All Questions
1,114 questions
3
votes
0
answers
65
views
p-torsion in the Tate-Shafarevich group of supersingular elliptic curves
Let $E$ be a supersingular elliptic curve over $\mathbb{F}_p(t)$. Is something known on the $p$-torsion of the Tate–Shafarevich group in this case? In particular, I would like to know if (or if known ...
- 103
1
vote
0
answers
89
views
Generic reducedness of geometric generic fibre
Let $f:X\to Y$ be a surjective morphism between two projective schemes over a field of characteristic $p>0$. Also assume that $X$ is smooth,$Y$ smooth & irreducible and $f_*\mathcal{O}_X=\...
- 5,986
5
votes
1
answer
332
views
$\ell$-adic analogue of Kedlaya–Mochizuki
There is a well-known analogy between holonomic $\mathcal{D}$-modules on complex algebraic varieties and $\ell$-adic perverse sheaves on varieties over finite fields. Many theorems in one setting have ...
- 773
1
vote
0
answers
82
views
Galois group of shimura varieties with different level structure
Let $(G,X)$ be a shimura data, and $K$ an open compact neat subgroup of $G(\mathbb A_f)$. Suppose $K'\subset K$ is open and normal, then, I see in many references that the finite etale cover $Sh(G,X)_{...
- 785
0
votes
0
answers
41
views
Descend local system to the canonical model of Shimura varieties
Suppose $(G,X)$ is a Shimura data, and $E$ be its reflex field. In page 33 of this paper, it constructs an etale local system on the canonical model $\mathrm{Sh}(G,X)_{K,E}$ (variety over $E$) for any ...
- 785
2
votes
0
answers
138
views
Leray spectral sequence for étale homology
Let $X$, $Y$, $Z$ be quasi-projective varieties over an algebraically closed field $k$, $f: X \to Y$ and $g: Z \to X$ proper (even projective) maps with $f$ smooth, and $h: Z \to Y$ their composite. ...
- 658
0
votes
0
answers
190
views
About Chern classes via Atiyah class
I am trying to understand a construction of the Chern classes of a vector bundle $\mathcal{E}$ via the Atiyah class, like is done in this text and here in section 1.4. I am interested in the case ...
- 348
1
vote
0
answers
113
views
Analytic vector bundle from an etale local system is algebraic?
Suppose $X$ is an algebraic variety over $\mathbb C$, and $\mathbb L$ is a $\mathbb Q_p$-local system on $X_{et}$, then it corresponds to a representation $\pi_1(X_{et})\to GL_n(\mathbb L)$. Since ...
- 785
1
vote
0
answers
82
views
Behavior of translation functors in characteristic $p$
Let $G$ be a semisimple and simply connected algebraic group over an algebraically closed field of characteristic $p>0$, and let $\mathfrak g$ be the Lie algebra of $G$. Let $U_\chi(\mathfrak g)$ ...
- 2,974
2
votes
0
answers
125
views
Topological Hochschild Homology and $p$-adic étale cohomology of $\mathbb{Q}$-schemes
Recent progress in $p$-adic geometry has produced an interesting comparison isomorphism between the crystalline cohomology of a smooth algebra $A$ over a perfect field $k$ in characteristic $p$, and ...
- 2,907
3
votes
0
answers
69
views
How would you call morphisms of varieties that induce isomorphisms on etale cohomology in low degrees?
In our text we have several statements of the following sort: for a certain morphism $f:X\to Y$ of varieties over an (algebraically closed) field of characteristic $p$ and some $c>0$ the ...
- 16.9k
4
votes
1
answer
154
views
Semistability of the $\ell$-adic representation of variety with semistable reduction
The question is in the title, but here's some quick background.
It's easy to show (assuming smooth-proper base change) that the $\ell$-adic cohomology of a variety over the fraction field of a DVR ...
- 141
3
votes
0
answers
152
views
Descent of classifying stack
Let $X$ be a variety over $k$ and $G$ be a finite abelian group. Then we know that $H_{fppf}^{2}(X,G)$ is in bijective correspondence with isomorphism classes of $G$-banded gerbes.
Now we consider a ...
- 253
2
votes
0
answers
96
views
Confusion about the Lefschetz standard conjectures for abelian varieties in the integral setting
Let $(A,\theta)$ be a principally polarized abelian variety of dimension $d$ over a number field $k$. By the hard Lefschetz theorem
$$H^2(A,\mathbb{Q}_{\ell}) \xrightarrow{\theta^{d-2}} H^{2d-2}(A,\...
- 679
2
votes
0
answers
168
views
When do we have $\bigoplus_{i + j = n} R^i f_* \mathbb{Q}_\ell \otimes_{\mathbb{Q}_\ell} R^j g_* \mathbb{Q}_\ell \cong R^{i + j} h_* \mathbb{Q}_\ell$?
Milne, Étale Cohomology, theorem 8.5 states the following version of the Künneth formula (in slightly greater generality). Let $\Lambda$ be a finite commutative ring. Let $X, Y, S$ be schemes with $S$ ...
- 531
2
votes
1
answer
237
views
Does there exist a scheme $X/{\operatorname{Spec}(\mathbb{Z})}$ such that $\pi_1^\text{ét}(X)=\smash{\hat{\mathbb{Z}}}^2$?
$\DeclareMathOperator\Spec{Spec}$Does there exist a scheme $X/{\operatorname{Spec}(\mathbb{Z})}$ such that $\pi_1^\text{ét}(X)=\smash{\hat{\mathbb{Z}}}^2$?
It's well known that $\pi_1^\text{ét}(\Spec(\...
- 131
3
votes
1
answer
205
views
Inertia Action on Kummer Sheaves
In 7.0.2 of Katz's book "Gauss Sums, Kloosterman Sums, and Monodromy Groups", Katz states the following (when $x=0$).
Let $\chi:\mathbb{F}_q^\times\to\mathbb{Q}_\ell^\times$ be a ...
- 65
1
vote
1
answer
211
views
The Étale Cohomology from the Variety to its Generic Point
Let $n\in \mathbb{Z}_{>0}$ and $X$ be a smooth projective variety over $k$, where $k$ contains all $n$-th roots of unity and $\operatorname{char}(k)=p$. Here $p$ and $n$ are coprime. I wonder ...
- 13
1
vote
1
answer
216
views
Flatness of "derived local system sheaves"
Let $f: Y\longrightarrow X$ be a smooth proper map of smooth proper schemes over $\mathbb{Q}$, and let $\mathcal{F} = R^1_\text{ét}\overline{f}_*\mathbb{Q}_p$ denote the derived pushforward of $\...
- 2,907
1
vote
1
answer
211
views
Characterize descents of geometric finite étale cover by means of homotopy exact sequence
Let $X/k$ be a geometrically connected $k$-variety (=separated of finite type, esp quasi-compact; the base field $k$ assumed to be separable, so $\overline{k}=k^{\text{sep}}$), $\overline{X} := X \...
- 5,986
2
votes
1
answer
211
views
Splitting of composition of trace and counit in derived setting
Let $X,Y$ be varieties (separated of finite type schemes) over base field $k$, $\mathcal{F}$ be constructible sheaf on $Y_{\mathrm{et}}$ and assume that we have a finite morphism $f: X \to Y$, which ...
- 5,986
3
votes
1
answer
241
views
Could I get an interpretation for application of Euler characteristics in number theory？
As a beginner who just get in touch with Euler characteristics in this field, could I get some intuition for the arithmetic meaning of Euler characteristics of bounded complexes, for example Selmer ...
- 319
4
votes
0
answers
205
views
Splitting in additive categories
Let $\mathcal{A}$ be an additive category and $B \to C$ a nonzero map. Are there say "standard" techniques & criteria one should keep in mind when working with additive categories to ...
- 5,986
1
vote
0
answers
137
views
Crystalline at $\ell=p$ implies unramified at $\ell\neq p$
Let $X$ be a smooth, projective variety defined over some $p$-adic field $K$. Is it true that if the etale cohomology $H^i_{et}(X_{\overline{K}},\mathbb{Q}_\ell)$ is crystalline at $\ell=p$, then $H^...
- 141
1
vote
0
answers
138
views
Syntomic f-cohomology for open varieties
Syntomic cohomology $H^{i+j}_{\mathrm{syn}}(X,n)$ of a proper variety $X$ with good reduction over a $p$-adic field $K$ is computed via a spectral sequence in terms of $H^i_{\mathrm{f}}(G_K;H^j_{\...
- 15.4k
3
votes
0
answers
192
views
How can I prove this stronger version of Fedder's Criterion?
I was reading Fedder's original paper which proved what is now known as "Fedder's criterion". I noticed that the abstract stated something which is a priori stronger than what is proved in ...
- 317
8
votes
0
answers
333
views
Triple comparison of cohomology in algebraic geometry
Let $X$ be a smooth proper variety over $\mathbb{Q}$ and $p$ a prime number. For an integer $k$, we have:
a finitely-generated abelian group $H^k(X^{\mathrm{an}}(\mathbb{C});\mathbb{Z})$
a finitely-...
- 15.4k
2
votes
0
answers
137
views
details of a dévissage argument for constructible sheaves
I am working on the following Künneth-type isomorphism from [SGA5, exposé III, 2,3]:
$\mathrm{Settings}.$ Let $X_1, X_2$ be separated finite type schemes over the spectrum of a field $S=\mathrm{Spec}...
- 375
3
votes
1
answer
384
views
Tate twist and cohomology groups
I am reading Milne's lecture notes on etale cohomology and I'm hoping someone could help me clear up some minor confusion.
Let $X$ be a nonsingular variety over an algebraically closed field $k,$ say $...
- 65
1
vote
0
answers
108
views
Do étale coordinates give rise to a regular sequence of diagonal elements?
Fix an algebraic extension $k\subseteq K$ of fields of characteristic zero and consider a map of commutative rings $\phi\colon K\left[T_{1}^{\pm},\dots,T_{n}^{\pm}\right]\to A$ which is étale. Now ...
- 507
2
votes
0
answers
177
views
Eigenspaces of complex conjugation on étale cohomology of a smooth projective curve
Let $X$ denote a smooth projective curve defined over $\mathbb{Z}[1/N]$, and its base change $ \overline{X} $ to $ \overline{\mathbb{Q}} $. Let $ V $ be a $ p $-adic local system on $X$ ($p\mid N$), ...
- 2,907
3
votes
1
answer
219
views
How to show this last condition is equivalent to saying the bilinear form in the proposition is nondegenerate?
I'm reading Lei Fu's "Etale Cohomology Theory".
How to show this last condition is equivalent to saying the bilinear form in the proposition is nondegenerate?
- 245
5
votes
1
answer
344
views
Surjection onto endomorphisms of multiplicative group of a field
Let $k$ be an algebraically closed field of characteristic $p > 0$. Denote by $k^\times$ the multiplicative group of $k$. There is a ring homomorphism given by restriction to $k^\times$
$$
\mathbb{...
- 53
4
votes
0
answers
128
views
Introduction to the theory of $D$-modules and the role of the characteristic cycle
I am seeking recommendations for a concise introduction to the theory of $D$-modules suitable for an algebraic geometer. Specifically, I am interested in understanding:
The role of the characteristic ...
- 2,871
2
votes
1
answer
319
views
Bounding $H^4_{\text{ėt}}$ of a surface
Let $X\longrightarrow X'$ be a smooth proper map of smooth proper schemes defined over $\mathbb{Z}[1/S]$, where $S$ is a finite set of primes. Assume $X'$ is a curve of positive genus, and $X$ is a ...
- 2,907
4
votes
0
answers
129
views
How does one compute the group action of the automorphism group on integral cohomology?
Suppose I have a curve $X$ (for concreteness, we can take $X$ to be a smooth, projective curve over a finite field $\mathbb F_q$, and even more concretely consider the family of curves described by ...
- 7,746
3
votes
1
answer
250
views
Action of complex conjugation on etale cohomology
Let $X$ be a genus $g$ smooth projective curve, defined over $\mathbb{Q}$, and let $\overline{X}$ denote the base change of $X$ to $\overline{\mathbb{Q}}$.
It is well known that $H^1_{\text{ét}}(\...
- 2,907
1
vote
1
answer
159
views
Zeta function of variety over positive characteristic function field vs. local zeta factor of variety over $\mathbb{F}_p$
Let $X = Y \times_{\mathbb{F}_q} C$, with $Y, C / \mathbb{F}_q$ smooth projective varieties, $C$ a curve. Let $d = \dim_{\mathbb{F}_q} X$. We can consider the local zeta function $Z(X, t) = \prod\...
- 658
2
votes
0
answers
80
views
Lift of nearby cycles functor
Let $S$ be the spectrum of a Henselian discrete valuation ring (called a Henselian trait). Let $f:X\to S$ be a finite type, separated morphism of schemes. Let $\eta\in S$ be the generic point. Let $s\...
- 615
0
votes
0
answers
133
views
Higher direct images of locally constant etale sheaf under smooth proper map locally constant
Let $f:X \to Y$ a surjective smooth proper map between Noetherian schemes and $F$ a locally constant sheaf on small etale site of $X$.
Question: Refering to Donu Arapura's answer here, how to see that ...
- 5,986
7
votes
0
answers
148
views
Is the $\ell$-adic cohomology ring of a cubic threefold a complete invariant?
The only interesting $\ell$-adic cohomology of a smooth cubic threefold $X$ is $H^3(X,\mathbb{Z}_{\ell}(2))$, which is isomorphic as a $\mathrm{Gal}_k$-module to $H^1(JX,\mathbb{Z}_{\ell}(1))^{\vee}$ ...
- 679
2
votes
1
answer
132
views
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
2
votes
0
answers
70
views
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 ...
- 2,907
2
votes
0
answers
184
views
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
0
votes
1
answer
111
views
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: \...
- 2,907
2
votes
1
answer
176
views
Isomorphisms $S^d(S^m(V)^*) \cong \Lambda^d(S^{m+d-1}(V)^*)$
$\DeclareMathOperator\SL{SL}$Let $K$ be an algebraically closed field. Let $V$ be a 2-dimensional vector space over $K$. The group $G=\SL_2(K)$ acts naturally on $V$ by left-multiplication on column ...
- 185
1
vote
0
answers
117
views
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 ...
- 2,907
7
votes
1
answer
549
views
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
3
votes
0
answers
177
views
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 ...
- 7,746
3
votes
1
answer
260
views
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}...
- 2,907