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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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1answer
110 views

Classes of hyperplane sections in cohomology

Let $X$ be a smooth projective variety over the algebraic closure of a finite field with Galois group $G$. Is it true that the vector space $H^{2k}(X,\mathbf{Q}_{\ell}(k))^G$ has always positive ...
5
votes
1answer
145 views

Frobenius eigenvalues algebraic numbers

Let $X$ be a smooth projective variety over $\mathbf{F}_q$ and $\overline{X}$ its base change to $\overline{\mathbf{F}_q}$. By Deligne’s Weil I, the eigenvalues of the geometric Frobenius acting on $...
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votes
0answers
98 views

The weight filtration on etale cohomology and Berkovich analytic geometry

If $X$ is a smooth projective curve over $\mathbb C_p$, then its first etale cohomology $\mathrm H^1_{et}(X,\mathbb Q_\ell)$ (with $\ell\neq p$) carries a certain weight filtration $W_\bullet$ -- also ...
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vote
0answers
70 views

Hodge-Tate weights of etale cohomology groups

Given a smooth algebraic variety $X$ over a number field $F$, its $p$-adic cohomology groups $H^i(X \times_F \bar F, \mathbb Q_p)$ carries an action of $\mathrm{Gal}(\bar F/F)$, which gives a ...
4
votes
1answer
286 views

etale higher direct image sheaf

Let $f:X\rightarrow Y$ be a smooth morphism of schemes such that all the fibres (for geometric points) are affine spaces. Let $F$ be a coherent sheaf on $X$. Is $R^i_{et}~f_*F=0~~~\forall i>0$? ...
6
votes
1answer
239 views

Cokernel of map of étale sheaves

Let $p:\mathbb{G}_m\to \operatorname{Spec} k$ be the structure map, and let $T$ be an algebraic $k$-torus viewed as an étale sheaf over $k$. Why is the cokernel of the canonical map $T\to p_*p^*T$ ...
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votes
1answer
93 views

Smooth loci and formal neighborhoods

Let $R$ be a Noetherian local ring with maximal ideal $I$. Suppose we have a morphism of smooth $R$-algebras $f : A\to B$ such that its reduction modulo $I^n$ $$f_n : A/I^n \to B/I^n$$ is an ...
5
votes
1answer
202 views

What is the relationship between the $\ell$-adic cohomology of a DM stack and that of its coarse moduli space?

Let $\mathscr{X}$ be a smooth proper DM stack over a field $k$ (perhaps assumed to be separably closed and/or of char. $0$) and let $\pi \colon \mathscr{X} \rightarrow X$ be its coarse moduli space. ...
14
votes
2answers
428 views

When is the etale cohomology of $\mathrm{Sym}^n(X)$ isomorphic to the $\Sigma_n$-invariants in the étale cohomology of $X^n$?

Suppose $X$ is a smooth projective variety defined over an arbitrary algebraically closed field $k$, and consider the action of $\Sigma_n$ on the $n$-fold product $X^n$. Is it true that $H_{\acute{e}t}...
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vote
0answers
121 views

Group scheme representation from action of a group scheme on a variety?

Let f be some homogenous polynomial of d. Let $X = \operatorname{Proj} (k[x,y,z]/(f))$ where $k$ is algebraically closed field of characteristic $p>0$. Now $G$ is a group scheme acting on $X$. ...
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vote
0answers
157 views

Cohomology and base change

Let $f$ be some homogenous polynomial of $d$. Let $X = \operatorname{Proj} (k[x,y,z]/(f))$ where $k$ is algebraically closed field of characteristic $p >0$. Now let $R$ be a $k$-algebra. What ...
4
votes
1answer
225 views

Interpolation of families of local fields

Let $\Sigma$ be the set of all places of $\mathbf{Q}$. We’ll call, for any $v\in\Sigma$, a local field complete with respect to an absolute value lying over $v$ and of finite degree over $\mathbf{Q}_v$...
11
votes
1answer
302 views

“Algebraization" of $p$-adic fields

Part 1: a single finite place. Let $K$ be a finite extension of $\mathbf{Q}_p$. Does there exist a number field $F/\mathbf{Q}$ and a finite place $v$ lying over $p$, such that for the completion ...
7
votes
1answer
367 views

Vector space objects in schemes - confusion

Let $R$ be the ring $\mathbf{C}\times\mathbf{C}$, and consider the affine line $\mathbf{A}^1_R$. $\mathbf{A}^1_R$ can be given the structure of additive group scheme over $R$, denoted $(\mathbf{G}_a)...
15
votes
0answers
311 views

Étale cohomology of varieties in positive characteristic via singular cohomology

Suppose $X$ is a smooth scheme, not necessarily projective, over $\mathbb{Z}[1/N]$ for some integer $N\neq 0$. I would like to understand the cohomology groups $H^i_{ét}(X_{\overline{\mathbb{F}}_p}, \...
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votes
0answers
125 views

Specialization map on geometric points

Let $\mathcal{X}$ be a proper and smooth scheme over $\text{Spec}(\mathbf{Z}_p)$, and let’s call $X$ the geometric generic fiber of $\mathcal{X}$, and $X_0$ the geometric special fiber of $\mathcal{X}$...
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0answers
105 views

Lefschetz trace formula and Frobenius elements

Let $\mathcal{X}$ be a smooth and proper scheme over $\text{Spec}(\mathbf{Z}_p)$ (with $\mathbf{Z}_p$ the $p$-adic integers). Let’s call $X$ its geometric generic fiber $X := \mathcal{X}_{\overline{\...
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votes
0answers
141 views

Weaker version of smooth base change for étale sheaves

Consider the cartesian square of schemes $$ \require{AMScd} \begin{CD} X' @>{g'}>> X \\ @V{f'}VV @VV{f}V \\ S' @>>{g}> S \end{CD} $$ and the base change map $$ \eta : ...
2
votes
0answers
129 views

Lefschetz trace formula over truncated Witt ring

Let $k$ be a finite field, $W_n(k)$ be its $n$-th truncated Witt ring. We have a Frobenius on $W_n(\bar{k})$ whose fixed point is exactly $W_n(k)$. Let $X$ be a finite type separated scheme over $W_n(...
8
votes
1answer
303 views

Picard group and reduced schemes

If $A$ is a ring, then we know that $Pic(A)=Pic(A_{red})$, but for a scheme $X$ it is false in general. On the other hand, we have that $Pic(X)=H^{1}_{et}(X,\mathbb{G}_m)$ and étale cohomology doesn'...
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vote
0answers
86 views

Special formal lifts of smooth algebras

Let $A$ be a smooth algebra over $k$ a finite field. Say $B$ is a $p$-adically complete smooth algebra over the Witt ring $W(k)$, lifting $A$. Assume $B$ is of the form $W(k)\langle t_1,\ldots, t_n\...
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votes
0answers
64 views

Twisted sheaves on tower of $\mathbb{P}^n$

Take the projective space $\mathbb{P}^n$ over a ring $W$. We call $\mathcal{O}(q)$ the usual twisted line bundle. Now take the map $f: \mathbb{P}^n\to\mathbb{P}^n$ defined by $$[x_0,\ldots, x_n]\...
3
votes
0answers
146 views

Non algebraizable formal abelian schemes

I'd like to collect a good number of examples of formal schemes over $\text{Spf}(\mathbf{Z}_p)$, whose special fiber is a projective variety over $\mathbf{F}_p$, but that are not algebraizable. If ...
6
votes
1answer
201 views

Smooth algebras always lift

Let $k$ be a finite field, $A$ a smooth $k$-algebra. Does there exists a smooth algebra $B$ over the Witt vectors $W(k)$, such that $B/p\simeq A$? How is it constructed?
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votes
0answers
113 views

Do residues commute with transverse base change?

Fix a number $n > 0$. Given a smooth $\mathbb{Z}[1/n]$-scheme $X$ (i.e., a smooth scheme such that $n$ is invertible in its ring of functions), we may consider the étale sheaf $\mu_n$ on $X$ which ...
3
votes
1answer
113 views

Liftings and closed immersions

Let $A$ be a flat $\mathbf{Z}_p$-algebra, $\overline{I}\subset A/p$ an ideal. Can we find an ideal $I\subset A$ such that $I$ mod $p$ = $\overline{I}$ $I$ does not contain $p$. It's harder than it ...
4
votes
1answer
222 views

Coherent modules over complete adic rings: counterexamples

Let $A$ be a coherent ring, complete with respect to the adic topology generated by a finitely generated ideal $I$. Define the category $Coh(A,I)$ whose objects are inverse systems $\{M_n\}$ of $A$-...
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votes
0answers
117 views

Absolute approximation of formal schemes

Let $\mathfrak{X}_j$ be an inverse system of qcqs $p$-adic formal scheme, flat over $\mathbf{Z}_p$, with affine transition maps, and assume $\mathcal{O}_{\mathfrak{X}_j}$ is a coherent sheaf of ...
2
votes
0answers
156 views

Liftability of varieties, after fpqc base change

Let $X$ be a smooth projective variety over a finite field, with a closed immersion to some other smooth projective variety $S$, with $S$ liftable. Suppose there exists an fpqc cover $S'\to S$, such ...
1
vote
1answer
154 views

Approximation of constructible abelian étale sheaves

Let $F$ be a constructible abelian étale sheaf of modules over a finite ring $\Lambda$ on a scheme $X$ over a field $k$, with the size of $\Lambda$ invertible on $X$. Suppose $X = \varprojlim X_j$, ...
3
votes
1answer
171 views

Projective embeddings and quasi-compactness

Let $X$ be a projective scheme over a ring $R$, and $p : X\to\mathbf{P}^n_R$ a projective embedding. Does there exist $n$ large enough so that the complement $U\subset \mathbf{P}^n_R$ of $p(X)$ in ...
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votes
0answers
132 views

What is the map induced by Frobenius on the cohomology group $H_{fppf}^1(X, \mathbb{G}_a)$?

Let $X$ be a projective curve of degree $p$ in $\mathbb{P}^2_k$, where $k$ is a field of charcteristic $p>0$. $X$ may not be reduced. I'm trying to compute $H_{fppf}^1(X, \alpha_p)$ using the ...
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vote
0answers
35 views

Push-forward along closed immersion

Let $X$ be a scheme, $p : Z\to X$ a closed immersion, $\mathcal{F}$ a locally free sheaf of modules on $Z$ of finite rank. Assume both $\mathcal{O}_Z$ and $\mathcal{O}_X$ are coherent sheaves of $\...
3
votes
1answer
111 views

Étale torsors and equivariant structures

Let $X$ be a scheme over a separably closed field, equipped with an action of a constant group scheme $G$. Let $H$ be a finite group whose size is invertibile on $X$, and $Y\to X$ an $H$-torsor with ...
3
votes
1answer
413 views

Descent of étale torsors

Let $X$ be a scheme over a field $k$, $G$ a finite abelian group of size invertible on $X$. Suppose $K/k$ is a Galois field extension and let $Y\to X_K$ be an étale $G$-torsor. For what field ...
4
votes
0answers
193 views

Comparison for cycle class maps for de Rham and etale cohomology via p-adic Hodge theory

Let $K$ be a p-adic local field, $X$ a smooth projective variety over $Spec(K)$, $CH^k(X)$ the Chow group of pure codimension $k$. Then there are cycle maps $cl^X_{DR}:CH^k(X)\to H^{2k}_{DR}(X/K)$ ...
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votes
1answer
135 views

Integral morphism between universally closed and separated schemes

Let $f : X\to Y$ be a morphism between schemes over $\text{Spec}(\mathbf{Z}_p)$. Assume: $f$ is integral both $X$ and $Y$ are universally closed and separated over $\mathbf{Z}_p$ $f$ mod $p^n$ is an ...
1
vote
1answer
163 views

Relative approximation of morphisms

Let $S$ a qcqs scheme, and let $f : X := \varprojlim_j X_j \to S$ be an inverse limit of qcqs schemes $f_j : X_j \to S$ with affine transition maps. Suppose $f$ is (P). Is $f_j$ also (P) for all $j$ ...
8
votes
1answer
666 views

Commutative algebra counterexample

Let $M$ be an $R[x]$-module, such that $M$ is finitely generated as an $R$-module. Does there exist one such $M$, such that $M\otimes_{R[x]}R[x,x^{-1}]$ is not finitely generated as an $R$-module?
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votes
0answers
145 views

Usage of Leray spectral sequence

Maybe this is an elementary stuff for experts, I could not figure it out by myself. Let $\pi:G\to S$ be an elliptic curve with zero section $e:S\to G$. Take $\mathcal{H}=(R^1\pi_*\mathbb{Q}_\ell)^\vee$...
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0answers
204 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 ...
4
votes
1answer
119 views

Gluing finitely presented quasi coherent sheaves

Let $X$ be a quasi-compact, separated scheme, and $\{\text{Spec}(A_i)\subset X\}_{i=1,\ldots, n}$ a finite affine open cover. Suppose a quasi-coherent $\mathcal{O}_X$-module $\mathcal{F}$ is such ...
7
votes
0answers
285 views

Reference request: Motivic Cohomology and Cycle class maps

For a smooth projective variety $X$ over any field $K$, Voevodsky showed in his paper ``Motivic Cohomology Groups Are Isomorphic to Higher Chow Groups in Any Characteristic" that the motivic ...
2
votes
0answers
191 views

A question on direct limits of rings, and descent of ideals

Motivated by an étale cohomology calculation I am going to do, here is a question that should have positive and not too hard answer. Let $A$ be a ring, $A_0$ a ring such that $A_0$ is equipped with ...
5
votes
0answers
245 views

Geometric Frobenius on $\ell$-adic cohomology

Let $X$ be a smooth projective variety over a finite field $k$, and $F$ the geometric Frobenius on $H^*_{et}(X_{k^{sep}}, \mathbf{Z}_{\ell})$. Is $F$ only rationally bijective, or integrally ...
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votes
0answers
63 views

Smoothness in formal GAGA

Suppose $X$ is a noetherian scheme over a complete noetherian local ring $(A,\mathfrak{m})$, and assume $X$ is projective. If $\widehat{X}$, the $\mathfrak{m}$-adic formal completion, is smooth in ...
9
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1answer
197 views

Liftable rational varieties

Is there an example of a rational smooth projective variety over a perfect field of characteristic $p$, that is not liftable to characteristic zero?
2
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0answers
108 views

Set theoretic complete intersections in toric varieties

Is it expected that every smooth projective variety over the complex numbers, is a set-theoretic complete intersection into a smooth projective toric variety? Is there an example of a smooth ...
4
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1answer
332 views

Complete intersections in toric varieties

Let $X$ be a smooth projective variety over the complex numbers. Is $X$ a global complete intersection inside a smooth projective toric variety?
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140 views

Algebraization of open formal subschemes

Let $\mathfrak{X}$ be a locally noetherian adic formal scheme over $\text{Spf}(A)$, with $A$ an $I$-adically complete and separated noetherian ring. Suppose the mod $I$-fiber of $\mathfrak{X}$ is an ...