# Tagged Questions

the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conjectures.

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### Are these connecting homomorphisms commutative?

Are the connecting homomorphism induced by Kummer sequence and that of localization sequence commutative? In other words, is the following statement true? If it is true, then, how can one prove it? ...
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### why is the local field Q_l not an etale sheaf over a scheme X?

I would like to know the reason why the local field Q_l is not an etale sheaf over a scheme X while its ring of integers Z_l can be regarded as a constant etale sheaf?
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### Lemmas on etale cohomology with compact support from the book 'Arithmetic Duality Theorems'

I was reading Milne's book "Arithmetic Duality Theorems". On page 166 there are a lot of useful lemmas on the etale cohomology with compact support on S-integers. However, I get confused when I tried ...
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### Quotients of Tate modules

Let $p$ be a prime number, let $K$ denote a finite extension $\mathbb{Q}_{p}$ and let $\overline{K}$ be an algebraic closure of $K$. Let $A$ be an ellitpic curve over $K$ and denote by $T_{p}A$ its ...
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### Formality of algebraic varieties via l-adic cohomology?

The cohomology with say real coefficients of any smooth projective algebraic variety is formal, by Deligne, Griffiths, Morgan, Sullivan: Real homotopy theory of Kähler manifolds, Inv. Math. 29, 245-...
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### etale homotopy and Adams conjecture

I was reading Quillen's paper on "Some remarks on etale homotopy theory and a conjecture of Adams". Up to some fact about etale version of spherical fibrations which was later proved by Friedlander, ...
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### Long exact sequence of cohomology and fibration

In an article I'm reading, the author is stating : $O$ is isomorphic to the complement of a zero section of a line bundle over $X$. We have a long exact sequence of (étale) cohomology associated ...
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### classify \mu_n torsors

Recently I read in Milne's book "etale cohomology" that the set $H^1(X,\mu_n)$ ($X$ a scheme, $n$ a nature number, the cohomology is flat cohomology) can be described as the set of pairs $(L,\phi)$, ...
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### is generically split Azumaya algebra locally split?

Let $A$ be an Azumaya algebra over a scheme $X$ (or maybe more specifically a scheme of finite type over a field). Suppose that the restriction of $A$ to $U=X\setminus Z$ (where $Z$ is a closed set) ...
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### Detecting zero morphisms via an open subscheme and its complement.

In the setting described in Bernstein, Beilinson, and Deligne, associated to a scheme $X$, a closed subscheme $i: Z \to X$ and its open complement $j: U \to X$ we have six functors between the ...
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### Poincare pairing and polarization of Hodge structure. Kuga-Satake construction.

If $X$ is a K3 surface over the complex (algebraic) then I wonder if the poincare pairing induces a polarization on the Hodge structure $H^2(X,\mathbb Z)$? The point is that I see that when ...
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### project proposal: English translation of Deligne's “La conjecture de Weil : II” [closed]

First of all, I hope this "question" is appropriate here. If not, please delete it. I would like to propose a translation project of Deligne's "La conjecture de Weil : II" 52_137_0">http://www.numdam....
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Let $X$ be an affine variety of dimension $n$ (say, over complex numbers). Then Artin's vanishing theorem yields: both singular and etale cohomology $H^i(X):=H^i(X,\mathbb{Z}/l\mathbb{Z})=0$ for $i>... 3answers 605 views ### étale cohomology with values in an abelian scheme is torsion? Let$A/X$be an abelian scheme. Is$H^n(X,A)$torsion for$n > 0$? Perhaps this can be proved analogously as Proposition IV.2.7 of Milne's Étale cohomology (where it is proved that the ... 0answers 342 views ### General cohomology groups and motives Let$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})$. In certain cases we can say a lot about this ... 1answer 331 views ### vanishing of étale cohomology groups with small support with values in an abelian scheme Let$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$... 1answer 737 views ### Etale cohomology in the$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$-adic ... 1answer 401 views ### is the presheaf of automorphisms a sheaf? In Chapter III,$\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$be a ... 5answers 6k views ### Textbook for Etale Cohomology What is the best textbook (or book) for studying Etale cohomology? 0answers 155 views ### Galois cohomology of generic points of formal completions (of components of a hypercovering of the subvariety): examples or general statements? Let$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$i.e. about the ... 3answers 701 views ### Is the Gelfand-Graev character isomorphic to a cohomology group for some sheaf on a Deligne-Lusztig variety? Deligne-Lusztig theory is awesome. You take a maximal torus$T$, you take a character$\theta$, construct a variety$X_T^*$, take etale cohomology, get a virtual character$R_T^\theta$, maybe it's ... 1answer 1k views ### Why does a group action on a scheme induce a group action on cohomology? This is probably totally obvious but I have no clue how this is done: Say you have an endomorphism$f:X \rightarrow X$of schemes. Why (if true, perhaps some additional assumptions are necessary!) do ... 1answer 573 views ### The etale fundamental group and etale cohomology with compact support Before me, the following was asked: etale fundamental group and etale cohomology of curves However, that question dealt only with projective curves. Question Let$X$be any scheme (or if you prefer ... 1answer 562 views ### Basic properties of Nisnevich cohomology;$l'$-topology? I would like to know more about Nisnevich cohomology (especially, on its properties that could be easily formulated). In particular, I would like to know which of the following statements are true, ... 0answers 112 views ### reading off invariants of a scheme$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$, see e.g. [Kiehl-Weissauer])? 0answers 246 views ### The restriction of the Gersten resolution (for etale cohomology) onto a closed subvariety. There is a very important result of Bloch and Ogus: for any smooth variety$X$and fixed$r\in \mathbb{Z}$,$r\ge 0$,$l$is prime to the residue field characteristic, the Zariski sheafification of ... 0answers 250 views ### Ordered Cech(-like) complexes that compute etale cohomology (of fields!) It is well known (cf. Equivalence of ordered and unordered cech cohomology. ) that for 'usual' topologies one can compute the cohomology of sheaves either using unordered Cech complexes or ordered ... 0answers 214 views ### On inverse images with respect to Zariski-etale topology. For a variety$X$I define its Zariski-etale site as follows: the category is the category of etale$X$-schemes, and the coverings are Zariski ones. Note that this topology is more coarse than the ... 0answers 317 views ### What are the easiest cases of base change (for sheaves on sites)? I have a closed embedding of schemes$i:X'\to X$, and for each of them I consider three Grothendieck topologies for the category of the corresponding (relatively) \'etale schemes: the \'etale one, the ... 0answers 270 views ### Do inverse images respect flabby sheaves? Let$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*...
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 ...
### 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, ...