All Questions
1,114 questions
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
4
votes
1
answer
502
views
Motivic integration in positive characteristic: how much is known?
It seems that in papers on motivic integration people usually assume the base field to have characteristic $0$ (and algebraically closed?). My question is: how much can one prove over a positive ...
- 16.9k
0
votes
1
answer
158
views
If $J$-coverings can be glued $I$-locally is $J$-locality an $I$-local property? (Reducing descent problems to simpler ones)
Let $(C,J)$ be a category with a grothendieck topology. For every object $X \in C$ there's (I hope) a little site which is the full subcategory of the slice category $C_{/X}$ whose objects are the ...
- 7,799
11
votes
1
answer
675
views
Extended Deformation Theory (dg-Lie algebra principle in positive characteristic?)
Recently, I looked at articles that make use of Deligne's idea that "in characteristic 0 every deformation problem is governed by a differential graded
Lie algebra" as explained first in Goldman-...
- 113
7
votes
1
answer
2k
views
Frobenius weights on etale cohomology and purity
Let $X_0$ be a smooth variety (for simplicity I'm willing to assume that X is a curve) over a finite field $k$, $X$ its geometric base change, and $\mathcal{F}$ an $l$-adic etale sheaf on $X$ with $\...
- 469
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
4
votes
2
answers
429
views
étale cohomology via Cech cocycles for a quasi-projective scheme
I am looking for the explicit reference to the fact that for a quasi-projective scheme a class in the étale cohomology of a sheaf of a certain degree can by computed using Cech cocycles.
- 41
3
votes
0
answers
243
views
Relation between Galois and etale cohomologies
Let $D$ be the ring of integers of a number field $F$.
Let $X=\mathrm{Spec} ~D$, and let $\pi$ be the etale fundamental group of $X$.
There are natural maps from $H^i(\pi, \mathbf{Z}/n)$ to $H^i_{...
- 367
6
votes
3
answers
912
views
étale cohomology with values in an abelian scheme is torsion?
Let $A/X$ be an abelian scheme. Is $H^n(X,A)$ torsion for $n > 0$?
Perhaps this can be proved analogously as Proposition IV.2.7 of Milne's Étale cohomology (where it is proved that the ...
user19475
2
votes
0
answers
121
views
Global invariant cycles in positive characteristic
Let $k$ be an algebraically closed field of positive characteristic $p$ and fix a prime $\ell\neq p$. Let $X$ be a smooth connected $k$-variety and $f:Y\rightarrow X$ a smooth projective morphism. ...
- 728
5
votes
1
answer
505
views
Flat cohomology for finite infinitesimal group scheme over a perfect field
Let $G$ be a finite infinitesimal group scheme (e.g.$\mu_p,\alpha_p) $ over a perfect field $k$, how much is known about $H^1_{fppf}(k,G)$?
- 147
2
votes
0
answers
345
views
Examples of semi-stable models of curves
Let $R$ be a discrete valuation ring with fraction field $K$ of characteristic zero and residue field $k$ of characteristic $p>0$. Assume $k$ is algebraically closed. I want to produce examples of ...
- 2,323
2
votes
0
answers
286
views
Does the sheaf of locally exact differential forms splitting in positive characteristic
Let k be an algebraically closed field of characteristic $p>2$, $X$ a smooth projective curve of genus $g>1$ over $k$, and $F_X:X\rightarrow X$ be the absolute Frobenius morphism. Let $B^1_X$ be ...
- 85
2
votes
1
answer
1k
views
Stalks of higher direct image under open embedding
Let $U$ be an open subset of $\mathbb P^1$ without two points (say $t=0$ and $t=\infty$) and $j: U\to \mathbb P^1$ be an open immersion. Ground field $k$ is algeraically closed. Let $G$ be the group ...
- 23
5
votes
2
answers
287
views
an explicit weak equivalence between $B{\mathbb G}_m$ and ${\mathbb P}^\infty$
OK, so I asked a similar question before; $B{\mathbb G}_m$ is a simplicial presheaf over number field $k$. I see that there is some $A^1$-homotopy equivalence between the sheaf represented by ind-...
- 193
4
votes
1
answer
242
views
$l$-dependence of the group of homologically zero cycles
Consider the class map $$cl:CH^i(X)\to H^{2i}_{cont}(X,\mathbb{Z}_l(i))$$ where the RHS is the continuous etale cohomology(defined by Jannsen in his paper "Continuous etale cohomology"). In this paper ...
- 7,377
4
votes
0
answers
197
views
Inequalities between numerical invariants of nonsingular projective Varieties in positive Characteristic
It is well-known that Miyaoka and Yau-type inequalities do not hold in positive characteristic. In "a note on Bogomolov-Gieseker’s inequality in positive characteristic", however, we can ...
- 4,902
3
votes
0
answers
113
views
Cohomologies of $[V/GL_n]$ in characteristic $p$ for a representation $V$ of $GL_n$
Let $V$ be a representation of $G=GL_n$ (or more generally any reductive group $G$) over an algebraically closed field $\mathbb k$ of characteristic $p$. Let $[V/G]$ be the corresponding quotient ...
- 790
2
votes
0
answers
282
views
Can one compute the (etale) cohomology with support at a point for a "big" regular $k$-scheme via limit arguments?
I am trying to understand the coniveau spectral sequence for the cohomology of a "big" regular scheme over a field. This involves cohomology with support at points, and I am getting some strange ...
- 16.9k
14
votes
1
answer
1k
views
The "Level N modular equation for delta" in characteristics 3, 5, 7 and 13
When $N > 1$, the modular forms $\Delta(z)$ and $\Delta(Nz)$ are algebraically independent over the complexes, and the same then is true of their expansions at infinity. But using the fact that
the ...
- 5,422
8
votes
1
answer
1k
views
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
5
votes
1
answer
1k
views
Excellent schemes
I noticed that many results in positive characteristic assumes that the object of the theorem is excellent. I have looked up the definition of excellent and have tried to get a feeling for it, but all ...
- 53
2
votes
0
answers
237
views
Intrinsic notions of étale?
The usual notion of trivial covering morphism is in a sense intrinsic to the adjunction $\Pi_0\dashv H$ between connected components and discrete spaces: a continuous map $f$ is a trivial covering ...
- 10.5k
8
votes
0
answers
301
views
Intuition for local Lefschetz theory in SGA2
In SGA2,Grothendieck introduced two important examples（EXP X 2.1） which satisfy Grothendieck-Lefschetz condition.My question is what's the intuition for the "local" version of Lefschetz's theory ?
I ...
- 388
8
votes
0
answers
286
views
Functorial classes in Brauer group
For a smooth variety $X$ over a perfect field of characteristics $p$ the sheaf of differential operators is an Azumaya algebra(etale locally is isomorphic to endomorphisms of its center, which is ...
- 7,377
3
votes
1
answer
328
views
Étale coverings of cubics and gluings
Consider the plane nodal cubic $X$ given by the equation $y^2=x^2(x+1)$ over a field $k.$ It is not too hard to show that $Y$ has a finite étale covering $X$ of degree $2.$ One does this in the ...
- 31
6
votes
1
answer
1k
views
Comparison of cycle maps
Let $X$ be an algebraic variety over $\bar{\mathbb{Q}}$ of dimension $d$, then there is the l-adic cycle map $\mathrm{cl}_{et}:\mathrm{CH}^i(X)\rightarrow\mathrm{H}^{2i}(X,\mathbb{Q}_\ell(i))$ from ...
- 273
4
votes
0
answers
247
views
Is there an analogue of the linking pairing in etale, crystalline, etc cohomology theories?
Let $M$ be a compact oriented manifold of dimension $n$. It is well-known that there is a perfect intersection pairing $$H_k(M;\mathbb{Z})_{torsion\,\,free}\otimes H_{n-k}(M;\mathbb{Z})_{torsion\,\,...
- 1,493
3
votes
1
answer
343
views
Two spaces attached to mod 2 level 9 modular forms--a conjectural Hecke isomorphism
MOTIVATION
Nicolas and Serre have analyzed the structure of the space of mod $2$ modular forms of level $1$, viewed as a "Hecke-module". They show that for each $p>2$, the operator $T_p$ acting on ...
- 5,422
8
votes
1
answer
570
views
Continuity of l-adic cohomology: is the cohomology of the generic point isomorphic to the completion of the limit of cohomology of open subvarieties?
Let $X$ be a variety over an algebraically closed field $k$. Denote by $\eta$ its generic point; it is the inverse limit of the open subvarieties $X_i$ of $X$. It is well known that the etale ...
- 16.9k
2
votes
1
answer
377
views
When is a $\overline{\mathbb{Q}}_{\ell}$-local system the inverse image of a $\overline{\mathbb{Q}}_{\ell}$-local system?
I am trying to learn character sheaf theory, and encounter the following question:
(*) Let $f\colon X\rightarrow Y$ be a morphism of quasi-projective smooth varieties over $\overline{\mathbb{F}}_q$, ...
- 1,666
1
vote
0
answers
187
views
Unitary dual of the Heisenberg group over non-archimedean local fields of characteristic two
What is the unitary dual of the Heisenberg group over non-archimedean local fields k of characteristic two? This is well-known for the real Heisenberg group and in fact, when local fields have ...
- 1,702
2
votes
1
answer
275
views
A homotopy argument in etale topology
Suppose everything below is defined over $k=\overline{\mathbb{F}}_q$.
Let $H$ be a connected algebraic group acting on a separated variety $Y$. Denote the morphism $H\times Y\rightarrow H\times Y; (h,...
- 1,666
2
votes
0
answers
72
views
Support of étale sheaves
Let $X$ be a scheme, $i: Z\to X$ a closed subscheme, $j: U\to X$ its complement in $X$. Assume the codimension of $Z$ in $X$ is large (at least $2$).
Let $A$ be an étale sheaf on $U$, $B$ an étale ...
user120812
8
votes
1
answer
426
views
When does the continuous Galois(=etale) cohomology of fields coincide with the naive one? Often true by the Bloch-Kato conjecture?
For a field $F$ I am interested in its $l$-adic (Galois=\'etale) cohomology; here $l$ is a prime distinct from the characteristic of $F$ (for simplicity one may assume that the latter is $0$).
For $...
- 16.9k
1
vote
0
answers
251
views
When is $\mathbb{Q}_X$ pure?
I'll ask this question in the language of mixed Hodge modules, since that's where I'm coming from, but the question has an exact analogue for mixed l-adic complexes on schemes over fields of positive ...
- 351
2
votes
1
answer
414
views
Etale cohomology and topological invariance
Let $X$ be a projective scheme and $X_0 \subset X$ a subscheme defined by a nilpotent ideal. Denote by $i:X_0 \to X$ the closed immersion. Let $\mathcal{F}$ be a locally free sheaf sheaf on $X_{\...
- 503
2
votes
1
answer
505
views
Higher direct image of locally constant torsion sheaf (étale cohomology)
Let $\phi:X\rightarrow Y$ be a generically smooth projective surjective morphism of algebraic varieties over $k=\bar k.$ Is it possible for $R^1\phi_*(\mathbb Z/l)$ to be supported on a divisor of $Y$ ...
- 155
8
votes
1
answer
747
views
Deligne's exterior power
In "Catégories Tannakiennes", Deligne defines the $n$th exterior power of an object $A$ of an abelian tensor category $\mathcal{C}$ as the image of the morphism
$$p : A^{\otimes n} \to A^{\otimes n}, ...
- 63.1k
8
votes
1
answer
1k
views
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
1
vote
0
answers
251
views
Stalks of derived tensor product (in the Kunneth formula)
So, essentially here's what I'm curious about. Suppose that $k$ is a (separably closed/algebraic closed) field $X_i,Y_i/k$ are finite type and $f_i:X_i\to Y_i$ are $k$-maps (all of this for $i=1,2$). ...
- 843
9
votes
2
answers
656
views
How does the order of a pole of a zeta function indicate any geometric information?
Here, I'm primarily concerced about zeta functions of hypersurfaces over fields of finite characteristic.
Assume $F_q$ to be a finite field with q elements. Consider the zeta function of the ...
- 141
23
votes
1
answer
2k
views
Wanted: Quadratic Space in Characteristic 2 as a Counterexample to a Theorem of Arf
Hi. Peter Roquette sent me an email asking for an example of a quadratic space in characteristic 2 having certain features. I have no idea on this, but maybe someone reading this does.
He would ...
- 50.8k
22
votes
3
answers
813
views
A hypersurface with many points
Ok, it's time for me to ask my first question on MO.
Consider the affine curve $Y+Y^q=X^{q+1}$ over the finite field $\mathbf{F}_q$. It's interesting because it has the largest number of points over ...
- 4,487
8
votes
0
answers
566
views
Gysin exact sequence for a singular subvariety
Let $k$ be an algebraically closed field (I'm interested in a characteristic $p>0$ specific example) and let $X$ be a (smooth if needed) algebraic variety.
Let $Y \subset X$ be a (possibly) ...
- 2,521
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
1
vote
0
answers
132
views
Independence of $\ell$-adic homological equivalence of $\ell$
Let $X$ be a smooth projective variety over an algebraically closed field $k$. Are there examples of $X$ and $\ell\neq char(k)$ for which homological equivalence with coefficients $\mathbb Z_{\ell}$ ...
- 1,463
7
votes
1
answer
2k
views
Is there an integral version of Faltings' isomorphism in p-adic Hodge theory between etale and Hodge cohomologies
Let $K$ be a $p$-adic field, that is a complete discrete valuation ring of characteristic $0$ with a perfect residue field $k$ of characteristic $p > 0$ (to simplify one could also take $K$ to be a ...
- 5,019
9
votes
1
answer
1k
views
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
5
votes
1
answer
2k
views
A Kunneth formula for the etale cohomology of the product of ('simple') varieties over not (necessarily) algebraically closed field
If $X$ and $Y$ are varieties over an algebraically closed field, then in the corresponding derived category of complexes we have $RH_{et}(X\times Y,\mathbb{Z}/l^n\mathbb{Z})\cong RH_{et}(X,\mathbb{Z}/...
- 16.9k