Questions tagged [etale-cohomology]

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

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Stalks of limit sheaves

Let $\{\mathcal{F}_i\}_{i\in \mathbb{N}}$ be an inverse system of sheaves of abelian groups on a space $X$. Then for any $x\in X$ we have a natural map $$\left(\lim_i \mathcal{F}_i\right)_x\rightarrow ...
curious math guy's user avatar
8 votes
2 answers
525 views

A very elementary question on the definition of sheaf on a site

I'm now studying the etale cohomology with the book 'Introduction to Etale Cohomology' by Tamme. In the page 26 of the book, 'a family of effective epimorphisms' is introduced. 'A family $\{ U_{i} \...
gualterio's user avatar
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1 vote
0 answers
117 views

Higher pushforwards of roots of unity

Suppose $X,Y$ are smooth projective varieties over a field $k$ with $Y$ one-dimensional, connected, and normal, and $f\colon X\to Y$ a proper, flat, surjective morphism so that its fibers are ...
whetham's user avatar
  • 199
4 votes
0 answers
329 views

Absolute purity for intersection cohomology

If $i:Z\hookrightarrow X$ is a closed embedding of codimension $c$, then $$i^*k_X\ =\ k_Z , \ \ \ i^!k_X\ \stackrel{(\star)}{=}\ k_Z[2c]$$ where $(\star)$ is true when $i$ is in addition regular. Here ...
Pulcinella's user avatar
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15 votes
0 answers
393 views

Applications of the Weight Monodromy conjecture

I think of the Weight Monodromy conjecture as an analogue of the Weil conjectures in the case of bad reduction. The Weil conjectures of course have lots of applications, from point counting to ...
Asvin's user avatar
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3 votes
0 answers
221 views

Braverman-Gaitsgory definition of local acyclicity

In the Appendix B of the published version of ‘Geometric Eisenstein Series’ (but not found in the arXiv version), Braverman and Gaitsgory compare their notion of local acyclicity with Deligne’s. I ...
Tomo's user avatar
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5 votes
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Čech nerve $C(U)$ corresponds to $BG$ in same manner as a hypercover $\mathcal{H}(U)$ to

We can via the bar construction canonically associate to a monoid $A$ the nerve $N(B A)$, a simplicial set with $N(\mathbf{B}A)_k := \times^{k+1} A $ and canonical face maps and degeneracy maps ...
user267839's user avatar
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1 vote
1 answer
580 views

Fpqc-locally constant if and only if étale-locally constant?

Also in SE. Let $\mathcal{F}$ be sheave over $S_\mathrm{fpqc}$. We say $\mathcal{F}$ is a fpqc-locally constant sheaf (of finitely generated abelian groups) if there exists a fpqc covering $(S_i\to S)...
Z Wu's user avatar
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4 votes
0 answers
178 views

Sato-Tate over function fields

Suppose we have an elliptic surface $\pi: \mathscr E \to C$ over a curve over a finite field $\mathbb F_q$. We consider only the places on $C$ over which we have good reduction. Over any point $x\in C$...
Asvin's user avatar
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4 votes
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nearby cycles map for affine formal schemes

Assume that $X=Spf R$ is p-adic formal scheme over $O_{C_p}$ with generic fiber $X_{\eta}$. I want to know why the nearby cycles map $Ru^\star \mathbb{Z/p}$ is equal to $R\Gamma_{et}(spec R[1/p],\...
ali's user avatar
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6 votes
1 answer
426 views

Etale fundamental group of an order in a number field

Let $\mathcal{O}$ be an order in a number field $K$, that is a subring of $K$ with rank as abelian group equal to $[K:\mathbb{Q}]$. What is known about the SGA3-étale fundamental group of $X=\mathrm{...
Adrien MORIN's user avatar
5 votes
1 answer
585 views

What is the natural motivation for smooth/étale/unramified morphisms restricting from formally smooth/étale/unramified morphisms?

(I asked it first in MathStackExchange but I haven't get an answer yet) Smooth (resp. étale) morphisms are just locally finitely presented + formally smooth (resp. étale) morphisms. For unramified ...
Z Wu's user avatar
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3 votes
0 answers
290 views

A Künneth formula for relative fiber products

There is a Künneth formula for the cohomology of a product of spaces $X\times Y$ in quite a lot of generality. Is there a Künneth formula for relative fiber products $X\times_S Y$? The case I am most ...
Asvin's user avatar
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4 votes
0 answers
177 views

Gysin map and $B\mathbf{G}_m$, confusion

Write $\text{Sh}(X)$ for the triangulated/stable $\infty$ category of $\ell$-adic sheaves on $X$, and $k\in\text{Sh}(X)$ fo the unit object. In playing around with $\text{Sh}(B\mathbf{G}_m)$ I've ...
Pulcinella's user avatar
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6 votes
0 answers
238 views

Computing Hodge numbers by point counting

In the lecture note of Bhatt from Arizona winter school 2017, there is an exercise which claims if X is a proper smooth scheme defined over $\mathbb{Z}[1/N]$ and if there is a polynomial $P$ such that ...
ali's user avatar
  • 1,043
4 votes
1 answer
492 views

Etale cohomology and Kummer theory

If $K$ is a field and $n \geq 1$ is such that $n \in K^{\times}$, then $H^1_{et}(\mathrm{Spec}(K),\mu_n)=K^{\times} / (K^{\times})^n$. This is easy to prove, see for instance Tamme, Etale Cohomology, ...
Martin Brandenburg's user avatar
3 votes
0 answers
203 views

Étale homotopy equivalent varieties are deformation equivalent

Let $k$ be an algebraically closed field of characteristic $p>0$. Let $V_1$ and $V_2$ be étale simply-connected smooth proper varieties over $k$. Assume there is an isomorphism between the prime-to-...
user avatar
2 votes
0 answers
130 views

Surjectivity of Étale cohomology

I've come across https://mathoverflow.net/q/172523, which says that if $D\subset Y$ is a contractible divisor, then we have a surjection $$H^n(Y,\mathbb{C})\rightarrow H^n(D,\mathbb{C})$$ for $n\geq \...
curious math guy's user avatar
1 vote
0 answers
157 views

To see that the fundamental class of a local complete intersection is independent of choice of regular sequence

In SGA 4½ ‘Cycle,’ Grothendieck defines (among other things) the fundamental class of a local complete intersection $Y\subset X$ ($X$ simply a noetherian scheme) of codimension $c$ locally as the cup-...
Tomo's user avatar
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10 votes
0 answers
439 views

How do I produce a basis of cohomology?

Suppose I am discussing a smooth projective variety over an algebraically closed field with my friend on the phone and I want to make a statement about its $l$-adic cohomology (integral or rational). ...
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2 votes
0 answers
229 views

What unramified Galois representations come from geometry?

I think we don't know what crystalline representations come from geometry. What about the unramified ones? Specifically let $\phi:\mathrm{Gal}(\overline{\mathbb{Q}_p}/\mathbb{Q}_p)\to GL_n(\mathbb{Q}...
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4 votes
1 answer
230 views

Suspension Theorem in $\mathbb{A}^1$-homotopy

In algebraic topology, the suspension theorem tells us that for a topological space $X$, we have $$\tilde{H}^n(X,F)\cong \tilde{H}^{n+k}(S^k\wedge X,F).$$ So I'm wondering if this has an analogue in ...
curious math guy's user avatar
3 votes
0 answers
226 views

$l$-adic Galois representations factor through a common finite quotient

Let $X$ be a smooth projective geometrically connected variety over $\mathbb{Q}$. Assume that for some $m>0$ we have $h^{i, 2m-i}(X)=0$ unless $i=m$. Does there exist a number field $E$ such that ...
user avatar
1 vote
0 answers
113 views

Essential Image of the Étale Homotopy type

For any scheme $X$ we can associate the étale homotopy type $Et(X)$, which is a pro-object in the homotopy category of CW-complexes. My question is, do we have a good understanding of the essential ...
curious math guy's user avatar
6 votes
1 answer
871 views

Homology of the étale homotopy type

$\DeclareMathOperator\Et{Et}$Let $X$ be a scheme and denote by $\Et(X)$ the associated étale homotopy type. Then by the work of Artin–Mazur, we know that for an abelian group $A$, we have $$H^n(\Et(X),...
curious math guy's user avatar
1 vote
0 answers
253 views

$p$-adic Galois representation and Étale homology

Let $X$ be a smooth proper scheme over some $p$-adic field $K$. The "usual" way to get a Galois representation out of this is to consider the étale cohomology (either $p$ or $\ell$-adic). ...
curious math guy's user avatar
9 votes
0 answers
295 views

How did Jouanolou define the cup product with no finiteness hypotheses in SGA 5?

In SGA 5 Exposé VII, at the beginning of §2, Jouanolou lets $X$ and $Y$ denote two schemes, $f:X\rightarrow Y$ a morphism, and $A$ the ring $\mathbf{Z}/\nu\mathbf{Z}$ where $\nu$ is an integer prime ...
Tomo's user avatar
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4 votes
0 answers
264 views

Explicit linear object underlying $l$-adic cohomology for almost all $l$

If you are working with closed manifolds you can consider cohomology with any coefficients you like but ultimately everything is determined by the singular cohomology with $\mathbb{Z}$-coefficients. ...
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5 votes
0 answers
400 views

Étale cohomology and normalization?

I have an argument, which I wonder if someone could check: Let $X$ be an irreducible reduced scheme over a field $k$. Then we have a normal scheme $X^{norm}$ with a finite birational $f:X^{norm}\...
curious math guy's user avatar
4 votes
1 answer
236 views

Pushout of schemes and étale cohomology

Let $k$ be an algebraically closed field and $X,Y$ two $k$-schemes. We fix a $k$-point in $X$ and in $Y$ each, which we denote by abuse of notation by $P$. Since the pushout of schemes along closed ...
curious math guy's user avatar
3 votes
1 answer
307 views

Homotopy Ehresmann and deformation invariance of $l$-adic Chern classes

Let $S$ be a connected scheme of finite type over $\overline{\mathbb{F}_p}$. Let $\pi:X\to S$ be a smooth proper morphism such that each fiber over a closed point has a trivial étale fundamental group....
user avatar
3 votes
1 answer
458 views

Example of an intersection complex not concentrated in a single degree

I'm having trouble finding references for in-depth examples of perverse sheaves, so answers in the form of such a reference would be most helpful. I want to construct an example of an intersection ...
jackson's user avatar
  • 133
4 votes
1 answer
578 views

"Universal coefficent theorem" for pro-étale cohomology

In algebraic topology, for any space with finite homology type, the universal coefficient theorem states that for any abelian group $G$, we have $$H^n(X,G)\cong \left( H^n(X,\mathbb{Z})\otimes G\right)...
curious math guy's user avatar
3 votes
0 answers
379 views

Galois representations and pro-étale Site

On a scheme, we can define the pro-étale site. This is an improvement over the étale site in that we can define the $\ell$-adic cohomology as the sheaf cohomology of the constant sheaf $\underline{\...
curious math guy's user avatar
5 votes
1 answer
400 views

Does the étale topos determine the Hodge numbers?

Does the small étale topos of a smooth proper variety over a perfect field of positive characteristic determine its Hodge numbers? We consider it as a Grothendieck topos over the étale topos of the ...
user avatar
7 votes
1 answer
363 views

On a quasi-separated assumption in a lemma for the homotopy exact sequence of the etale fundamental group

Background: I've seen two versions of the homotopy exact sequence for etale fundamental groups. One from Stacks: Stacks 0BTX: Let $k$ be a field with algebraic closure $\overline{k}$. Let $X$ be a ...
KReiser's user avatar
  • 659
1 vote
0 answers
309 views

Any finite flat commutative group scheme of $p$-power order is etale if $p$ is invertible on the base

This question is immediately related to Discriminant ideal in a member of Barsotti-Tate Group dealing with Barsotti–Tate groups and here I would like to clarify a proof presented by Anonymous in the ...
user267839's user avatar
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4 votes
0 answers
197 views

Galois action of Weil restriction

Let $K/\mathbb{Q}$ be a quadratic field. Let $E$ be an elliptic curve defined over $K$ but not over $\mathbb{Q}$, and let $\bar{E}$ be the Galois conjugate of $E$. Then by the descent theory (for ...
Leo D's user avatar
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1 vote
0 answers
159 views

Galoisian perspective on local system tamely ramified along a smooth divisor

This question is about (1.7.8) and (1.7.11) in Deligne’s Weil II paper. Let $X$ be a regular scheme and $D\subset X$ a smooth principal divisor cut out by the function $t$. Let $\mathcal F$ be a ...
Tomo's user avatar
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3 votes
0 answers
67 views

Family of Lie algebras parametrized by a discrete valuation ring

I have a family of Lie algebras parametrized by a discrete valuation ring, whose generic fiber is reductive and whose special fiber is nilpotent. I'd like to learn about the relationship between the ...
mayflowers46's user avatar
1 vote
0 answers
160 views

Discriminant ideal in a member of Barsotti-Tate Group

Let $S = \operatorname{Spec} R$ an affine scheme (in our case latter a complete dvr) and $p$ a prime. Then Barsotti-Tate group or $p$-divisible group $G$ of height $h$ over $S$ is an inductive system ...
user267839's user avatar
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6 votes
0 answers
214 views

A lisse mixed sheaf as an extension of pure lisse sheaves

I am trying to understand Corollary 1.8.11 in Deligne's Weil II paper. The statement is that for a normal scheme $X_0$ that is of finite type over $\mathbb{F}_q$, every lisse $\ell$-adic $\iota$-mixed ...
Lisa S.'s user avatar
  • 2,623
2 votes
0 answers
136 views

Decomposition of cohomology via algebraic cycles (or correspondences)

In the paper " On the calculation of local terms in the Lefschetz-Verdier trace formula and its application to a conjecture of Deligne" by Richard Pink, he wrote in the first page that To obtain ...
yzchen's user avatar
  • 149
4 votes
0 answers
143 views

Counting the image of a map of varieties using the trace formula

Suppose $f: X\to Y$ is a finite map of varieties over a finite field $\mathbb F_q$. Is there an etale constructible $\mathbb Q_\ell$ sheaf $\mathscr F$ on $Y$ which counts the number of rational ...
Asvin's user avatar
  • 7,648
3 votes
0 answers
108 views

Čech for $\ell$-adic sheaves

On the étale site of a scheme $X$, there is a spectral sequence associated to the data of an étale hypercover $K$ of $X$ and an abelian étale sheaf $\mathcal F$ on $X$: $$E_2^{p,q}=\check H^p(K,\...
delgato's user avatar
  • 153
1 vote
1 answer
157 views

Does a morphism of etale sheaves restricting to a closed subscheme $Z$ induce a morphism of their subsheaves of sections supported on $Z$?

Let $X$ be a locally Noetherian scheme and $i:Z\to X$ be an immersion of closed subschemes. Let $\mathcal{F},\mathcal{G}$ be two etale abelian sheaves over $X_{et}$. We can define the subsheaf $\...
Z Wu's user avatar
  • 340
1 vote
0 answers
56 views

local acyclicity when restricting to an hypersurface

Let $X$ be a smooth scheme over $\mathbb{C}$ and a constructible sheaf $K$ of complex vector spaces on $X\times\mathbb{A}^1$ and a function $g:X\rightarrow \mathbb{A}^1$. Suppose that $K$ is locally ...
prochet's user avatar
  • 3,432
3 votes
0 answers
129 views

Where general mixed Galois representations are defined?

I am interested in etale cohomology of varieties, and respectively, in mixed $\mathbb Q_{\ell}$-adic Galois representations over finitely generated fields. What is the canonical reference for this ...
Mikhail Bondarko's user avatar
2 votes
1 answer
276 views

Help with $\mathbf{Q}_{\ell}$ sheaves

Let $X\to S$ be a morphism of smooth connected varieties over an algebraically closed field $k$; let $j:\eta\to S$ be the inclusion of the generic point into $S$ (not a geometric generic point) and ...
delgato's user avatar
  • 153
5 votes
1 answer
374 views

Which $p$-adic valuations of Weil numbers (that is, eigenvalues of Frobenius) are possible?

Let $C$ be a smooth projective curve over a finite field $\mathbb F_q$, $q$ is a power of the characteristic $p$. It is well-known that if $\alpha$ is an eigenvalue of Frobenius acting on $H^1_{et}(C,\...
Mikhail Bondarko's user avatar

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