Questions tagged [etale-cohomology]

For questions on the étale cohomology groups of an algebraic variety or scheme, algebraic analogues of the usual cohomology groups with finite coefficients of a topological space.

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78 views

Equivalent characterizations of Henselian Rings (Theorem 4.2 in James Milne's "Étale Cohomology")

I am stuck on a step in the proof of Theorem 4.2 in Chapter I of James Milne's "Étale Cohomology". The particular implication is (c) $\Rightarrow$ (d). Let $X=\text{Spec} (A)$, where $A$ is ...
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36 views

What does henselization do in etale cohomology theory?

When I asked my professor they say: A polynomial of a ring gives an etale cover of the spectrum of that ring, and the descent with respect to this cover is the henselian codition. How to interpret ...
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1answer
45 views

Proposition 3.2.7 of Etale cohomology theory by Lei Fu

I have trouble understanding the proof in several pieces of the proposition below. Proposition 3.2.7. Let $(S, \gamma)$ be a pointed connected noetherian scheme, $X_{1}$ and $X_{2}$ two etale ...
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Why does Weil cohomology have to be on $\mathbb{Q}_l$ for $l\neq p$?

Specifically, I saw a brief explanation: The construction of Weil cohomology isn't easy. Here's an example by Serre: Consider the endomorphism ring $\operatorname{End}(C)$ of an elliptic curve $C$, ...
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36 views

Computation using projection formula for $\ell$-adic cohomology

I'm not familiar with derived tensor product, hence it may be a stupid question. Let $f: X\to Y$ be a separated morphism of finite type between schemes. Let $A$ be a torsion ring, and $D^-(X,A)$ (resp....
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105 views

Why can't we compute the singular cohomology of schemes/varieties with the Zariski topology?

I have been reading etale cohomology. The book says that it is algebraic analogue of singular cohomology. My question is that why can’t we compute the singular cohomology of schemes/varieties over the ...
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41 views

Milne's proof of weak lefschetz

I was reading Milne's proof of Weak Lefschetz theorem from his book on Étale cohomology [VI.7] and certain parts of theorem 7.9 did not make sense to me. I shall give some definitions first. For any ...
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76 views

Purity of the $\mathrm{H}^0$ of a pure $\overline{\mathbb Q_{\ell}}$-sheaf

Let $q$ be the power of some prime number $p$ and let $\overline{\mathbb F}$ be a fixed algebraic closure of $\mathbb F_q$. Let $X$ be a variety over $\mathbb F_q$ ; it is equivalent to the data of ...
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55 views

characterisation of etale morphism of rings

I am reading Freitag’s etale cohomology and Weil conjecture. He says that A finitely generated flat algebra $A \rightarrow B$ is etale if and only if the following condition is satisfied- For every ...
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31 views

Fiber product with henselisation

Let $X$ be a scheme over an algebraically closed field. Consider a point $x \in X$ and an open subscheme $U = X \setminus x$. Is there a good description of $\text{Spec} (\mathcal{O}_{X, x}^{h}) \...
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26 views

One to one correspond between decomposition of Galois cover $Y\to X$ and subgroup of ${\rm Aut}_X Y$.

Sorry for my bad English. I'm reading Jacob Tsimerman's lecture note in lecture 6 where is written etale fundamental group. Cor.2.9 of Lecture 6 is next proposition; For any connected finite etale $Z$...
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Etale sheaf $\mu^{\otimes d}$

I see everywhere the etale sheaf $\mu^{\otimes d}$, but I cannot see the precise definition. Concretely, what is the sheaf $\mu_3 \otimes \mu_3$ over ${\rm Spec}\mathbb{Q}$, for example described as ${...
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Cup product of cohomology.

Suppose $[K \colon {\Bbb Q}_p] < \infty$ and that $\mu_p \in K$. We shall consider the cup product $$ H^1(G, {\Bbb Z}/p) \times H^1(G, {\mu_p}) \overset{\cup}{\to} H^2(G, \mu_p) \cong {\Bbb Z}/p{\...
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50 views

global section of inverse image and etale cohomology

Let $f: X\to Y$ be a morphism of schemes. If it is needed, we can assume that $X,Y$ are quasi-compact and quasi-separated, and $f$ is affine. Consider the two small etale sites $X_{et}$ and $Y_{et}$. ...
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1answer
94 views

Local systems on a punctured line

How to describe the category of local systems on $\mathbb{G}_{m, \mathbb{Z}}$ in different topologies? I think that in etale topology there is a version of Riemann-Hilbert correspondence which says ...
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Determining what "moduli scheme" means in Katz's paper in regards to his Corollary

I'm trying to "decode" Corollary 2.6 of Katz's text in the case $m = 1$. A shortened background: Fix a prime $p > 5$ and $N \in \mathbb{N}$ prime to $p$. Let $F := \mathbb{F}_{p}(\zeta_{N}...
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66 views

How does the Frobenius really act on Weil sheaves in $\ell$-adic cohomology?

Let $X_0$ be a connected scheme defined over $\mathbb F_{p}$ and let $X$ be the product $X_0 \times_{\mathbb {F}_p} \overline{\mathbb F_p}$, as usual, with the natural map $\pi:X \to X_0$. Given an ...
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1answer
47 views

Etale sheaves locally constant around a geometric point

Let $X$ be a scheme and $\bar{v}$ a geometric point of $X$. Say that a sheaf $F$ on the étale site is locally constant around $\bar{v}$ if its restriction to an étale neighbourhood of $\bar{v}$ is ...
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1answer
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Confusion about Milne -- locally constant sheaves on étale site?

In Lectures on Étale Cohomology by J. S. Milne, Example 8.5(b) on Page 60, it is claimed that locally constant sheaves on [the étale site of] $U := \Bbb A^1_k \setminus \{0\}$, where $k$ is an ...
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1answer
97 views

Divisor exact sequence in the etale topology?

In Milne's book Lectures on étale cohomlogy(link), there is the following proposition. Proposition 13.4. Let X be a connected regular scheme with generic point $\eta$. There is a short exact sequence ...
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1answer
51 views

Equivalent definitions of n-acyclic morphisms / Problem with spectral sequence

I'm trying to understand section (VI.4) on smooth base change in Milne's Étale Cohomology. He defines a morphism $g \colon Y \longrightarrow X$ to be $n$-acyclic (I only care about the case $n \geq 0$)...
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1answer
81 views

Cohomology of De Rham complex with respect to different sites.

Sheaf cohomology of algebraic De Rham complex gives the algebraic De Rham cohomology groups. This coincides with the singular cohomology if for example the variety is defined over $\mathbb{C}$. I was ...
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50 views

About the name "mapping cylinder" in etale sheaf

I learned etale sheaf on $X$ can be described by the triplet $(\mathcal{F}_U, \mathcal{F}_Z, \phi:\mathcal{F}_z \to i^* j_* \mathcal{F}_U )$, where $j:U \to X$ is open immersion and $i:Z \to X$ is the ...
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1answer
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Sheaf of ideals of composition of closed immersion with clopen immersion

I have a question ; Let $ U \overset{i} \hookrightarrow Y \overset{j} \hookrightarrow Z $ , where i is closed open inclusion ( ; i.e, closed open subscheme), and j is closed inclusion, and $ k:= j \...
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2answers
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Uniqueness of a local certain homomorphism (Etale Cohomology and the Weil Conjecture by Freitag, Kiehl)

(All rings in the context are presumed to be commutative, unital and noetherian.) Let $(A,n), (B_1, m_1),$ and $(B_2, m_2)$ be local rings with their maximal ideals, $k$ a field. For $i=1,2$ there are ...
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Lei Fu' s Etale Cohomology theory, Proposition 2.5.3

I have a question on Lei Fu's Etale Cohomology theory, p.75, proposition 2.5.3. First see the next two linked piles enter image description heres enter image description here In this proof, the author ...
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14 views

Reference request- Galois Coverings over genus 1 curves

Would anyone have any good recommendations on learning about Galois coverings of genus 1 curves/torsors over genus 1 curves? More specifically learning about $H^{1}_{ét}(E,\mathbb{Q}/\mathbb{Z})$ ...
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58 views

Some question on the answers (in a previous mathstackexchange page) on why the data of the sheaf on the empty set must be terminal

I'm recently reading on the arguments that for the category of left $G$-sets with the canonical topology. The sheaves are all representable. To understand the proof, I needed the fact that for any ...
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50 views

Local etale cohomology for sheaves of abelian groups.

I have seen local cohomology (cohomology supported on a closed subspace) in different contexts, like for topological spaces and for quasi-coherent sheaves on a scheme. I was wondering whether the same ...
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59 views

Preimages of non-zero ideals under Unramified morphism

Let $\varphi:A \to B$ be a flat, unramified be ring morphism between Noetherian, local, normal rings $A,B$. Assume $\mathfrak{a} \subset B$ is a non-zero ideal of $B$. Question: Is it true that if $\...
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60 views

Notation for $G$-sets

For a group $G$ and a $G$-set $M$, I am not entirely sure what the notation $M^G$ means. For context: Over a field $k$, if $\mathcal{F}$ is the sheaf of abelian groups over $X=(\operatorname{Spec}k)_{\...
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1answer
97 views

Pullback and etale cohomology

I am pretty sure I don't understand well the action of a pullback of some etale map on the first etale cohomology group. In fact, let $f : T \rightarrow X$ (etale map) be a $X$-torsor for some ...
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1answer
41 views

Unramified local homomorphism

A local homomorphism $f:A \longrightarrow B$ of local rings is unramified if $B/f(m_A)B$ is finite separable field extension of $A/m_A$, or, equivalently, if (1) $f(m_A)B=m_B$, and $\\$ (2) the field $...
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How should I think of the nearby cycle $\mathrm R\Psi\overline{\mathbb Q_{\ell}}$ with its Galois action on a smooth projective scheme?

Let $F$ be a $p$-adic field with ring of integers $\mathfrak o$ and residue field $k$. Let $X$ be a smooth equidimensional projective scheme over $\mathrm{Spec}(\mathfrak o)$ of dimension $d$. Denote ...
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1answer
66 views

Filtered colimits in $D(X_{\text{proét}})$

In §5.2 of their pro-étale paper, Bhatt and Scholze define a functor $L$ in terms of a sequential colimit, and, separately, a full subcategory $D’$ of $D(X_{\text{proét}}):=D(X_{\text{proét}},\mathbf ...
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Contraction of loops on algebraic surfaces.

Suppose ${\Bbb C}$ be a complex number field and $S$ be a projective smooth surface over ${\Bbb C}$. We consider a finite etale Galois covering $\pi \colon T \to S$ of degree $d$. We consider the ...
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Locally quasi-finite descent datum is effective.

As in stack project Tag 02W8, Let $X \rightarrow S$ be faithfully flat morphism between affine scheme. Let $V \rightarrow X$ be a locally quasi-finite morphism with $\varphi: V\times_SX \rightarrow X\...
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Simplification of links between idele class group and étale cohomology

For interest I have been looking at links between class field theory and étale cohomology. Let $k$ be a global field. I started with the link between étale cohomology and Galois cohomology, $H^i(\...
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132 views

fppf/ etale Cohomology calculate with Cech cohomology

Let $R$ be a commutative ring with one (so living in standard commutative algebra setting) and let $\phi: R \to S$ a faithfully flat. Then the so called Amitsur complex $R \to S^{\otimes \bullet +1}$ ...
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1answer
73 views

An étale morphism that restricts to an isomorphism on a closed subvariety.

If an étale morphism $f:X\rightarrow Y$ induces an isomorphism $f:f^{-1}(Z)\rightarrow Z$ for some closed subvariety $Z$ of $Y$. Doesn't it imply that $f$ is an isomorphism (I believe the answer ...
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Zariski cohomology of an etale sheaf vs etale cohomology.

Let $\mathcal{F}_{Zar}$ be a Zariski sheaf (on the big site of schemes or $S$-schemes) and $\mathcal{F}_{et}$ be its etale sheafification on the big site. Let $X$ be a scheme. If we know that $H_{Zar}^...
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Is the trace computable via mod $p$ techniques?

One nice feature of étale cohomology is that it can be used to compute the Betti numbers via a reduction modulo $p$, as for smooth proper schemes $X\to \text{Spec}(\mathbb{Z}[\frac{1}{N}])$ we have ...
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20 views

Recovering the cohomology of non-locally constant sheaves on anabelian schemes.

Connected affine schemes in char $p$ are $K(\pi_1,1)$, it is expected that in the aforementioned case the schemes are anabelian (These are stated in this). I was wondering since it is expected to be ...
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Simple Calculation in Flat Cohomology

Question: Let $R$ be a PID and let $n \geq 2$ be even. What is $H^0_{\operatorname{fppf}}(R,\operatorname{SL}_n/\mu_2)$? Attempt at an Answer: From the short exact sequence of algebraic groups $1 \to \...
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open subscheme and étale fundamental group

Let $X$ be an irreducible scheme, $U\subset X$ be a nonempty open subscheme. Is $\pi^{ét}_1(U)\to \pi^{ét}_1(X)$ surjective? For example, if $X$ is normal integral scheme, then this follows from ...
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19 views

Universal cover with respect to étale topology of scheme

Let $X$ be a connected quasi-compact quasi-separated scheme. I tried to define a universal cover as follows (thanks to the help of one friend). Consider $I$ the directed set of all open normal ...
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58 views

$l$-adic cohomology of fields.

I had a number of basic questions about $l$-adic cohomology about some examples that can be calculated. Is there calculations/interpretations of $l$-adic cohomology of fields? Let's say for a field ...
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1answer
87 views

Why is the functor that associates to a scheme $S$ the set of $S$-isomorphism classes of elliptic curves over $S$ not representable?

A few days ago I heard an online presentation about elliptic curves and the presenter claimed that the functor which assigns to a scheme $S$ the isomorphism classes of elliptic curves over $S$ is not ...
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1answer
169 views

The Etale Fundamental Group of $\mathbb{A}_{\mathbb{F}_q}$ and $\mathbb{G}_{m,\mathbb{F}_q}$

I am learning the etale fundamental group of a scheme. And I am hugely confused by the etale fundamental group of the additive and multiplicative group scheme $\mathbb{G}_{a}$ and $\mathbb{G}_{m}$ ...
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1answer
51 views

Question About Proposition 2.3.12 in Lei Fu Etale Cohomology

I have some trouble with the last step of the proof of the following theorem in Lei Fu's Etale Cohomology. The statement of the theorem is: Proposition 2.3.12: Let $S$ be a scheme, $S_0$ a closed ...

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