All Questions
1,114 questions
9
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885
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Computing the etale cohomology of spheres
$\newcommand\Z{\mathbb{Z}}$
Let $K$ be an algebraically closed field of characteristic $\ne 2$. We have the unit sphere $S^n:~x_0^2 + \ldots + x_n^2 = 1$.
What are the $\Z/2\Z$ cohomology groups of ...
- 274
9
votes
1
answer
763
views
Restriction theorems over finite fields
A short while ago, Dvir proved the Kakeya conjecture over finite fields. Does this have any implications for restriction theorems over finite fields? I am aware only of implications going in the ...
- 20.2k
9
votes
1
answer
718
views
Example of a variety over a number field with non-semisimple Galois representation on $\ell$-adic cohomology
This question is inspired by the question: Example of non-projective variety with non-semisimple Frobenius action on etale cohomology?
Let $K$ be a number field (or finitely generated field of ...
- 5,504
9
votes
1
answer
983
views
Is the $\ell$-adic cohomology of a non-proper variety unramified at good primes?
Let $X$ be a smooth variety of finite type over a number field $k$. Let $\overline{X} = X \times_{k} \overline{k}$, and let $\ell$ be a prime. It's well known that if $X$ is proper, then the é...
- 37k
9
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1
answer
1k
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Top chern class in positive characteristic
Given a nonsingular, projective variety $X$ of dimension $n$ over an algebraically closed field $k$.
Over $k=\mathbb{C}$, the top chern class $c_n(T_X)$ of the tangent sheaf is the Euler ...
- 4,902
9
votes
1
answer
235
views
Liftable rational varieties
Is there an example of a rational smooth projective variety over a perfect field of characteristic $p$, that is not liftable to characteristic zero?
user120812
9
votes
1
answer
2k
views
Under what conditions is the induced map of etale fundamental groups surjective?
Let $f:X \to Y$ be a morphism of schemes. I am interested in sufficient conditions on $f$ which would ensure that the induced map $\pi_1^{et}(X) \to \pi_1^{et}(Y)$ of etale fundamental groups is ...
- 2,974
9
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1
answer
2k
views
Comparison of etale and singular cohomology for varieties over number fields
Whilst reading Hartshorne's appendix C I came across the comparison theorem for etale cohomology and singular cohomology:
Let $X$ be a smooth projective variety over a number field $K$ and $\ell$ a ...
- 93
9
votes
4
answers
1k
views
cohomology of moduli spaces
Does anyone know if there's any reference on the $\ell$-adic cohomology of some simple moduli spaces/Shimura varieties, like Siegel moduli varieties $A_{g,N}$ of genus $g$ and level $N,$ for small $g$ ...
- 4,265
9
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1
answer
1k
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deformation theory in positive characteristic
The idea "Formal deformation theory in characteristic zero is controlled by a differential graded Lie algebra (dgla)" goes back to Goldman-Millson, Deligne, Drinfeld among others; see Lurie's ICM talk....
- 528
9
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1
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2k
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Is first etale cohomology of a variety always (dual to) a Tate Module?
The two examples I have in mind are curves and abelian varieties. To be precise, if $C$ is a smooth projective algebraic curve over a number field $K$, then, for a prime $l$
$H^1_{et}(C_{\bar{K}},\...
- 643
9
votes
2
answers
523
views
Concrete interpretations of higher (sheaf) cohomology groups
$H^1$ has an interpretation as torsors. But what about the higher $H^i$ (in the setting of algebraic geometry, and étale or flat cohomology)? For example $H^2(X, \mathbf{G}_m)$ is (often) isomorphic ...
user19475
9
votes
1
answer
1k
views
Picard group and reduced schemes
$\DeclareMathOperator\Pic{Pic}$If $A$ is a ring, then we know that $\Pic(A)=\Pic(A_\text{red})$, but for a scheme $X$ it is false in general.
On the other hand, we have that $\Pic(X)=H^{1}_{et}(X,\...
- 3,472
9
votes
1
answer
665
views
Group schemes, adeles, double cosets, and étale cohomology
Let $K$ be a number field, $R$ the ring of integers of $K$,
${\mathbf{A}^f}$ the ring finite adeles of $K$, and ${\widehat{R}}\subset {\mathbf{A}^f}$ the ring of integral adeles.
Let $G$ be an affine ...
- 14.2k
9
votes
2
answers
868
views
Nakano vanishing in positive characteristic
Let $X$ be a smooth projective variety defined over a field $k$.
In characteristic zero, the following is a special case of the (Kodaira-Akizuki-)Nakano vanishing theorem:
$(\ast) \quad$ $\mathrm H^...
- 1,072
9
votes
1
answer
833
views
Endomorphism ring of simple ordinary abelian variety
Is there an example of an ordinary and simple abelian variety $A$ over an algebraically closed field $K$ (of characteristic $p>0$) such that ${\rm End}(A)$ is not commutative? Note that the answer ...
- 3,076
9
votes
1
answer
704
views
is the presheaf of automorphisms a sheaf?
In Chapter III,$\S 4$ of Milne's Etale cohomology a correspondence between twisted forms and Cech cohomology cocycles is described.
Fix some Grothendieck topology, say, etale, and let $A$ be a ...
- 4,070
9
votes
1
answer
566
views
algorithm for calculating the Chow groups of a variety over a finite field
Is there an algorithm for calculating the Chow groups of a variety over a finite field?
It is know that $H^{2i,i}_\mathrm{mot}(X,\mathbf{Z}) = CH^i(X)$. In how many cases does this help us?
user19475
9
votes
1
answer
546
views
Showing subgroups with equal Lie algebras are equal
Let $k$ be a field. It might as well be algebraically closed, but I do not want to assume that it has characteristic $0$. I will write "group" for "affine group scheme over $k$", ...
- 12.9k
9
votes
1
answer
893
views
Universal homeomorphism of stacks and etale sites
A morphism between schemes is a universal homeomorphism if it is integral, surjective, universally injective. For morphism between algebraic stacks, this notion also make sense.
It is well know that ...
- 313
9
votes
1
answer
777
views
Geometric (or intuitive) interpretation of additional derivatives in characteristic p > 0
In characteristic $p > 0$ there are "extra" differential operators, i.e., ones that are outside the algebra generated by first-order derivations.
Is there any interpretation of these operators in ...
- 139
9
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1
answer
1k
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étale cohomology with G_m coefficients
Most calculations of étale cohomology in Milne's book deal with constructible or torsion sheaves. Are there references where the cohomology of varieties with $\mathbf{G}_m$ coefficients are calculated?...
user19475
9
votes
1
answer
381
views
Lifting of families of curves to characteristic 0
Let $k$ be a finite field, $X_0$ be a smooth affine variety over $k$ and $C\rightarrow X$ a smooth projective family of curves of genus $\geq 2$.
By a result of Elkik we can always lift $X_0$ to a ...
- 728
9
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1
answer
1k
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Coderivations of $S(V)$ correspond to linear maps $S(V) \to V.$ Only over characteristic $0$?
Definition. Let $k$ be a commutative ring. Let $V$ be a $k$-module. We turn the symmetric algebra $\mathrm{S}\left(V\right)$ of $V$ into a graded Hopf algebra by defining the comultiplication
\begin{...
- 33.8k
9
votes
1
answer
819
views
Giraud's proper base change for Gerbes - Elimination of Noetherian hypotheses
I was looking through Giraud's book Cohomologie Non-abelienne, and there is a very nice theorem that Giraud proves in the Noetherian case (Cohomologie Non-Abelienne VII.2.2):
Let $f:X\to Y$ be a ...
- 19.6k
9
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1
answer
430
views
Existence of certain endomorphism of supersingular elliptic curve
Let $E$ be a supersingular elliptic curve over an algebraically closed field $K$ of characteristic $p$. Let $R = \operatorname{End}(E)$ be its ring of endomorphisms. Then, it is known $R \otimes_{\...
9
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1
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356
views
Reference request for indecomposable representations of $\mathfrak{sl}(2)$ over an algebraically closed field of characteristic $p > 0$
Is there a good reference on the classification of indecomposable representations of the Lie algebra $\mathfrak{sl}(2)$ over an algebraically closed field of characteristic $p > 0$ (of course, in ...
- 239
9
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1
answer
429
views
Is there a Poincare residue in characteristic $p$?
The Poincare residue I mean is there one here:
https://en.wikipedia.org/wiki/Poincar%C3%A9_residue
Basically, I would like a nice way to use a meromorphic $n$-form on $\mathbf{P}^n_{\mathbf{F}_p}$ ...
- 4,164
9
votes
1
answer
449
views
Structure of the variety of $n$-tuples of $m \times m$ matrices with zero product
Consider the functor sending a commutative ring $R$ to $\{(A_1,\dots,A_n) \in ( M_m(R) )^n | A_1 \dots A_n =0 \}$ which defines a scheme over $\mathbb Z$, let $X$ be its base change to $\mathbb C$.
...
- 6,622
9
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0
answers
692
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 ...
- 773
9
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0
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300
views
How did Jouanolou define the cup product with no finiteness hypotheses in SGA 5?
In SGA 5 Exposé VII, at the beginning of §2, Jouanolou lets $X$ and $Y$ denote two schemes, $f:X\rightarrow Y$ a morphism, and $A$ the ring $\mathbf{Z}/\nu\mathbf{Z}$ where $\nu$ is an integer prime ...
- 1,217
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0
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355
views
Does etale homotopy type see the existence of rational points?
Do there exist two smooth projective schemes over $\mathbb{Q}$ that are etale homotopy equivalent and only one of them has a $\mathbb{Q}$-point?
user137767
9
votes
0
answers
703
views
étale vs syntomic vs flat cohomology
Let $\mathscr{A}/X$ be an abelian scheme over $X$ of characterisitic $p$. The étale topology is not fine enough for the Kummer sequence for $\mathscr{A}$ to be (right) exact, but the syntomic and flat ...
user19475
9
votes
0
answers
206
views
Eilenberg-Moore spectral sequence in etale cohomology?
Let $X,Y \rightarrow S$ be schemes over an algebraically closed field $k$. (Actually I'm interested in the case where they are stacks, but I'll ignore that for now.) The vague form of my question is: ...
- 2,809
9
votes
0
answers
201
views
Etale maps and local intersection cohomology
Suppose that $f:(X,x) \to (Y,y)$ is etale at $x$, meaning that it induces an isomorphism $C_xX \to C_yY$ on tangent cones. Then $f$ induces an isomorphism from the cohomology of $IC_{X,x}$ (the stalk ...
- 2,537
9
votes
0
answers
560
views
Function fields of characteristic p modular curves, and mod p reductions of the classical modular equation
Let l and p be distinct primes, l>2. There are "characteristic p modular curves" X_0(l) and X(l), defined over an algebraic closure, K, of Z/p, solving moduli problems for elliptic curves with some ...
- 5,422
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0
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1k
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Ample vector bundles, $H^1=0$ and global generation in characteristic $p$
This is a follow up from Ample vector bundles on curves question, in characteristic $p>0$. First, some background. It is known that in characteristic $0$, a vector bundle $E$ on say a projective ...
- 2,976
8
votes
2
answers
8k
views
What does "supersingular" mean?
Are supersingular primes and supersingular elliptic curves related?
(this was essentially a subquestion in my earlier question, but still looks sufficiently different to me to deserve a separate post)...
- 15.1k
8
votes
2
answers
2k
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The Mumford-Tate conjecture
The Mumford-Tate conjecture asserts that, via the Betti-étale comparison isomorphism, and for any smooth projective variety $ X $, over a number field $ K $, the $ \mathbb{Q}_{ \ell } $-linear ...
- 595
8
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1
answer
787
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Commutative algebra counterexample
Let $M$ be an $R[x]$-module, such that $M$ is finitely generated as an $R$-module.
Does there exist one such $M$, such that $M\otimes_{R[x]}R[x,x^{-1}]$ is not finitely generated as an $R$-module?
user120812
8
votes
2
answers
532
views
A very elementary question on the definition of sheaf on a site
I'm now studying the etale cohomology with the book 'Introduction to Etale Cohomology' by Tamme.
In the page 26 of the book, 'a family of effective epimorphisms' is introduced.
'A family $\{ U_{i} \...
- 1,013
8
votes
2
answers
1k
views
Lefschetz on étale fundamental group for quasi-projective varieties
If $X$ is a smooth projective variety of dimension at least $3$ over $\mathbb{C}$, Lefschetz's Hyperplane theorem says that for every hyperplane section $H$
$$\pi^1(H)\to\pi^1(X)$$
is an isomorphism, ...
- 483
8
votes
2
answers
2k
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etale cohomology of an abelian variety and its dual
Let $A$ an abelian variety over a field $k$ and $A^{*}$ the dual abelian variety.
How can we relate the étale cohomology of $A$ with etale cohomology of $A^{*}$?
- 3,472
8
votes
1
answer
683
views
Etale Cohomology of Punctured Spectra of Local Rings
Let $R=\mathbb{C}[[x,y]]$ be a power series ring in two variables (or maybe more generally a strictly Henselian local ring) with maximal ideal $\mathfrak{m}$.
What is $H^*_{et}(\operatorname{Spec}...
- 23k
8
votes
1
answer
2k
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Cohomology of Grothendieck topology
My naïve cartoon picture of the construction of étale cohomology is this:
start with a scheme, associate to it a Grothendieck topology (making a site).
A functor from the Grothendieck topology to ...
- 2,050
8
votes
1
answer
1k
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obstruction to smooth lifting of smooth schemes
According to general theory, for a square zero thickening defined by an ideal I: SpecA -> SpecA', there is an obstruction of lifting a smooth scheme X over A to a smooth scheme over A' living in H^2(X,...
- 5,052
8
votes
1
answer
2k
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Cohomology of a constant etale sheaf
Let $X$ be a smooth proper algebraic variety over $\mathbb{C}$.
I know that in the analytic world, there is an isomorphism between the de Rham cohomology and the cohomology of the constant sheaf $\...
- 1,500
8
votes
1
answer
1k
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motivic t-structure and realisations
Let $k$ be a field and $DM_k$ denote the triangulated category of geometric motives with $
\mathbb{Q}$ coeffients over $k$. Recall that there exists a motive functor $M: Var_k\rightarrow DM_k$, which ...
- 81
8
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1
answer
639
views
Etale fundamental group of the circle
What is the étale fundamental group of the circle $X({\bf R})$, where
$$
X(k) = \{(x,y) \in k^2 \mid x^2+y^2 = 1\}?
$$
I know that there is a sequence
$$
1 \rightarrow \pi_1^{et}(X({\bf C})) \...
- 18.7k
8
votes
1
answer
403
views
Is $H_{et}^1(X,F) = H^1(\pi_1^{et}(X), F(\bar{k}))$ true?
Let $X$ be a smooth geometrically connected scheme over a field $k$ of characteristic 0 (but not necessarily algebraically closed, I can take it to be a number field). Let $F$ be a finite algebraic ...
- 83