Questions tagged [etale-cohomology]
for questions about etale cohomology of schemes, including foundational material and applications.
700
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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{...
2
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0
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104
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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 $\ell$ ...
2
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0
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113
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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 $...
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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}\...
8
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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 ...
2
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207
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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 $U \to X$,...
3
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172
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When inverse image presheaf is already a sheaf
Following proof from Milne's Étale Cohomology (page 94) contains an equality I not understand.
Setting: assume $X$ is a variety (=absolutely reduced, irreducible scheme of finite type over base field ...
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202
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Some questions about splitting of sequence $0\to I\to\mathrm{Gal}_K\to\mathrm{Gal}_k\to 0$ for Henselian val field $K$
I have a couple of questions about following proof by Peter Scholze on splitting of the ses (...does it have a name?...)
$$0\to I\to\mathrm{Gal}_K\to\mathrm{Gal}_k\to 0$$
for $K$ henselian valuation ...
2
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0
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195
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Using the Dold-Thom Theorem to define \'etale cohomology
For reasonable spaces $X$, the Dold-Thom Theorem states that $\pi_i(SP(X)) \cong \tilde{H}_i(X)$ where $SP(X) = \bigsqcup_i \mathrm{Sym}^i(X)$. There is a purely algebro-geometric realization of this ...
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221
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Locally acyclic morphism which is not flat
Let $k$ be a closed field of characteristic $p \geqq 0$ and $\Lambda = \mathbf{Z}/\ell$, $\ell \neq p$. Recall that a morphism $f \colon X \to S$ of $k$-varieties is said to be locally acyclic if for ...
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124
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A stalk criterion for unit map to be an isomorphism on étale site
Let $f: X \to Y$ be a morphism of schemes and $\mathcal{F}$ sheaf of sets/Abelian groups on the small étale site $Y_{ét}$. Assume we manage somehow to show thatat every geometric point $\overline{y} \...
3
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1
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399
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Characterization of étale locally constant sheaves over a normal scheme
I have a question about the verification of remark 1.2 in James Milne's book Étale Cohomology stated on page 156:
Assume $X$ be a normal & connected scheme with generic
point $g: \eta \to X$.
Then ...
1
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0
answers
240
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A hard-Lefschetz theorem with torsion coefficients?
Let $X$ be a smooth projective surface over $\overline{\mathbb{F}_{q}}$. Let $\ell$ be a prime distinct from the characteristic.
Assume we have a Lefschetz pencil of hyperplane sections on $X$. Let $...
4
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317
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Unit map on étale site under $(f^*,f_*)$ adjunction
Let $f: X \to Y$ be a morphism between two irreducible schemes and $\mathcal{F}$ sheaf on the small étale site $Y_{ét}$. My question is more or less "dual" to this one:
Question: Under which ...
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1
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315
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Diagonal morphism of henselization is an open immersion?
Let $(R,\mathfrak{m})$ be a local ring, denote by $R \rightarrow R^h$ its henselization. Write $S = \operatorname{Spec} R$ and $S^h = \operatorname{Spec} R^h$. Is it true that the diagonal morphism $\...
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1
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116
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An etale cover of a semiperfect ring
Assume that $R$ is a semiperfect ring in characteristic $p$, i.e the frobenius is surjective on $R$. I think one can prove that an etale cover of $R$ should again be semiperfect by considering the ...
5
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373
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Cohomology of Shimura varieties before and after completion at some prime
Let $(G,X)$ be a Shimura datum with reflex field $E\subset \mathbb C$. For any neat open compact subgroup $K \subset G(\mathbb A_f)$, let $\mathrm{Sh}_K$ denote the associated Shimura variety. It is a ...
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0
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141
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Multiplicity and the perfect projective line
Let $\mathbf{F}_p$ be the field with $p$ elements, and $X = (\mathbf{P}^1_{\overline{\mathbf{F}}_p})^\text{perf}$ the inverse perfection of the projective line over $\mathbf{F}_p$.
Let $\Gamma$ be the ...
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1
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290
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Self-intersection of the diagonal on a surface
Let $X$ be a smooth projective curve over the complex numbers, and take $\Delta$ the diagonal divisor on $X\times X$. Using the adjunction formula, one computes $\Delta\cdot\Delta =2-2g$ for $g$ the ...
3
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171
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Beilinson-Lichtenbaum conjecture for algebraic extensions of $\mathbb{Z}/m$
Let $X$ be smooth over some field $k$ and $m\in\mathbb{Z}$ so that $m$ maps to a unit in $k^{\times}$. By Beilinson-Lichtenbaum one has an isomorphism of cohomology groups
\begin{equation*}
\...
3
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0
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163
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Relations between some categories of étale sheaves
I asked this question on math.stackexchange but nobody answers, so I try here even if I'm not sure my question is a research level one..
Let $X$ be a scheme over a number field $k$. Feel free to add ...
3
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1
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286
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Purity of Frobenius on cohomology of a projective variety over $\mathbb F_q$ with isolated singularities
Let $X_0$ be a projective variety of dimension $n>0$ over a finite field $\mathbb F_q$ of characteristic $p$. Let $X$ denote its base change to an algebraic closure. Let $\ell$ be a prime number ...
2
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0
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230
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Tate's conjecture for arithmetic schemes
Tate's conjecture is about a map from Chow groups of a smooth projective variety $X$ to the $l$-adic cohomology i.e. $CH^n(X)\rightarrow (H^{2n}(\bar{X}, \mathbb{Q}_l(n)))^G$ where $G$ is the Galois ...
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0
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119
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Is the Frobenius semisimple on the de-Rham cohomology?
Suppose $K$ is a unramified finite extension of $\mathbb Q_p$, and $X$ is a projective smooth curve defined over $K$. By $p$-adic Hodge theory we know $D_{cris}(H_{et}^i(X,\mathbb Q_p))=H_{dR}^i(X)$. ...
2
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1
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249
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Deligne finitude and finiteness of etale cohomology
This probably is a very straightforward question. Does Deligne finitude imply etale cohomology with $\mu_l^{\otimes n}$ ($l$ is invertible) for finite type schemes over a finite field is finite? This ...
3
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0
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183
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Tate isogeny theorem over varieties?
Let $X$ be a nice scheme, $\pi:E\to X$ an elliptic curve, and $\ell$ a prime invertible on $K$. Then we can consider the "Tate module" $(R^1\pi_*\mathbb{Z}_{\ell})^\vee=\hbox{''}\varprojlim\...
3
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$l$-adic cohomology of hyperplane arrangements
Consider an arrangement of hyperplanes given by the faces of a simplex. Let's consider it as a scheme (a non-regular scheme) and let's also work over a finite field. Has the rational $l$-adic ...
2
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1
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259
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Commutative group scheme cohomology on generic point
Setup:
Let $k$ be an algebraically closed field.
Let $C$ be a smooth connected projective curve over $k$.
Let $J$ be a smooth commutative group scheme over $C$ with connected fibers.
Let $j:\eta\to C$ ...
1
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1
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179
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Confusion about relative Poincaré duality in the context of $\ell$-adic cohomology
I have recently learned about relative Poincaré duality in the book Weil conjectures, perverse sheaves and $\ell$-adic Fourier transform by Kiehl and Weissauer (2001). The reference is section II.7. ...
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1
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153
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Field of fractions of etale stalk of Dedekind domain (Example from Milne's LEC)
Let $X=\operatorname{Spec}(A)$ be an affine Dedekind domain with field of fractions $K$. Let $\widetilde{A}$ be the integral closure of $A$ in separable closure $ K^{\text{sep}}$. A closed point $x$ ...
2
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1
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221
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Calculate stalk of etale derived pushforward sheaf (Milne's LEC)
Assume $X=\operatorname{Spec}(A)$ is connected and normal (especially integral), and let $g:\eta \hookrightarrow X$ be the inclusion of the generic point of $X$. In Milne's LEC script on Etale ...
2
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0
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94
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etale cohomology and algebric K theory for algebraic stack
Let $X$ be a smooth variety over a perfect field $k$. Fix a prime $p$ which is invertible in $k$.
Thomason proved that there is Atiyah-Hirzebruch type spectral sequence that computes $K(1)$-local $K$ ...
3
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1
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219
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$\mathbf{Z}$-points of quasi-projective schemes
Let $U\subset\mathbf{P}^n_{\mathbf{Z}}$ be an open subscheme such that the smooth morphism $U\to\text{Spec}(\mathbf{Z})$ is surjective. Suppose $U(\mathbf{Q})\neq\varnothing$ and $U(\mathbf{Z}_p)\neq\...
2
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0
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219
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Zero dimensional varieties and the L-function $1/(1-p^{-n})$
I am interested in positive characteristic varieties which produce an L-function of the form $\frac{1}{1-χ} = \frac{1}{1-p^{-s}} = \sum_{n = 0}^\infty p^{-ns}$. It seems related to the positive ...
2
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1
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198
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Composition of Gysin and restriction maps on $\ell$-adic cohomology
I already posted this question on mathstackexchange there, but I figured that it may have more replies here.
I follow the notations of Milne's lectures notes on etale cohomology, most specifically ...
1
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1
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263
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Cohomology of singular curves
Suppose $X$ is a singular quasi-projective curve over the complex numbers, and $X'$ is a good nonsingular compactification of a resolution of singularities $Y\to X$. Let $D$ be the complement of $Y$ ...
3
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168
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Nearby cycles of a constant $\Lambda$-sheaf
Let $X$ be a scheme over a henselian trait $S = (S,s,\eta)$. Let $\ell$ be a prime number which is invertible on $S$ and let $\Lambda := \mathbb Z_{\ell}/\ell^k\mathbb Z_{\ell}$ where $k\geq 1$. Let $\...
4
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0
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181
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Is there a simple counterexample to étale proper base change on the unbounded derived category?
The best non-derived version of proper base change on the étale site of a scheme I know is that for $f : X \to Y$ proper and $g : Y' \to Y$ arbitrary, the base change morphism $g^{-1} R f_\star \...
2
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Base change for fundamental group prime to p in mixed characteristic?
I found the answer to this question while typing it up, but since I've already written it, it is probably worthwhile to post-and-answer in case someone finds it useful.
Let $S=\operatorname{Spec}\...
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Is anything known about de Rham $K(\pi,1)$'s?
Let $X$ be a connected qcqs scheme. We say that $X$ is a (étale) $K(\pi,1)$ if for every locally constant constructible abelian sheaf $\mathscr{F}$ on $X$ and every geometric point $\overline{x}$ the ...
2
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1
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Images of smooth schemes under lci morphisms
Let $S$ be a Noetherian scheme, $f : X\to S$ a smooth quasi-projective morphism, $g : X\to Y$ a morphism of finite type, and $h : Y\to S$ a smooth projective morphism with $h\circ g =f$.
Can we say ...
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123
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Why do we have $X/G\cong \coprod_jX_j/G_j$?
I'm reading Lei Fu's "Etale Cohomology Theory".
Corollary 3.2.6. Let $S$ be a noetherian scheme,let $X\to S$ be an etale covering space, and let $G$ be a finite group acting on the right of ...
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0
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245
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Base points and geometric points of schemes
Let $k$ b a field and $X$ a geometrically connected variety over $k.$ Let us fix an algebraic closure $$\ast: \text{Spec } \bar{k} \rightarrow \text{Spec } k$$ and denote by $\bar{X} = X \times_k \bar{...
4
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375
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Étale group schemes and specialization
If $A$ is an abelian variety over a finite field $\mathbf{F}_q$, then $A(\mathbf{F}_q)$ (resp. $A(\overline{\mathbf{F}}_q)$) is a finite (resp. infinite torsion) group, but $A(\mathbf{F}_q(t))$ is a ...
2
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229
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Flat scheme-theoretic closure
Suppose $R$ is a discrete valuation ring with fraction field $K$. Let $X\subset \mathbf{P}^n_{C_K}$ be a closed subscheme, flat over $C_K$, a smooth projective curve over $K$.
Let $C_R$ be a flat ...
22
votes
1
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Is there a ring stacky approach to $\ell$-adic or rigid cohomology?
Ever since Simpson's paper [Sim], it was observed that many different cohomology theories arise in the following way: we begin with our space $X$, we associate to it a stack $X_\text{stk}$ (which ...
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0
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202
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$p$-adic étale cohomology group of open smooth varieties
Let $K$ be a finite extension of $\mathbb{Q}_p$ and let $X$ be a smooth variety over $K$.
Dr. Yamashita announced that he had proved the Galois representation of $p$-adic étale cohomology group $H^*_{\...
1
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0
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155
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Interpretation of Tate conjecture using motivic homotopy
For a smooth projective variety $X$ over a field $k$ the Tate conjecture says that the cycle class maps
$$CH^i(X)\otimes \mathbb{Q}_l \to H^{2i}(X_{\bar{k}},\mathbb{Q}_l(i))^{G_k}$$
are surjective. To ...
5
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0
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319
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Calculating étale fundamental groups from the usual fundamental group
$\newcommand{Spec}{\operatorname{Spec}}$Let $X$ be a connected affine smooth variety over $\mathbb{Q}$, with a point $x\in X(\Spec(\mathbb{Q})$.
For any algebraically closed field $K$ of ...
24
votes
1
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When (or why) is a six-functor formalism enough?
The six functor formalism in a given cohomology theory consists of for each space a derived category of sheaves and six different ways to construct functors between those categories (four involving a ...