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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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$l$-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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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 ...
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 $\mathbb{G}_{m}$ ...
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 ...