Questions tagged [etale-cohomology]
for questions about etale cohomology of schemes, including foundational material and applications.
724
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Specialization of w-contractible objects on intersections on the pro-étale site
I'm trying to understand sections [61.25] and [61.26] of Stacks Project on closed immersions and extension by zero on the pro-étale site.
Lemma [61.25.5] refers to affine weakly contractible objects $...
- 41
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0
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Finite dimensionality of Galois cohomology
Let $K_S$ denote the maximal extension of $\mathbb{Q}$, unramified outside a finite set of primes $S$, and let $G_S$ denote the Galois group of $K_S/\mathbb{Q}$.
It is known that for any finitely ...
- 2,113
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Unramified lisse $\overline{\mathbb{Q}}_{\ell}$-sheaves
Let $X$ be a connected noetherian scheme and $\ell$ a prime invertible on $X$. Let $D \subset X$ be a regular effective Cartier divisor (or more generally a normal crossings divisor, I suppose). Write ...
- 53
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Kernel of restriction in étale cohomology of curves over number fields
Let $X$ be a smooth projective curve defined over a number field $K$. Let $\overline{K}$ denote the algebraic closure of $K$, and set $\overline{X} := X\otimes \overline{K}$. Denote by $\iota: \...
- 2,113
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Second group cohomology of a twisted fundamental group
Let $X$ be a smooth hyperbolic projective curve defined over $\mathbb{Z}[1/S]$, where $S$ is a finite set of primes, and let $\pi:=\pi_1^{\text{ét}}(X, \overline{b})$ denote its étale fundamental ...
- 2,113
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Finiteness of the Brauer group for a one-dimensional scheme that is proper over $\mathrm{Spec}(\mathbb{Z})$
Let $X$ be a scheme with $\dim(X)=1$ that is also proper over $\mathrm{Spec}(\mathbb{Z})$. In Milne's Etale Cohomology, he states that the finiteness of the Brauer group $\mathrm{Br}(X)$ follows from ...
user122877
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A relative Abel-Jacobi map on cycle classes
I have a question about relativizing a classical cohomological construction that I think should be easy for someone well versed in such manipulations.
Background:
Suppose $X$ is a smooth projective ...
- 7,716
3
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230
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Etale cohomology of relative elliptic curve
Let $E_a: y^2 = x(x-1)(x-a)$ be a smooth proper relative elliptic curve over $\text{Spec}(A)$, with $a\in A$, and assume $\text{Spec}(A)$ is a $\text{Spec}(\mathbb{Q}_p)$-scheme.
Let $R^1f_*\mathbb{Q}...
- 2,113
6
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Reference request: good reduction equivalent to crystalline étale cohomology
Suppose $X$ is an abelian variety over a $p$-adic field $K$, and it's well known that $X$ has good reduction is equivalent to the étale cohomology of $X$ is crystalline, and $X$ has semistable ...
- 613
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Find stratification to decompose constructible sheaf to constant parts (example from Wikipedia)
I have a question about techniques used in determining the stratification over which a constructible sheaf falls into even constant pieces demonstrated on this example from Wikipedia.
Let $f:X = \text{...
- 5,780
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Some questions about $\ell$-adic monodromy
I'm stucking on the proof of the Lemma 3.12 of A p-adic analogue of Borel’s theorem.
Here $\mathcal A_{g,\mathrm K}$ is just a shimura variety defined over $\mathbb Z_p$, and full level $\ell$ ...
- 613
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A relative cycle class map
Suppose I have a smooth projective morphism $p: X \to S$ between varieties, and a relative cycle $Z \subset X \to S$ which is assumed to be as nice as can be (rquidimensional with fibers of dimension $...
- 7,716
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Etale local systems and proper base change
I am looking for a reference, or a proof, of the following statement:
Let $f:Y\longrightarrow X$ be a smooth proper map of quasiprojective $K$ schemes, and let $\overline{f}:\overline{Y}\...
- 2,113
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In Mann's six-functor formalism, do diagrams with the forget-supports map commute?
One of the main goals in formalizing six-functor formalisms is to obtain some sort of "coherence theorem", affirming that "every diagram that should commute, commutes". In these ...
- 1,139
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"Simple Limit Argument" in Freitag's and Kiehl's Etale Cohomology
I have a question about an argument used in Freitag's and Kiehl's Etale Cohomology and the Weil Conjecture in the proof of:
4.4 Lemma. (p 41) Every sheaf $F$ representable by an étale scheme $U \to X$,...
- 5,780