Questions tagged [etale-cohomology]
for questions about etale cohomology of schemes, including foundational material and applications.
704
questions
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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
$$\left(\lim_i \mathcal{F}_i\right)_x\rightarrow ...
8
votes
2
answers
525
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A very elementary question on the definition of sheaf on a site
I'm now studying the etale cohomology with the book 'Introduction to Etale Cohomology' by Tamme.
In the page 26 of the book, 'a family of effective epimorphisms' is introduced.
'A family $\{ U_{i} \...
1
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0
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117
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Higher pushforwards of roots of unity
Suppose $X,Y$ are smooth projective varieties over a field $k$ with $Y$ one-dimensional, connected, and normal, and $f\colon X\to Y$ a proper, flat, surjective morphism so that its fibers are ...
4
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0
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329
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Absolute purity for intersection cohomology
If $i:Z\hookrightarrow X$ is a closed embedding of codimension $c$, then
$$i^*k_X\ =\ k_Z , \ \ \ i^!k_X\ \stackrel{(\star)}{=}\ k_Z[2c]$$
where $(\star)$ is true when $i$ is in addition regular.
Here ...
15
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0
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393
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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 ...
3
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0
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221
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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 ...
5
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0
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297
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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 ...
1
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1
answer
580
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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)...
4
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0
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178
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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$...
4
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0
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264
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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],\...
6
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1
answer
426
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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{...
5
votes
1
answer
585
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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 ...
3
votes
0
answers
290
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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 ...
4
votes
0
answers
177
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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 ...
6
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238
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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 ...
4
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1
answer
492
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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, ...
3
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0
answers
203
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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-...
2
votes
0
answers
130
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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})$$
for $n\geq \...
1
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0
answers
157
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To see that the fundamental class of a local complete intersection is independent of choice of regular sequence
In SGA 4½ ‘Cycle,’ Grothendieck defines (among other things) the fundamental class of a local complete intersection $Y\subset X$ ($X$ simply a noetherian scheme) of codimension $c$ locally as the cup-...
10
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0
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439
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How do I produce a basis of cohomology?
Suppose I am discussing a smooth projective variety over an algebraically closed field with my friend on the phone and I want to make a statement about its $l$-adic cohomology (integral or rational). ...
2
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0
answers
229
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What unramified Galois representations come from geometry?
I think we don't know what crystalline representations come from geometry. What about the unramified ones? Specifically let $\phi:\mathrm{Gal}(\overline{\mathbb{Q}_p}/\mathbb{Q}_p)\to GL_n(\mathbb{Q}...
4
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1
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230
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Suspension Theorem in $\mathbb{A}^1$-homotopy
In algebraic topology, the suspension theorem tells us that for a topological space $X$, we have
$$\tilde{H}^n(X,F)\cong \tilde{H}^{n+k}(S^k\wedge X,F).$$
So I'm wondering if this has an analogue in ...
3
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0
answers
226
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$l$-adic Galois representations factor through a common finite quotient
Let $X$ be a smooth projective geometrically connected variety over $\mathbb{Q}$. Assume that for some $m>0$ we have $h^{i, 2m-i}(X)=0$ unless $i=m$.
Does there exist a number field $E$ such that ...
1
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0
answers
113
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Essential Image of the Étale Homotopy type
For any scheme $X$ we can associate the étale homotopy type $Et(X)$, which is a pro-object in the homotopy category of CW-complexes. My question is, do we have a good understanding of the essential ...
6
votes
1
answer
871
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Homology of the étale homotopy type
$\DeclareMathOperator\Et{Et}$Let $X$ be a scheme and denote by $\Et(X)$ the associated étale homotopy type. Then by the work of Artin–Mazur, we know that for an abelian group $A$, we have
$$H^n(\Et(X),...
1
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0
answers
253
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$p$-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). ...
9
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0
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295
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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 ...
4
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0
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264
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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.
...
5
votes
0
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400
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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}\...
4
votes
1
answer
236
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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 ...
3
votes
1
answer
307
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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....
3
votes
1
answer
458
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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 ...
4
votes
1
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578
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"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)...
3
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0
answers
379
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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{\...
5
votes
1
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400
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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 ...
7
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1
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363
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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 ...
1
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0
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309
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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 ...
4
votes
0
answers
197
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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 ...
1
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0
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159
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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 ...
3
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0
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67
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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 ...
1
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0
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160
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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
...
6
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0
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214
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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 ...
2
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0
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136
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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 ...
4
votes
0
answers
143
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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 ...
3
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0
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108
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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,\...
1
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1
answer
157
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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 $\...
1
vote
0
answers
56
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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 ...
3
votes
0
answers
129
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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 ...
2
votes
1
answer
276
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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 ...
5
votes
1
answer
374
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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,\...