All Questions
1,114 questions
0
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0
answers
196
views
Question regarding étale sheaf under finite étale surjective morphism
Let $X$ and $Y$ be smooth projective varieties over $\mathbb{C}$, and suppose we have a surjective finite étale morphism $f:X\rightarrow Y$ (actually $Y=X/G$ for a free action of a finite group $G$), ...
- 63.9k
5
votes
1
answer
483
views
Does p-adic etale cohomology know the variety has ordinary reduction or not?
For a smooth proper variety $X$ over discrete valuation ring $\mathcal{O}$ of mixed characteristic $(0,p)$, let $X_K$ be the generic fibre over a generic point $\textbf{Spec} K$ and let $X_k$ be the ...
- 16.2k
7
votes
2
answers
2k
views
Weights for etale cohomology: why does Deligne's definition work?
For a field $K$ and a variety $X/K$ (whose characteristic could be $0$) I need a 'simple' explanation for the (Deligne's) method of defining weights of the $l$-adic etale cohomology of $\overline{X}$ (...
- 1,446
11
votes
2
answers
1k
views
Representations of $\mathrm{SL}(2)$ in characteristic 2
$\DeclareMathOperator\SL{SL}$In characteristic zero one can use the Clebsch-Gordan rule to decompose tensor products of $\SL(2)$-modules. In characteristic $p$, things are more complicated.
I am ...
- 63.9k
3
votes
1
answer
459
views
Frobenius functor and length of local cohomology
Let $(R,\mathfrak{m})$ be a Noetherian local ring of positive prime characteristic $p$ and let $F$ be the Frobenius functor. Write $d$ for dimension of $R$. Assume that for some $0\leq i< d $ the ...
- 2,821
4
votes
0
answers
215
views
Reference request: Radicial morphisms & Jacobson-Bourbaki correspondence
My name is Chemy (Przemysław). I am a PhD student at UvA (Amsterdam), and I work on projects related to foliations in algebraic geometry in positive characteristic. Therefore I am avidely reading two ...
- 485
2
votes
1
answer
690
views
Restricted universal enveloping algebra of Abelian p-Lie algebra
Question: Let $p$ be a prime. Let $k$ be a commutative ring such that $p=0$ in $k$.
Let $\mathfrak g$ be an abelian $p$-restricted Lie algebra over $k$. In other words, let $\mathfrak g$ be a $k$-...
- 4,492
6
votes
1
answer
506
views
Degree-2 étale covers of curves in characteristic 2 vs torsion points on the Jacobian
It can be found, at Hartshorne exercise 4.2.7 for example, that in the case where $\operatorname{char}(k) \neq 2$ we have a nice correspondence between etale degree 2 covers of a curve $C$ and 2-...
- 12.9k
4
votes
2
answers
560
views
Dimension is an invariant in the Grothendieck ring of algebraic varieties
$\DeclareMathOperator\Var{Var}$Let $k$ be any field and let $K_0(\Var_k)$ be the Grothendieck ring of $k$-algebraic varieties (i.e. algebraic varieties up to cut-and-paste relations). Given an ...
- 12.9k
3
votes
0
answers
149
views
What direction does the derivation of an inseparable algebraic variable point in?
I've been thinking about the geometry of inseparable field extensions lately, since I'm studying smoothness in commutative rings in an advanced topics course this semester. I've generally come to the ...
- 1,969
3
votes
1
answer
249
views
Sheafifcation for the étale site
Let $X$ be a scheme and $\mathcal{F}$ a presheaf on $X_{ét}$.
For each $x_{i}\in X$, pick a geometric point $\bar{x}_{i}$ over $x$ and denote by $i_{\bar{x}_{i}}:\text{Spec}(k_{i})_{\text{ét}}\...
- 149k
2
votes
0
answers
206
views
Stratified sites/topoi and constructible sheaves
Is it possible to define (possibly derived) categories of constructible sheaves over sites more general than those of open subsets of topological spaces while still retaining essential features, like ...
- 1,516
3
votes
1
answer
129
views
Schur multiplier of finite-dimensional simple Lie algebras in positive characteristic
The Schur multipliers of finite simple groups are known and easily accessible:
https://en.wikipedia.org/wiki/List_of_finite_simple_groups
Moreover, as a consequence of the second Whitehead's Lemma, if ...
- 5,878
6
votes
1
answer
351
views
Adèlic points and algebraic closure
Consider $\mathcal{X}$ a projective and flat scheme over $\text{Spec}(\mathcal{O}_K)$, with $\mathcal{O}_K$ the ring of integers of a number field $K$.
Let $F/K$ vary over all finite Galois number ...
- 8,374
2
votes
1
answer
515
views
Galois invariants and tensor products
Consider a number field $K$ and a finite Galois field extension $L/K$. Let $E$ be an elliptic curve over $K$ and consider the abelian group
$$E(L)\otimes L^{\times}.$$
Every element $g$ in $\text{Gal}(...
- 28.2k
4
votes
1
answer
419
views
Tate-Shafarevich groups under finite Galois field extensions
Suppose $L/F$ is a finite Galois extension of number fields. Let $E$ be an elliptic curve over $F$ and $E_L$ its base change to $L$.
Do we have that $\text{Sha}(E/F)$ is finite if and only if $\text{...
- 8,945
6
votes
1
answer
643
views
Classification of simple Lie algebras over finite fields
Classification of simple (or simple-restricted) Lie algebras over algebraically closed fields in positive characteristic is studied for a long time. Today, we know all finite-dimensional simple (or ...
- 5,878
2
votes
0
answers
171
views
Monogenic function fields
Recall that a number field $K$ is said to be monogenic if its ring of integers is of the form $\mathbb{Z}[\alpha]$, or equivalently, if it has a power integral basis. There are many references one can ...
- 103
27
votes
4
answers
3k
views
Have people successfully worked with the full ring of differential operators in characteristic p?
This question is inspired by an earlier one about the possibility of using the full ring of differential operators on a flag variety to develop a theory of localization in characteristic $p$. (Here ...
- 1,228
3
votes
0
answers
317
views
About the inverse function theorem in the étale topology
It's clear that the étale topology is closed related to some form of inverse function theorem. Let me give some reasons. (Also, see the comments here.)
(1) A morphism $f:X\to S$ between smooth ...
- 773
4
votes
1
answer
198
views
Simple restricted but not restricted simple Lie algebras
Let $F$ be a field which has a positive characteristic $p \ge 2$ and $(\mathfrak{g},[p])$ be a restricted Lie algebras over a field $F$ where $[p]$ is a $p$-th power map on $\mathfrak{g}$. $(\mathfrak{...
- 5,878
2
votes
1
answer
185
views
Finite, normal subgroups of reductive groups in positive characteristic
Consider the following statement about a connected, reductive group $G$ over a field $k$:
Every finite, normal subgroup $N$ of $G$ is central.
In characteristic $0$, this is true, and the proof is ...
- 12.9k
8
votes
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})) \...
- 2,866
3
votes
1
answer
355
views
Puncturing a curve and étale fundamental groups
Let $X$ be a smooth projective curve over $\mathbb{C}$ and puncture it at a point $x$. We get a map from $spec(\mathbb{C}((z)))$ to the punctured curve $X^{°}$ corresponding to this puncture. Is the ...
- 18.7k
2
votes
0
answers
216
views
$G$-torsor over $\mathbb{A}^1_S$ where characteristic of $S$ does not divide $|G|$
I am reading the paper $\mathbb{A}^1$-homotopy theory of schemes by Morel and Voevodsky from 1999. There is a proposition saying that
Let $G$ be a finite étale group scheme over $S$ of order prime to ...
- 1,168
6
votes
0
answers
470
views
Étale cohomology of the field with one element
In the function field - number field analogy, some expect progress on RH to come from reproducing various aspects of the Grothendieck program in a way where $\mathbb{Z}$ could be treated as a function ...
- 3,446
2
votes
0
answers
287
views
Frobenius endomorphism is not flat
I am actually going through "Twenty four hours of local cohomology". They discuss the Frobenius endomorphism in Chapter 21. Here is the Exercise 21.6 which I am finding hard to solve:
Find a ...
- 63.9k
2
votes
0
answers
161
views
Nearby cycle is tamely ramified?
Let $S$ be a henselization of a closed point $s$ in a smooth algebraic curve $C$ over some finite field $\mathbb{F}_q$. Then we can consider nearby cycles over $S$. Let $s$ be the closed point of $S$ ...
- 71
3
votes
0
answers
173
views
Smooth proper varieties over the integers that are not toric
Does there exist a smooth proper variety $X$ over $\operatorname{Spec} \mathbb Z$ that is not toric?
By Fontaine, we know that there is no Abelian scheme over $\operatorname{Spec} \mathbb Z$. Also by ...
- 7,746
2
votes
1
answer
146
views
Should the identity labelled by red line be $\overline{f(Z)}=X$?
The above picture is from Milne's Etale Cohomology.
Suppose $A=\Bbb Z, \mathfrak q=(2T+3)$, consider $Z=\operatorname{Spec} \Bbb Z[T]/(2T+3)\to \operatorname{Spec} \Bbb Z[T]\to\operatorname{Spec} \Bbb ...
- 755
16
votes
2
answers
1k
views
Is the tangent space functor from PD formal groups to Lie algebras an equivalence?
The previous version of this question was rather badly broken, and I hope this version makes some sense.
There have been a few questions on MathOverflow about how much representation-theoretic ...
- 35.5k
3
votes
0
answers
658
views
Most general form of Poincaré duality in étale cohomology
I am interested in Poincaré duality from the point of view of Grothendieck's 6-functor formalism. I am predominantly interested in the proof that Poincaré duality holds in étale cohomology from this ...
- 63.9k
4
votes
0
answers
231
views
How big are small inverse powers of 2 mod powers of 3?
Let $a \bmod b$ take value as an integer in $[0, b)$. For any $T \ge 1$, for what $R \in [0, 3^n)$ is
$$\min \{2^{-t}\bmod 3^n: t =1, \dotsc, T\} := \min A_T > R?$$
When $T$ is fixed as $n$ ...
- 459
5
votes
0
answers
139
views
Liouville property in the Bost theorem on foliations
Let $X$ be a smooth algebraic variety over a number field $K$, and let $\mathcal{F}$ be an involutive coherent subsheaf of the tangent bundle. After a pullback to $\mathbb{C}$, $\mathcal{F}_\mathbb{C}$...
- 485
2
votes
1
answer
156
views
Extending the domain of the yoneda embedding map from étale schemes to the small étale topos so that it is still fully faithful
Let $X$ be a scheme. For $Y$ a scheme over $X$, the representable presheaf $h_Y : U\mapsto \mathrm{Hom}_X(U,Y)$ on the small étale site $X_{et}$ is actually a sheaf, and by the Yoneda lemma the ...
- 4,492
10
votes
2
answers
2k
views
Is there a "universal" cohomology theory for varieties over p-adic fields?
Let $K$ be a $p$-adic field, $X$ a smooth proper algebraic variety over $K$, and $0 \le i \le 2 \dim X$. For a prime $\ell \ne p$ one can consider the $\ell$-adic cohomology $H^i(\overline{X}, \mathbb{...
- 12.9k
5
votes
1
answer
323
views
On realizing a topos of sheaves as a topos of equivariant sheaves
This question is motivated by the following example : let $X$ be a variety over a field $k$, with algebraic closure $\bar{k}$. The Galois group $G_k:=\mathrm{Gal}(\bar{k}/k)$ acts on $X_{\bar{k}}:=X\...
- 1,329
6
votes
0
answers
173
views
Orlik-Solomon algebra and hyperplane complements in positive characteristic
Let $k$ be an algebraically closed field of characteristic $p\geq 0$, $\underline H:=\{H_1,\dots, H_m\}$ a set of hyperplanes in $\mathbb A_k^n$ and $X:=\mathbb A^n-(\bigcup H_i)$.
Given a ring $R$ ...
- 728
11
votes
4
answers
3k
views
What does ramification have to do with separability?
Does ramification have anything to do with inseparability? It feels like an extension of Q in which p ramifies should somehow correspond to an extension of F_p(t). Does totally ramified <--> purely ...
- 411
3
votes
0
answers
274
views
Ind-etale vs weakly etale
In this article Bhatt and Scholze consider ind-etale and weakly etale maps of affine schemes. We have two (easy) statements, proven in Prop.2.3.3(1) and (5):
-- any ind-etale map is weakly etale,
-- ...
- 755
6
votes
0
answers
113
views
$S_n$-invariant polynomials on the dual of reflection representation
Let $W=S_n$ (the symmetric group) acting on $V=k^n$ via permutation of the indices, where $k$ is an algebraically closed field. Closely related to this are the reflection (sub-)representation $V_0=\{ (...
- 1,336
21
votes
5
answers
5k
views
Mirror symmetry mod p?! ... Physics mod p?!
In his answer to this question, Scott Carnahan mentions "mirror symmetry mod p". What is that?
(Some kind of) Gromov-Witten invariants can be defined for varieties over fields other than $\mathbb{C}$...
- 35.5k
5
votes
1
answer
533
views
$\mathbf{A}^1$-invariance of Brauer groups and $H^2_{\mathrm{et}}(-;\mathbb{G}_m)$
The $\mathbf{A}^1$-invariance of vector bundles have been discussed in, for example, this paper by Asok, Hoyois and Wendt. This of course implies storng $\mathbf{A}^1$-invariance results for the first ...
- 2,017
2
votes
0
answers
167
views
Explicit construction of a presentation of a constructible sheaf of $\mathbb{Z}$-modules
This question was prompted by the two following:
Constructible étale sheaves on X are étale algebraic spaces over X
Naive question about constructing constructible sheaves
If I have a ...
- 617
1
vote
0
answers
128
views
understanding higher direct images of $\mathbb{G}_m$ for a finite Galois map
Let $X$ be a smooth quasi-projective variety over $\mathbb{C}$, and let $\mu_r$ denote the group of $r$-th roots of unity, and moreover suppose $\mu_r$ (algebraically) acts on $X$ freely. Then $Y:= X/\...
- 737
1
vote
1
answer
680
views
Cohomology with coefficients in $\mu_\infty$
I'm encountering a lot of problems when dealing with the root of unity sheaf $\mu_\infty := \mathrm{colim}_n\mu_n$.
Let $X$ be a smooth geometrically integral variety over a number field $k$. Although ...
- 1
1
vote
1
answer
220
views
Lowest weight of compactly supported cohomology with coefficients
Let $X_0/\mathbb F_q$ be a variety, and let $\mathcal F$ be a Weil sheaf on $X := (X_0)_{\overline{\mathbb{F}_q}}$ that is pure of weight $n$. If $j < n$, does the weight $j$ piece of $H^i_c(X,\...
- 149k
3
votes
0
answers
174
views
When the Leray spectral sequences for nice compactifications give the Deligne's weight ones?
Assume that $X$ is a proper smooth variety over an algebraically closed field $k$, $U=X\setminus (\cup D_i))$ where $D_i$ are closed subvarieties such that the set-theoretic intersections of all sets ...
- 16.9k
1
vote
0
answers
241
views
Higher direct images of $\mathbb{G}_m$ under a projective bundle
Let $X$ be a smooth projective rational variety over $\mathbb{C}$, and let $\pi:Y\rightarrow X$ be a principal projective bundle with fibers isomorphic to $SL(n,\mathbb{C})/P$, where $P$ is a ...
- 737
19
votes
1
answer
2k
views
The Infinitesimal topos in positive characteristic
This question was inspired by and is somewhat related to this question.
In his article "Crystals and the de Rham cohomology of schemes" in the collection "Dix exposes sur la cohomologie ...
- 19.4k