The etale-cohomology tag has no usage guidance.

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### Algebraic proof without using comparison theorem for étale cohomology

Let $X$ be some smooth scheme over $\mathbf C$ equipped with an action of $\mu_n$ (the group of $n$th roots of unity).
The étale cohomology groups of X are therefore equipped with an action of ...

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**1**answer

194 views

### Motivic cohomology and pushforward maps

I have a question about pushforward maps for the motivic cohomology groups $H^p(X, \mathbf{Q}(q))$, for $X$ a smooth variety over a characteristic 0 field.
According to Mazza--Voevodsky--Weibel ...

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**1**answer

183 views

### How do non-trivial global differentials give non-trivial cohomology classes in positive characteristic

Let $k$ be an algebraically closed field and let $X$ be an $n$-dimensional smooth projective variety over $k$.
If $k= \mathbb C$, there is a natural injective morphism of vector spaces
...

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189 views

### A problem on universally locally acyclic

Let $k$ be an algebraically closed field of characteristic $p>0$. Let $X$ and $S$ be two smooth varieties over $k$ and $\mathcal F$ a constructible \'etale sheaf of $\mathbb F_\ell$-modules on $X$ ...

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95 views

### Cycle with integral coefficients from cycle with $\mathbb Z_l$-coefficients

Let $X$ be a $n$-dimensional ($n>2$) smooth projective variety over $k=\bar k$ of positive characteristic. Take a divisor $D\in Pic(X).$ Suppose we know that $\frac{[D]^2}{2}\in H^4(X,\mathbb ...

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186 views

### Birational morphism and first cohomology group

Let $f:X\rightarrow Y$ be a proper birational morphism of smooth projective varieties over $k=\bar k$ ($char(k)>0$). Is it true that $H^1(Y,\mathbb Z_l)\stackrel{f^*}\simeq H^1(X,\mathbb Z_l)$ ...

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**1**answer

136 views

### Higher direct image of locally constant torsion sheaf (étale cohomology)

Let $\phi:X\rightarrow Y$ be a generically smooth projective surjective morphism of algebraic varieties over $k=\bar k.$ Is it possible for $R^1\phi_*(\mathbb Z/l)$ to be supported on a divisor of $Y$ ...

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92 views

### Cycle map and flat cycle

Let $\mathcal X\rightarrow C$ be a smooth projective morphism over an open subset of $\mathbb A_k^1$ ($k$ algebraically closed of characteristic $p>0$, one can suppose $C$ to be the spectrum of a ...

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**1**answer

232 views

### Etale cohomology and topological invariance

Let $X$ be a projective scheme and $X_0 \subset X$ a subscheme defined by a nilpotent ideal. Denote by $i:X_0 \to X$ the closed immersion. Let $\mathcal{F}$ be a locally free sheaf sheaf on ...

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220 views

### Picard groups of Fano varieties in positive characteristic

Let $k$ be an algebraically closed field of characteristic $p \geq 0$. Let $X$ be a smooth Fano variety over $k$ and let $\ell \neq p$ be a prime.
Is the natural morphism $\mathrm{Pic}(X) \otimes ...

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162 views

### Is there a Hochschild-Serre spectral sequence for unramified cohomology?

Similar to the Hochschild-Serre spectral sequence for etale cohomology ($H^p(G, H^q_{et}(X_L, \mathcal F|_{X_L})) \Rightarrow H^{p+q}_{et}(X, \mathcal F)$ for a Galois field extension $L/k$ with ...

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50 views

### Picard sequence for sujective morphisms

Given $\phi:X\rightarrow Y$ a surjective morphism of $k$-algebraic varieties ($k$ separably closed), I wanted to find how the write an exact sequence involving Pic(X) and Pic(Y). We can use the long ...

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355 views

### Group schemes, adeles, double cosets, and étale cohomology

Let $K$ be a number field, $R$ the ring of integers of $K$,
${\mathbf{A}^f}$ the ring finite adeles of $K$, and ${\widehat{R}}\subset {\mathbf{A}^f}$ the ring of integral adeles.
Let $G$ be an affine ...

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**1**answer

100 views

### formally etale deformations of algebras

Let $A$ be a local artinian ring with residue field $k$, $S$ a $k$-algebra. Suppose there is a formally etale deformation $B$ of $S$ over $A$, i.e. a flat $A$-algebra $B$ such that $S\cong ...

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47 views

### constructibility for pushforward

Let consider a quasicompact open $j:U\rightarrow\mathbb{A}^{\mathbb{N}}$ over a field $k$, Is there an example where $Rj_{*}\mathbb{Z}/n\mathbb{Z}$ is not constructible, where $n$ is prime to the ...

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238 views

### étale cohomology via Cech cocycles for a quasi-projective scheme

I am looking for the explicit reference to the fact that for a quasi-projective scheme a class in the étale cohomology of a sheaf of a certain degree can by computed using Cech cocycles.

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71 views

### Stalks of higher direct image under open embedding

Let $U$ be an open subset of $\mathbb P^1$ without two points (say $t=0$ and $t=\infty$) and $j: U\to \mathbb P^1$ be an open immersion. Ground field $k$ is algeraically closed. Let $G$ be the group ...

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196 views

### Is it possible to assume that an étale neighborhood is connected?

I am new to étale topology (though I've seen Grothendieck's sites before).
Let $S:=\mathcal{O}^\textrm{sh}_{X,x}$ be the strict local ring of a point $x$ of a scheme $X=\operatorname{Spec}R$ (over a ...

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269 views

### When does the continuous Galois(=etale) cohomology of fields coincide with the naive one? Often true by the Bloch-Kato conjecture?

For a field $F$ I am interested in its $l$-adic (Galois=\'etale) cohomology; here $l$ is a prime distinct from the characteristic of $F$ (for simplicity one may assume that the latter is $0$).
For ...

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437 views

### Étale cohomology versus classical cohomology

Let $X$ be an algebraic variety over $\mathbb{C}$. If $X$ is smooth, the étale cohomology $H^p_{\textrm{ét}}(X,\mathbb{Z}/n)$ is isomorphic to the singular cohomology ...

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880 views

### When is “independence of l” known?

My question is for which varieties over local fields is "independence of l" known for
etale cohomology. Say $X/{\mathbb Q}_p$ is a complete non-singular variety and $W_l$ is the (complex) ...

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128 views

### cohomology of an intermediate extension of a local system

Let $V$ be affine $n$-space over a field $k$; and $j \colon U \to V$ an open subscheme of $V$. Let $L$ be an $\ell$-adic local system on $U$.
Suppose the cohomology of $H^{\bullet}(U,L)$ does not ...

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179 views

### $H^2(S, f_* \mathbb{G}_m)$ in the fppf versus etale topology for proper $f$

Let $f\colon X \rightarrow S$ be a proper morphism of schemes. Is the cohomology group $H^2(S, f_* \mathbb{G}_m)$ the same regardless of whether it is computed in the etale or the fppf topology? And ...

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312 views

### “Weight-monodromy” for open varieties

Suppose that $X/\mathbb{Q}_p$ is a smooth, projective variety, and choose a prime $\ell\neq p$. Then the weight-monodromy conjecture says that the graded pieces $\mathrm{Gr}_k^M$ of the monodromy ...

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384 views

### Is the Weil–Deligne representations coming from $\ell$-adic cohomology independent of $\ell$?

Let $F$ be a $p$-adic field. Let $(G_{F}, W_{F}, I_{F})$ denote the (absolute Galois group, Weil group, inertia group) of $F$.
Let $X/F$ be a proper smooth variety. Let $\ell$ be a prime number $\ne ...

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539 views

### Are there known cases of the Mumford–Tate conjecture that do not use Abelian varieties?

(For a formulation of the Mumford–Tate conjecture, see below.)
The question
As far as I know, all non-trivial known cases of the Mumford–Tate conjecture more or less depend on the Mumford–Tate ...

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110 views

### Hodge numbers of l-adic sheaves?

Assume first that $C$ is a curve, say over $\mathbb{Q}$ and $(E, \nabla)$ is a vector bundle with a flat connection. Assume further that $(E, \nabla)$ has regular singularities at $S=\overline{C}-C$. ...

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### Tate's conjecture and symmetry of Hodge-Tate weights

I'm reading Bellaiche's notes on the Block-Kato conjecture (Hawaii summer school). Here is the link http://people.brandeis.edu/~jbellaic/BKHawaii5.pdf
At page 10 he claims that an indirect ...

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243 views

### Analogy between connections and $\ell$-adic sheaves: what happens with the residue?

There are many analogies between $\ell$-adic sheaves on varieties over finite fields and vector bundles with connections on varieties over fields of characteristic zero. I would like to know what is ...

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139 views

### Descent datum for a line bundle

Let $\pi:C \to \mathbb P^1$ be a double cover branched at $r$ points. To understand the theory of descent better, I would like, if possible, to construct by hands the descent datum of a line bundle ...

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197 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 ...

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131 views

### Compute higher direct image for Gm under open embedding

Let $U \subset \mathbb P^1$ be an open subset of projective line (over $\mathbb C$) after removing $r$ points and $j: U\hookrightarrow \mathbb P^1$ an open immersion. How do I compute $R^1j_*\mathbb ...

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342 views

### $\ell$-adic monodromy theorems (over $\mathbb{C}$)

This question is about $\ell$-adic monodromy theorems for families over a number field. ($\ell$-adic analogues of Corollaries 6.2.8 and 6.2.9 in [BBD].)
Notation
$H$ denotes étale cohomology.
Let ...

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380 views

### Twist in identification with singular cohomology

Let $X$ be a smooth projective variety over $\mathbb{Q}$ and $$V = H^m(X(\mathbb{C}), \mathbb{Q} \cdot (2\pi i)^r)$$ Then I've seen people write the comparison with complex cohomology (an isomorphism ...

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333 views

### Comparison of etale and singular cohomology for varieties over number fields

Whilst reading Hartshorne's appendix C I came across the comparison theorem for etale cohomology and singular cohomology:
Let $X$ be a smooth projective variety over a number field $K$ and $\ell$ a ...

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156 views

### Question about the specialization map for Etale Fundamental Groups

Let $A$ be a complete, discrete valuation ring, and let $s$ (resp $\eta$) be the special (resp. generic) point of $S=Spec(A)$. Let $\phi:X \rightarrow S$ be a proper morphism and fix geometric base ...

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237 views

### vanishing of étale cohomology of affine surface

Let $U$ be an affine smooth surface over an algebraic closure of a finite field. Let $\mathscr{A}/U$ be an Abelian scheme and $\ell \neq \mathrm{char}(k)$ be prime.
Are there vanishing results for ...

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198 views

### Unravelling some hypotheses on a variety

In Le group de Brauer II, Grothendieck states
Proposition 1.4.- Soit $X$ a préschéma noetherien. Supposon que les anneaux hensélisés stricts des anneaux locaux de $X$ soient factoriels, [...] ...

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### an explicit weak equivalence between $B{\mathbb G}_m$ and ${\mathbb P}^\infty$

OK, so I asked a similar question before; $B{\mathbb G}_m$ is a simplicial presheaf over number field $k$. I see that there is some $A^1$-homotopy equivalence between the sheaf represented by ...

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237 views

### etale cohomology of ${\mathbb P}^n_k$

Suppose $k$ is a number field. I want to compute $H^\ast({\mathbb P}^n_k,\mu_l^{\otimes r})$ where $l,r\in {\mathbb N}$. I know that Milne has some computations, but he assumes throughout that his ...

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407 views

### DG enhancements of $\ell$-adic derived categories

This question is similar in flavor to Existence of dg realization for 6 functors
Let $X$ be a complex variety and $D(X)$ the bounded derived category of constructible sheaves (the Euclidean topology ...

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309 views

### Cohomology of a constant etale sheaf

Let $X$ be a smooth proper algebraic variety over $\mathbb{C}$.
I know that in the analytic world, there is an isomorphism between the de Rham cohomology and the cohomology of the constant sheaf ...

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### Reference request: construction of Chern classes

I am looking for a reference on splitting principle for etale cohomology of simplicial schemes (over arbitrary field k). I found a paper by Schechtman, "On the delooping of chern character and Adams ...

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288 views

### Example of a variety over a number field with non-semisimple Galois representation on $\ell$-adic cohomology

This question is inspired by the question: Example of non-projective variety with non-semisimple Frobenius action on etale cohomology?
Let $K$ be a number field (or finitely generated field of ...

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### Degeneration of etale Hochschild--Serre exact sequence

Let $k$ be a field, $X$ a smooth $K$-variety and $\ell$ a prime not dividing the characteristic of $K$.
Then one can make sense of continuous $\ell$-adic etale cohomology (in the sense of Jannsen), ...

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325 views

### infinite grassmannian in algebraic geometry

Geometric realization of $B{\mathbb G}_{\mathfrak m}({\mathbb C})$ is ${\mathbb C}{\mathbb P}^\infty=\varinjlim_n~ {\mathbb C}{\mathbb P}^n_k$; what if one considers a separable field $k\neq ...

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### Étale cohomology with support and functoriality

Suppose we have a scheme $X$ and a closed subscheme $Z$, with complement $U$. Then, for any étale sheaf $F$ on $X$, we get a long exact sequence in cohomology
$\cdots H^i(X,F) \to H^i(U,F) \to ...

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### Which algebraic surfaces have non-trivial H^1?

Informally, my question is the following:
Is there an "inverse theorem" for the first cohomology group $H^1$ of (the projective completion of) an algebraic surface $S$? Namely, can we give a ...

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155 views

### integral hard Lefschetz

I am looking for examples $(X,\eta)$ where the integral hard Lefschetz is an isomorphism:
$X/k$ is a smooth projective variety of dimension $d$ over an algebraic closure of a finite field and $\eta ...

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209 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 ...