Questions tagged [etale-cohomology]

for questions about etale cohomology of schemes, including foundational material and applications.

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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 ...
kindasorta's user avatar
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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 ...
prince homer's user avatar
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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 ...
Asvin's user avatar
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3 votes
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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}...
kindasorta's user avatar
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6 votes
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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 ...
Richard's user avatar
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1 answer
168 views

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{...
user267839's user avatar
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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 $\ell$ ...
Richard's user avatar
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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 $...
Asvin's user avatar
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1 vote
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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}\...
kindasorta's user avatar
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8 votes
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490 views

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 ...
Gabriel's user avatar
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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 $U \to X$,...
user267839's user avatar
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3 votes
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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 ...
user267839's user avatar
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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 ...
user267839's user avatar
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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 ...
Asvin's user avatar
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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 ...
Thiago's user avatar
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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} \...
user267839's user avatar
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3 votes
1 answer
423 views

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 ...
user267839's user avatar
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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 $...
a17's user avatar
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1 answer
335 views

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 ...
user267839's user avatar
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3 votes
1 answer
326 views

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 $\...
Hugo Zock's user avatar
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1 answer
121 views

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 ...
ALi1373's user avatar
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1 answer
387 views

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 ...
Suzet's user avatar
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141 views

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 ...
Tim's user avatar
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1 answer
302 views

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 ...
Tim's user avatar
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3 votes
0 answers
175 views

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*} \...
The Thin Whistler's user avatar
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165 views

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 ...
Marsault Chabat's user avatar
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1 answer
290 views

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 ...
Suzet's user avatar
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2 votes
0 answers
240 views

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 ...
user127776's user avatar
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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 $D_{cris}(H_{et}^i(X,\mathbb Q_p))=H_{dR}^i(X)$. ...
Richard's user avatar
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2 votes
1 answer
250 views

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 ...
user127776's user avatar
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3 votes
0 answers
187 views

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\...
Curious's user avatar
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3 votes
1 answer
165 views

$l$-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 ...
user127776's user avatar
  • 5,851
2 votes
1 answer
261 views

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 $j:\eta\to C$ ...
lzww's user avatar
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1 vote
1 answer
182 views

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. ...
Suzet's user avatar
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1 vote
1 answer
157 views

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 $x$ ...
user267839's user avatar
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2 votes
1 answer
232 views

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 ...
user267839's user avatar
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2 votes
0 answers
95 views

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 $K$ ...
OOOOOO's user avatar
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3 votes
1 answer
222 views

$\mathbf{Z}$-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\...
Tim's user avatar
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2 votes
0 answers
219 views

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 ...
user avatar
2 votes
1 answer
205 views

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 ...
Suzet's user avatar
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1 vote
1 answer
273 views

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 $Y$ ...
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3 votes
0 answers
177 views

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 $\...
Suzet's user avatar
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4 votes
0 answers
192 views

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 \...
C.D.'s user avatar
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2 votes
1 answer
77 views

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}\...
Curious's user avatar
  • 301
7 votes
1 answer
374 views

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 ...
Gabriel's user avatar
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2 votes
1 answer
238 views

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 ...
user avatar
0 votes
0 answers
123 views

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 ...
Born to be proud's user avatar
1 vote
0 answers
258 views

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{...
Crystallineperiodic's user avatar
4 votes
1 answer
379 views

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 ...
user avatar
2 votes
1 answer
237 views

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 ...
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