All Questions
Tagged with characteristic-p or characteristic-p
393 questions
3
votes
0
answers
120
views
Resolving the "wild" singularities of $\mathbb A^n/C_n$
Let the cyclic group on $n$ elements, $C_n$, act on $\mathbb A^n$ by permuting the co-ordinates (over a field $k$). If $n \neq 0 \in k$, we can resolve the singularities of $X = \mathbb A^n/C_n$ by ...
- 7,746
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
0
answers
197
views
How to write down an explicit equation of given degree yielding a smooth hypersurface in a projective space?
Let F be a field of positive characteristic $p$ and let $d,n$ be two positive integers.
Can we explicitly write down an equation defining a smooth hypersurface $X_d⊂\mathbb P^n_F$ of degree d ?
This ...
- 417
3
votes
0
answers
218
views
Is there a proper smooth variety in characteristic $p$ whose Hodge-to-de Rham spectral sequence does not degenerate at $E_1$?
By Deligne-Illusie, such a variety has no lifting to $W_2(k)$. In their paper they state that they do not know if such an example exists. Has this question been answered since then?
- 4,164
3
votes
0
answers
232
views
Lifting a Frobenius endomorphism under an étale morphism
Let $X$ be a smooth affine scheme over $\mathbb{Z}/{p^2}$ that is a complete intersection, say $X$ is the spectrum of $\mathbb{Z}/{p^2}[x_1,...x_n]/(f_1, ... f_r)$, where $n-r$ is the dimension of $X$....
- 111
3
votes
0
answers
143
views
What kind of equivalences exist between categories of characteristic $0$ and characteristic $p$?
The tilting equivalence for perfectoid algebras gives an equivalence of categories $$K\text{-perf} \cong K^\flat\text{-perf}$$
where the left-hand-side are algebras in characteristic zero and the ...
- 4,164
3
votes
0
answers
112
views
Indecomposablity in purely inseparable extensions
Let $k$ be a field of characteristic $p$ (e.g. the separable closure of $\mathbb{F}_p(t)$) and consider the extension $k(x)/k$ where $x^p\in k$ but $x$ does not. Consider a (finitely generated) ...
- 31
3
votes
0
answers
307
views
Isotrivial factors of Jacobian
Let $k$ be an algebraically closed field of positive characteristic that it is not the algebraic closure of a finite field. Fix a smooth proper $k$-curve $C$ and write $J_C$ for its Jacobian abelian ...
- 728
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
3
votes
0
answers
742
views
Explicit description of O^{cris}_n in Fontaine/Messing
Let $k$ be a perfect field of characteristic $p$, $W(k)$ the Witt ring and $K$ its quotient field. In their article "$p$-adic periods and $p$-adic etale cohomology" Fontaine and Messing give in II.1.4 ...
3
votes
0
answers
308
views
Invertible Hasse-Witt for non-ordinary curves
Assume that $S$ is a smooth curve of over a field $k$ of characteristic $p>0$ and $f\colon X\to S$ is a relative curve over $S$ (i.e., the fibers are curves). It is well-known that when all the ...
- 395
3
votes
0
answers
281
views
What would be a characteristic-$p$ analogue for $C^{\infty}$-fiber bundles?
I'd like to know a notion for a morphism between algebraic varieties in characteristic $p$ that plays the role of a $C^{\infty}$-fiber bundle. It should be, in particular, flat. I'm not assuming the ...
- 4,265
2
votes
1
answer
332
views
Ample bundle under Frobenius morphism
Let $k$ be an algebraically closed field of char($k$)=p>0, $X$ a smooth projective variety over $k$, $F:X\rightarrow X^{(1)}$ the relative Frobenius morphism. Let $E$ be an ample vector bundle on $X$. ...
- 85
2
votes
1
answer
528
views
Is there an easy proof of the fact that the intermediate image functor respects weights?
It was proven in BBD (see Corollary 5.3.2) that for an open immersion $j$ the functor $j_{!*}$ preserves weights of mixed sheaves. The proof relies on several previous results; it is especially ...
- 16.9k
2
votes
1
answer
278
views
What is the family derived from the absolute Frobenius on the Hilbert scheme?
Let $f$ be a Hilbert polynomial, and $X := Hilb_h(P^d_{F_p})$ a Hilbert scheme defined over $F_p$. Then there is an absolute Frobenius map $F: X \to X$. I'm even interested in the case $f \equiv 1$, ...
- 27.9k
2
votes
1
answer
304
views
Connected extensions of finite by connected algebraic groups
Let $1\to H\to E\to G\to 1$ be a short exact sequence of algebraic groups defined over an algebraically closed field $k$ of characteristic $p$. Suppose $H$ is a finite group, and $G$ and $E$ are ...
- 631
2
votes
2
answers
2k
views
Reference of primitive root mod p
Can any body give me a reference of the result about primitive root mod p for a class of prime number p.
The result that I am looking for is something along this line:
$2$ is a primitive root mod $p$...
- 73
2
votes
1
answer
106
views
Does a polynomial $P(X,Y)$ that specializes to a polynomial $P(x_0,Y)$ with distinct roots in $\overline{k}$ have distincts roots in $\overline{k(X)}$
Let $k$ be a field, $P\in k[X][Y]$ be a monic polynomial of degree $n$ in $Y$.
I am looking for a simple proof of the following fact.
"If there exists $x_0\in k$ such that $P(x_0,Y) \in k[Y]$ has ...
- 2,521
2
votes
1
answer
606
views
Do we have Hodge symmetry for char $p$?
Let $X$ be a smooth projective variety over a field $k$. Let $h^{p,q}=dim_k H^q(X,\Omega_{X/k}^p)$ be the Hodge numbers.
If $k$ is of char $0$, by Lefschetz principle, we always have Hodge symmetry, i....
- 547
2
votes
2
answers
645
views
Supersingular elliptic curves and their automorphisms
If $E$ is a supersingular elliptic curve over a finite field of characteristic $p$, what is known about its automorphism group (i.e the stabilizer of a point in algebraic curves terminology).
Do all ...
2
votes
2
answers
480
views
Lifting to char 0, references and questions
Suppose that I have a surface $S$, smooth proper over an algebraically closed (perfect?)field $k$ that lifts algebraically to some $S_W$ defined over a field of char 0. I am interested in properties ...
- 645
2
votes
1
answer
170
views
Automorphism of positive characteristic field
Suppose $K$ is a field with $\text{char}(K) \geq 0$. Let $L$ be a cyclic extension of $K$ with degree $2n$. We consider the generator $\sigma$ of the Galois group $\text{Gal}(L/K)$.
I am interested in ...
- 923
2
votes
1
answer
406
views
Isomorphism between pull-backs of an F-crystal by different liftings of Frobenius
This might be a naive question. But since I haven't seen this in any reference, I'll try to ask it here. Let $T$ be a smooth scheme over the algebraically closed field $k$ of characteristic $p>0$ (...
- 637
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 $k$-...
- 33.8k
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
2
votes
1
answer
361
views
Lie algebroid in algebraic geometry
When I did net-surfing at home, I met some geometric backgrounds of Lie algebras and encountered the concept of Lie algebroids. In differential geometry, a Lie algebroid seems to be defined as ...
- 145
2
votes
1
answer
451
views
Tie-Breaking Trick for Log Canonical Pairs and F-pure pairs in Positive Characteristic
Let $X$ be a projective 3-fold in characteristic $p>0$. Let $(X, D)$ be a klt pair, and $D'$ a $\mathbb{R}$-Cartier divisor such that $D'=A'+B'$, where $A\geq 0$ is an ample $\mathbb{Q}$-divisor ...
- 313
2
votes
1
answer
207
views
Subschemes in group action
Let $k$ be a field, of any characteristic. Let $G$ be a smooth group scheme over $k$ and let $X$ be a smooth scheme of finite type over $k$. Let $Y\subseteq X$ be a smooth subscheme of $X$, and let $...
- 21
2
votes
1
answer
153
views
A question about decomposing mod 2 modular forms of level p^2
Fix an odd prime $p$. Each $f \in \mathbb{Z}/2[[x]]$ can be written as $f_{+} + f_{-} + f_0$ where each exponent k of $x$ appearing in $f_{+}$ (resp. $f_{-}$, $f_0$) has Legendre symbol $(k/p)$ equal ...
- 5,422
2
votes
1
answer
791
views
Branching rule for classical Lie algebras in positive characteristic
The restriction of an irreducible $\mathfrak{sl}_n(\mathbb{C})$-module to $\mathfrak{sl}_{n-1}(\mathbb{C})$ is described by a branching rule which says that if $L(\lambda)$ is the simple $\mathfrak{sl}...
- 2,721
2
votes
1
answer
267
views
On Non F-pure ideal and Sharp F-Purity for a pair $(X, \Delta)$ where $K_X+\Delta$ is NOT $\mathbb{Q}$-Cartier
Suppose $(X,\Delta\ge 0)$ is a pair such that $(p^g-1)(K_X+\Delta)$ is an Integral Weil Divisor for some $g>0$ and $X$ is a normal variety. Define $\mathcal{L}_{e,\Delta} = \mathcal{O}_X( (1-p^g)(...
- 165
2
votes
1
answer
303
views
Lifting vector fields to its resolution in char $p$
In this paper 4.7, the authors showed that on a normal variety $X$, if there is a tangent vector field on its smooth locus, then it can be lifted as a logarithmic tangent vector field on a log ...
- 656
2
votes
1
answer
510
views
hyperalgebras (positive characteristic)
The question is about commutator in integral forms. Let $A$ an associative algebra over a field of characteristic zero, $x\in A$ and $k\in \mathbb Z$, we denote $x^{(k)}=\frac{x^{k}}{k!}$.
How to ...
- 829
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
2
votes
1
answer
646
views
Quotient by p-th roots of unity in characteristic p
Let $X$ be a variety over $k$ of characteristic $p>0$ (you can assume $k$ algebraically closed and $X$ normal) with an action of the group scheme of $p$-th roots of unity $\mu_p = {\rm Spec}\ k[\...
- 16.2k
2
votes
1
answer
160
views
Finding the mistake in an argument concerning $F$-finite $F$-split local Cohen--Macaulay ring of dimension $1$
Let $R$ be a commutative Noetherian ring, and $\phi: R \to R$ be a ring homomorphism. For an $R$-module $M$, let $^{\phi}M$ be the $R$-module defined via restriction of scalars via $\phi$, i.e., as ...
- 412
2
votes
1
answer
307
views
On a Strongly F-regular Pair (X, \Delta)
Let $X$ be a normal projective variety over a field of characteristic $p>0$ and $(X, \Delta\geq 0)$ be a pair such that $K_X+\Delta$ is $\mathbb{Q}$-Cartier whose index is not divisible by $p$. ...
- 165
2
votes
0
answers
127
views
Classification of restricted Lie algebras of reductive groups
$\DeclareMathOperator\Lie{Lie}$Let $G/K$ be a reductive group over a field $K$. In characteristic $0$ the Lie algebra is invariant under base change of fields, so to understand $\Lie(G)$ it is enough ...
- 461
2
votes
0
answers
119
views
Resolution of singularities of the resultant locus
We consider projective space of dimension $n$ as the parameter space of degree $n$ polynomials in one variable. Then, I am interested in resolving the singularities of the "resultant locus" $...
- 7,746
2
votes
0
answers
145
views
How to compute the character of the Steinberg module for the group $\mathrm{SL}_n$ over a field of characteristic $p$?
It is known that the Steinberg representation $V$ of the group $\mathrm{SL}_n$ over a field $k$ of characteristic $p$ (maybe one needs to assume that $k$ is perfect, I am not sure) is the irreducible ...
2
votes
0
answers
47
views
Characters of simple $\mathfrak{sl}_n$-modules in positive characteristic with subregular nilpotent central character
Consider representations of $\mathfrak{sl}_n$ in positive characteristic with a subregular nilpotent central character $\chi$ (i.e. $\chi$ is a nilpotent matrix whose Jordan normal form has two blocks ...
2
votes
0
answers
177
views
How do characters of representations in cohomology depend on the (positive-characteristic) field?
The following sentence appears in Jantzen - Representations of algebraic groups, 2nd edition, p. x, where $G$ is a reductive group over an algebraically closed field $k$, $B$ is a Borel subgroup, $T$ ...
- 12.9k
2
votes
0
answers
253
views
Künneth formula for algebraic de Rham cohomology
Let $X$ and $Y$ be finite type schemes over a field $k$, and let $H^i(X/k)$ denote the $i$-th algebraic de Rham cohomology group of $X$ over $k$. I'm interested in the extent to which a Künneth ...
- 333
2
votes
0
answers
218
views
Borel-Weil-Bott theorem for wonderful compactification in characteristic p
Are there any known results for a Borel-Weil-Bott theorem for the wonderful compactifications over characteristic $p$ (i.e., theorems that classify the cohomologies of all line bundles on a wonderful ...
- 41
2
votes
0
answers
147
views
Automorphism groups of "reductive" Lie algebras in positive characteristic
I put "reductive" in quotes because, of course, in positive characteristic one should speak of Lie algebras of reductive groups, not of reductive Lie algebras.
Let $G$ be a reductive group ...
- 12.9k
2
votes
0
answers
97
views
Non-noetherian Cartier Isomorphism
A result in positive characteristic is that if $R/\mathbb{F}_p$ is a smooth ring, then we have a Cartier isomorphism
$$\Omega_{R}^\bullet\cong H^\bullet(\Omega_R^\bullet)$$
which is essentially ...
- 4,428
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
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 ...
- 183
2
votes
0
answers
100
views
Composing equal characteristic and mixed characteristic deformations
Suppose $k$ is an algebraically closed field of characteristic $p>0$ and $A$ is a $k$-point on a moduli space of certain objects over $k$. Suppose, moreover, that $A$ deforms to a $k[[t]]$-point $B$...
- 245
2
votes
0
answers
177
views
vanishing of higher algebraic de Rham cohomology and sheaves of differentials for singular curves
I'm looking for some results or references about de Rham cohomology of curves in less-than-optimal cases. The two vanishing results I care about are:
$H^{k}_{\text{dR}}(X) = 0$ for $k>2$, since $H^...
- 1,142