohomology groups are given by étale cohomology?
Example: $S = Spec(K)$ the spectrum of a field, $F \in Sh(K)$, then $H^p(K, F) = H^p(G_K, F_{\bar{K}})$.
What if $G = \...
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prove statement in Galois cohomology by étale cohomology
According to Milne's Arithmetic Duality Theorems, Proposition I.6.4: $0 \to H^1(G_S, A[m]) \to H^1(K, A[m]) \to \oplus_{v \not\in S}H^1(K_v, A)$ for an abelian variety $A$ and a nonempty set of primes ...
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When inverse image is conservative; a reference or a generalization?
I am interested in the following question: for $f$ being a morphism of schemes, which conditions ensure that $Rf^*_{et}$ is conservative? This is true if $f$ has a section or if $f$ is an \'etale ...
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Do all the main properties of constructible and perverse sheaves (in an 'arithmetic' situation) follow from results of Gabber?
This question is a continuation of Bad behaviour of perverse sheaves over 'general' bases?
Let $S$ (for example) be a finite type separated scheme over $\mathbb{Z}$. I would like: (1) to ...
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Stalks of etale sheaves
Let $\pi:X \rightarrow Y$ be a finite morphism of schemes and $\mathfrak{F}$ be an etale sheaf on $X$. Then for a $y \in Y$ we have the stalk $(\pi_{*}\mathfrak{F})_{y}=\prod_{\pi(x)=y}\mathfrak{F}_{x}...
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Bad behaviour of perverse sheaves over 'general' bases?
Could one define $\mathbb{Q}_l$-perverse etale sheaves over more or less general (excellent, separated) base scheme by combining the results of Gabber and Ekedahl? Would their functoriality properties ...
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Concrete interpretations of higher (sheaf) cohomology groups
$H^1$ has an interpretation as torsors. But what about the higher $H^i$ (in the setting of algebraic geometry, and étale or flat cohomology)? For example $H^2(X, \mathbf{G}_m)$ is (often) isomorphic ...
user19475
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If one wants to work with $Q_l$-adic sheaves, should the scheme be of finite type over a 1-dimensional one?
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 ...
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Could the Kunneth decomposition of a motif depend on the choice of $l$?
Suppose that over some (algebraically closed) field $K$ of characteristic $p>0$ we have: numerical equivalence of cycles coincides with homological one with respect to ${\mathbb{Q}}_{l}$ and ${\...
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