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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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
5 votes
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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 $\mathbb{C}$, $\mathcal{F}_\mathbb{C}$...
P. Grabowski's user avatar
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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
5 votes
0 answers
149 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 $R$ ...
Emiliano Ambrosi's user avatar
6 votes
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105 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
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4 votes
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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
206 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
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4 votes
0 answers
232 views

de Rham Bloch-Ogus theory in positive characteristic

In their famous paper Gersten's conjecture and the homology of schemes, one of the results that Bloch and Ogus prove is that the second page of the coniveau spectral sequence for $X$ smooth over a ...
xir's user avatar
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3 votes
1 answer
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Maximal closed subscheme stable under the action of a finite connected group scheme

Let $k$ be a field of characteristic $p>0$, $X$ a smooth projective $k$-variety and $Y\subseteq X$ a closed irreducible subvariety. Let $G$ be a connected finite $k$-group scheme acting on $X$. ...
Emiliano Ambrosi's user avatar
5 votes
1 answer
374 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
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2 votes
1 answer
494 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
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7 votes
1 answer
375 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
4 votes
1 answer
558 views

perfect fields in positive characteristic

Let $k$ be an infinite perfect field in positive characteristic $p$, i.e. every element of $k$ is a $p$th power. I am interested in properties of finite fields that can be extended to $k$. For example:...
JNS's user avatar
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8 votes
1 answer
315 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 ...
tyrese's user avatar
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19 votes
1 answer
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Perfectoid approach to resolution of singularities in char $p$

Since perfectoid techniques have built a bridge between char $0$ and char $p$ worlds, it is conceivable that they can be applied to resolution of singularities in char $p$ using their successful ...
Arna's user avatar
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Which algebra is $\mathbb{F}_p \otimes_{HH^{\cdot} (\mathbb{F}_p)} \mathbb{F}_p$?

Let $HH^{\cdot}(\mathbb{F}_p)$ be the Hochschild cohomology of $\mathbb{F}_p$ over $\mathbb{Z}$, which as a $E_1$ ring is simply $\mathbb{F}_p[x]$ with $x$ in cohomological degree $2$. Then it's clear ...
davik's user avatar
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4 votes
0 answers
124 views

Castelnuovo–Mumford regularity and wedge powers in positive characterisitc

A vector bundle on $\mathbb{P}^n$ is said to be $r$-regular if $$H^i(\mathbb{P}^n,F(r-i))=0$$ for all $i>0$. It is always true that if $F$ is $r$-regular and $G$ is $s$-regular (both vector bundles)...
davik's user avatar
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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
397 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
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8 votes
1 answer
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Automorphisms over finite field that do not lift to an automorphism in characteristic zero

My main question is the following: is there an automorphism of the affine space $\mathbb{A}^n$ (automorphism of an algebraic variety) defined over a finite field which does not lift to an automorphism ...
Jérémy Blanc's user avatar
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
988 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
136 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
377 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
1 vote
0 answers
51 views

Do we have $\cap_{P\in S}\left(\mathcal{O}_P+(1-\tau)(K)\right)=\left(\cap_{P\in S}\mathcal{O}_P\right)+(1-\tau)(K)$?

Let $\mathbb{F}$ be a finite field and let $q$ be its number of elements. Let $(C,\mathcal{O}_C)$ be a geometrically irreducible smooth projective curve over $\mathbb{F}$ and let $K=\mathbb{F}(C)$ be ...
Stabilo's user avatar
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3 votes
0 answers
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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?
Kim's user avatar
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12 votes
0 answers
630 views

Rings whose Frobenius is flat

Let $R$ be a ring of characteristic $p>0$. The (absolute) Frobenius is the map of rings $F_R:R\rightarrow R$ defined by $x\mapsto x^p$. I am interested in rings for which $F_R$ is flat (hence ...
user avatar
7 votes
1 answer
450 views

Weyl algebra as an Azumaya algebra over its centre

Assume that $k$ is an algebraically closed field of positive characteristic $p$. On page 3 (page 6 of the PDF file) of Bezrukavnikov, Mirković, and Rumynin - Localisation of Modules for a semisimple ...
user11235813's user avatar
9 votes
1 answer
448 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 $k$", ...
LSpice's user avatar
  • 11.3k
16 votes
1 answer
970 views

Reconstruct a variety from its crystalline topos

Let $k$ be a perfect field of positive characteristic. Let $X$ be a smooth projective geometrically connected $k$-scheme with a $k$-point. Can we reconstruct $X$ from its small crystalline topos $((X/...
user avatar
4 votes
2 answers
790 views

Katz's proof of Cartier's (descent) theorem

I am trying to understand the proof of Cartier’s theorem on pages 370-371 (pages 17-18 of the PDF file) of Katz’s “Nilpotent connections and the monodromy theorem: applications of a result of ...
clarkkent's user avatar
  • 121
3 votes
1 answer
244 views

Fourier transform on finite groups in characteristic $p>0$

Is there a Fourier theory for finite groups in characteristic $p>0$? Assume that $p$ divides the order $|G|$ of finite groups (or just work with $p$-groups), i.e., in a modular representation-...
l'etranger's user avatar
46 votes
1 answer
1k views

Summing infinitely many infinitesimally small variables makes sense in algebra

There is an identity $e^x=\lim_{n\to \infty} (1+x/n)^n$, and I always thought it is a purely analytic statement. But then I discovered its curious interpretation in pure algebra: Consider the ring of ...
Anton Mellit's user avatar
  • 3,572
2 votes
0 answers
92 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 $B$...
Huy Dang's user avatar
  • 245
4 votes
2 answers
416 views

Are Chow groups invariant under universal homeomorphisms?

Let $f:X\to Y$ be a universal homeomorphism of schemes, $R$ a coefficient ring. Which assumptions on $f$ and $R$ are suffient to ensure that the pullback map $f^*$ of $R$-linear Chow groups is ...
Mikhail Bondarko's user avatar
5 votes
1 answer
229 views

Hochschild cohomology of an Azumaya algebra

Let $k$ be a field. Given a commutative $k$-algebra $Z$ and an associative algebra $A$ that is Azumaya over $Z$, do we have an isomorphism of Hochschild cohomologies: $HH^*(A) \cong HH^*(Z)$? This is ...
MathManiac's user avatar
3 votes
1 answer
260 views

Frobenius splitting for an excellent, non $F$-finite, $F$-pure hypersurface

Let $p$ be an odd prime. Let $k$ be a field of characteristic $p$ such that $[k:k^p]=\infty$ (i.e. $k$ is not $F$-finite ) . Also assume that $-1$ is not a square in $k$ . Consider the homogeneous ...
sdey's user avatar
  • 642
2 votes
2 answers
527 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 ...
Marco Timpanella's user avatar
4 votes
0 answers
160 views

Extensions of fraction field and residue field

Let $A \subset B$ be integrally closed local domains, $K(A), K(B)$ be fields of fractions, and $k(A),k(B)$ be residue fields. How to prove $[K(B):K(A)] \ge [k(B):k(A)]$? This question should be easy ...
Hu Zhengyu's user avatar
3 votes
0 answers
213 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 $X$....
user11235813's user avatar
3 votes
1 answer
221 views

Do Neron-Severi groups of smooth projective unirational varieties contain $\ell$-torsion?

Let $X$ be a smooth projective unirational variety over an algebraically closed field of characteristic $p>0$, and $\ell\neq p$ a prime. My question: can the Neron-Severi group of $X$ contain (non-...
Mikhail Bondarko's user avatar
3 votes
1 answer
299 views

What is known about lower etale cohomology of unirational varieties?

Which information is currently known about $H^1_{et}(X,\mathbb{Z}_l)$ and $H^2_{et}(X,\mathbb{Z}_l)$, where $X$ is a smooth unirational variety over an algebraically closed field of finite ...
Mikhail Bondarko's user avatar
8 votes
1 answer
729 views

A variant on characteristic $p$ de Rham cohomology

I was thinking about de Rham cohomology in characteristic $p$, and in particular the recent question about Poincare residues, and I came up with the following construction. Let $k$ be a perfect field ...
David E Speyer's user avatar
1 vote
0 answers
216 views

Frobenius twist of a field

Let $k$ be a field of characteristic $p>0$ (not necessarily perfect). Consider the Frobenius endomorphism $F : k \to k$, $x \mapsto x^p$. I am curious about what happens when we take $k$ as a $k$-...
VerrückterPinguin's user avatar
3 votes
0 answers
142 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 ...
Kim's user avatar
  • 4,034
2 votes
0 answers
162 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^...
Somatic Custard's user avatar
5 votes
1 answer
220 views

Do regular (but non-smooth) conics over a discretely valued field of characteristic $2$ admit a regular model over the valuation ring?

Let $K$ be a non-perfect field of characteristic $2$. Let $T \subseteq K$ be a discrete valuation ring. Assume there exist $a,b \in K^{\times}$ such that the projective conic $C$ defined by $$aX^2 + ...
DGrimm's user avatar
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