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Questions tagged [characteristic-p]

Fields of characteristic $p$, i.e., fields for which there is a prime $p$ such that $px=0$ for each $x$. Do not use this tag for questions on characteristic polynomials of a matrix.

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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 ...
Jon Elmer's user avatar
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4 votes
0 answers
103 views

Shafarevich conjecture for Abelian varieties over global function fields

Let $S$ be a finite set of places of a global function field $K$. Are there finitely many Abelian varieties over $K$ with good reduction outside $S$? What if we exclude isotrivial families?
TCiur's user avatar
  • 557
2 votes
1 answer
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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 ...
Oblomov's user avatar
  • 2,501
3 votes
0 answers
110 views

Nilpotent orbits in characteristic $0$ vs. positive characteristics

Let $G_\mathbb{C}$ be a connected reductive group over $\mathbb{C}$ with Lie algebra $\mathfrak{g}_{\mathbb{C}}$. For any algebraically closed field $k$, let $G_k$ denote the connected reductive group ...
Dr. Evil's user avatar
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3 votes
0 answers
164 views

Characters of finite fields - Reference request

Let $\mathbf{F}_q$ ($q=p^f$) be a finite field. We are interested in the characters $\chi: \mathbf{F}_q\rightarrow \mathbf{K}$ ($\chi(0)=0$) where the $ \mathbf{K}$ is an alg.closed field of ...
Grad Student's user avatar
9 votes
0 answers
350 views

How large must the characteristic of $k$ be, for the cohomology of the Lie algebra $\mathfrak{sl}_n(k)$ to be exterior as in characteristic zero?

$\DeclareMathOperator\SU{SU}$In this question, all Lie algebra cohomology is of the form $H^*(\mathfrak{g}; k)$, with $k$ the trivial one-dimensional representation of $\mathfrak{g}$. All Lie algebra ...
user509184's user avatar
2 votes
0 answers
122 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 ...
Martin Ortiz's user avatar
1 vote
1 answer
115 views

What are the finite étale coverings of a quasi-hyperelliptic surface?

Let $X$ be a quasi-hyperelliptic surface in characteristic 3 where the canonical bundle $K_X$ is trivial. Question: Is there a finite étale covering $Y \rightarrow X$ such that $Y$ is an abelian ...
LeechLattice's user avatar
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11 votes
1 answer
368 views

Chromatic representation theory of the symmetric groups?

We know that in characteristic 0, the group ring of the symmetric group $\Sigma_n$ splits via one idempotent for each partition of $n$. In characteristic $p$, I believe the analogous statement is that ...
Tim Campion's user avatar
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2 votes
1 answer
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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 ...
uno's user avatar
  • 280
5 votes
1 answer
137 views

Derived subalgebra of a restricted Lie algebra

Let $L$ be a restricted Lie algebra over a field of characteristic $p>0$. It is well known that the commutator subalgebra $[L,L]$ is not necessarily restricted (that is, closed under the $p$-map). ...
Rocky Smith's user avatar
17 votes
0 answers
347 views

Ado's theorem and the reduction to positive characteristic

The synopsis: proofs of Ado theorem in positive characteristic are simple, and in characteristic $0$ are difficult. Can one infer the characteristic $0$ case from the positive characteristic case? The ...
Dmitrii Korshunov's user avatar
2 votes
1 answer
154 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 ...
Sky's user avatar
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0 answers
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Bad primes of twists of modular curves $X_E^{-1}(p)$

I am interested in a good reference for reading about primes of bad reduction for the modular curve $X_E^{-}(N)$, which is a twist of $X(N)$ parametrizing all elliptic curves $E'$ whose $N$-torsion ...
did's user avatar
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1 vote
0 answers
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Relative compactification without resolutions of singularities

Let $Y$ be a smooth proper variety over a field $k$, let $X$ be a smooth variety over $k$, $U\hookrightarrow X$ the complement of a strict normal crossing divisor and $\phi\colon U\to Y$ a map. By ...
user197402's user avatar
3 votes
0 answers
95 views

What are the possibilities of the general fibres in an Iitaka fibration?

This question is motivated by complex algebraic geometry. If $X$ is a complex algebraic variety with Kodaira dimension in $[1,\dim X-1]$, then the Iitaka fibration (the rational map induced by the ...
LeechLattice's user avatar
  • 9,421
4 votes
0 answers
101 views

Boundary divisor of an equivariant compactification of a non-trivial twist of $\mathbb{G}_a^n$

Let $K$ be a non-perfect field of positive characteristic. Then there exist non-trivial forms of $\mathbb{G}_a^n$ which split over finite purely inseparable extensions of $K$, i.e., algebraic groups $...
Abdulmuhsin Alfaraj's user avatar
3 votes
0 answers
90 views

Dimension of a kernel of a cocycle map

Inspired by a previous question (Dimension of a kernel of a linear map) and some of the answers I was given I thought wheter I can generalize the question to the following: Compute the kernel (or at ...
Marcos's user avatar
  • 577
3 votes
0 answers
108 views

Torsion of Fermat hypersurfaces

An interesting invariant of a rationally chain-connected variety $X/k$ is the exponent of the group, $$ \ker{(\mathrm{CH}_0(X_K) \xrightarrow{\deg} \mathbb{Z})} $$ where $K = k(X)$ is the function ...
Ben C's user avatar
  • 3,363
3 votes
0 answers
85 views

Obstruction for points to be contained in smooth hypersurfaces in tterms of inseparability degree of residue field

Let $k$ be an imperfect field of char $p>0$ and $x \in \mathbb{P}^n_k$ be closed point of projective space. In this discussion Qing Liu wrote that Over an imperfect field, a reduced point can not ...
JackYo's user avatar
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3 votes
0 answers
104 views

Points with residue fields having big inseparability degree cannot be contained in smooth hypersurfaces

Let $X$ be a $k$-scheme over imperfect field $k$ and $x \in X $ some (reduced) point with residue field $\kappa(x) = \mathcal{O}_{X,x}/ \mathfrak{m}_x$. How to check that if $\kappa(x)$ has "big ...
JackYo's user avatar
  • 543
2 votes
0 answers
113 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" $...
Asvin's user avatar
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4 votes
0 answers
71 views

Conjugacy of cocharacters from conjugacy of labelled diagrams

Everything to follow is over some fixed algebraically closed field $k$. Although all the definitions make sense regardless of characteristic, the meat of the question is about small positive ...
LSpice's user avatar
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1 vote
1 answer
78 views

Generic finite subgroups, associated to small finite fields, of reductive algebraic groups

Theorem 1 of [LS] Liebeck and Seitz - On the subgroup structure of exceptional groups says: Let $X = X(q)$ be a quasisimple group of Lie type in characteristic $p$, and suppose that $X < G$, where ...
LSpice's user avatar
  • 11.6k
2 votes
0 answers
125 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 ...
IntegrableSystemsEnthusiast's user avatar
3 votes
0 answers
120 views

Isogeny of elliptic curve over positive characteristic $p$ which does not come from characteristic $0$

Let $K$ be quadratic imaginary field. Let $E$ be an elliptic curve which has CM over $R_K$ ($R_K$ is ring of integers of $K$). According to SIlverman's ''ADvanced topics in the arithmetic of elliptic ...
Duality's user avatar
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2 votes
0 answers
167 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, $T$ ...
LSpice's user avatar
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1 vote
1 answer
139 views

When is $R$ a direct summand of Frobenius pushforwards?

Let $(R,\mathfrak m)$ be a reduced Noetherian local ring of prime characteristic $p$. For integer $e>0$, let $F^e_* R$ denote the $R$-module which is $R$ as an abelian group, but the $R$-module ...
Snake Eyes's user avatar
4 votes
1 answer
207 views

Compute de Rham-Witt sheaves

I am really new to this, but I am having a hard time understanding all the de Rham-Witt construction. It seems to be really difficult to compute anything with those beasts: like I cannot find any ...
user197402's user avatar
4 votes
1 answer
246 views

Relation between positive roots of $E_8$ and $\mathbb{F}_2^8 \setminus \{0\}$

There exists an explicit bijection (due to Cayley, that has built up a very nice table to describe this) between the positive roots of the lattice $E_7$ and $\mathbb{F}_2^6 \setminus \{0\}$ (where $\...
IMeasy's user avatar
  • 3,717
3 votes
1 answer
270 views

Is the theta characteristic attached to an etale double cover of a plane quintic arising from a cubic threefold even in characteristic two?

We work over an algebraically closed field $k$ of characteristic $2$. Let $X$ be a cubic threefold realized as a conic bundle via $f: X \to \mathbb{P}^2$ (after blowing up some line). Let $C \to \...
TCiur's user avatar
  • 557
2 votes
0 answers
202 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 ...
Merrick Cai's user avatar
4 votes
0 answers
264 views

de Rham Witt complex vs. de Rham complex of the Witt ring

I am reading the paper "Revisiting the de Rham-Witt complex" by Bhatt-Lurie-Mathew and I am a bit confused about the difference between $W\Omega_R^*$ and $\hat{\Omega}^*_{W(R)}$. Let $\...
Jun Koizumi's user avatar
3 votes
0 answers
132 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 ...
Doron Grossman-Naples's user avatar
6 votes
1 answer
442 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-...
TCiur's user avatar
  • 557
2 votes
1 answer
168 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 ...
LSpice's user avatar
  • 11.6k
2 votes
0 answers
252 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 ...
dongrugose's user avatar
4 votes
0 answers
228 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 $n$ ...
SorcererofDM's user avatar
6 votes
0 answers
107 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=\{ (...
Paul Levy's user avatar
  • 1,184
4 votes
0 answers
174 views

If the Frobenius endomorphism of a characteristic $p$ ring is epimorphic, is it surjective?

MO question 19282 is about integral epimorphisms of commutative rings, and a counterexample is given to surjectivity. What about the case of the Frobenius endomorphism of a commutative, characteristic ...
Matthieu Romagny's user avatar
3 votes
1 answer
221 views

Can non-geometrically reduced reduced subschemes happen for reductive groups?

The title is meant to be punchy, but also a tongue-in-cheek acknowledgement of the prevalence of ‘reduce’-derived words in this area. (Unfortunately, I overlooked the fact that the question in the ...
LSpice's user avatar
  • 11.6k
5 votes
1 answer
388 views

Lifting $\mathfrak{sl}_2$-triples

Let $k$ be an algebraically closed field, $G$ a (smooth, connected) reductive algebraic group over $k$, $H$ a (smooth, connected) reductive group of semisimple rank 1, and $T$ a maximal torus in $H$. ...
LSpice's user avatar
  • 11.6k
2 votes
1 answer
512 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....
Yuan Yang's user avatar
  • 547
7 votes
1 answer
379 views

Does perfect fraction field imply perfect residue field?

Let $A$ be a local integral domain of characteristic $p$. Let $K$ be the fraction field and let $k$ be the residue field of $A$. If $K$ is perfect, is $k$ necessarily perfect? Thoughts: If $A$ is ...
user2831784's user avatar
8 votes
0 answers
663 views

An Azumaya algebra from a vector bundle, and a construction of Belov-Kanel and Kontsevich

Let $S/k$ be a scheme over a perfect field $k$ of characteristic $p>0$. In Automorphisms of the Weyl Algebra, Belov-Kanel and Kontsevich write down the map $$\alpha: H^0(\Omega^1_{S/k}/d\mathcal O) ...
Joshua Mundinger's user avatar
4 votes
1 answer
404 views

p-torsion in the Picard group of a regular projective curve

Let $K$ be a field of characteristic $p>0$ and $C$ a regular projective geometrically integral curve over $K$. If $C$ is smooth, then the connected component ${\rm Pic}^0_C$ of the Picard scheme of ...
Arno Fehm's user avatar
  • 2,041
3 votes
0 answers
196 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 ...
lefuneste's user avatar
  • 417
11 votes
4 answers
1k views

Explicit large finite fields in characteristic $2$

Every finite field of characteristic $2$ ist given by $\mathbb{F}_2[x]/P(x)$ for some irreducible polynomial $P\in \mathbb{F}_2[x]$. For small degree, a simple algorithm gives a way to find $P$. Is ...
Jérémy Blanc's user avatar
5 votes
0 answers
138 views

algebraic de Rham cohomology of toric varieties (reference request)

I haven't been able to find anything workable yet, but I'm looking for a reference on the de Rham cohomology of toric varieties, where as many as possible of the following conditions are handled: ...
Somatic Custard's user avatar
3 votes
3 answers
387 views

Are there characteristic-dependent Betti numbers in characteristic not equal to two?

Let $\Delta$ be a finite simplicial complex on $n$ vertices. Let $S = \mathbb{k}[\mathbf{x}]$ be a polynomial ring over a field $\mathbb{k}$ in $n$ variables and $I$ be the Stanley-Reisner ideal of $\...
Aaron Dall's user avatar

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