Questions tagged [positive-characteristic]

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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 ...
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1 vote
0 answers
78 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 ...
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3 votes
0 answers
119 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 ...
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3 votes
1 answer
75 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 ...
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5 votes
1 answer
255 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 ...
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1 vote
0 answers
49 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 ...
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4 votes
1 answer
129 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{...
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3 votes
0 answers
91 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 $\mathbb{C}$, $\mathcal{F}_\mathbb{C}$...
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5 votes
0 answers
97 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 $R$ ...
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4 votes
0 answers
144 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 ...
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3 votes
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177 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 ...
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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 $X$. ...
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4 votes
1 answer
352 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:...
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7 votes
1 answer
218 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 ...
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  • 189
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 ...
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8 votes
0 answers
224 views

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 ...
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  • 1,703
4 votes
0 answers
111 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)...
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  • 1,703
8 votes
1 answer
818 views

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 ...
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9 votes
0 answers
301 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 ...
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7 votes
1 answer
273 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 ...
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9 votes
1 answer
347 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 $k$", ...
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  • 7,343
3 votes
1 answer
181 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-...
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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 ...
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  • 3,462
4 votes
2 answers
331 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 ...
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5 votes
0 answers
114 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 ...
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2 votes
2 answers
245 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 ...
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3 votes
0 answers
110 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 ...
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3 votes
1 answer
261 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 ...
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1 vote
0 answers
166 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 $k$-...
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3 votes
0 answers
139 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 ...
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2 votes
0 answers
125 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^...
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5 votes
1 answer
175 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 + ...
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  • 83
4 votes
0 answers
115 views

Non-cyclic Galois groups over the field of formal Laurent series in positive characteristic

This should be an easy question, but I am unfortunately not able to give an answer, so I am sorry if this is not the appropriate level for the site. Let $C$ be an algebraically closed field of ...
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4 votes
0 answers
167 views

Fibered surfaces degenerating to Frobenius

Let $R$ be a DVR with algebraically closed, positive characteristic residue field $k$. Let $X\rightarrow Spec(R)$ and $C\rightarrow Spec(R)$ be smooth projective morphisms of relative dimension 2 and ...
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  • 589
10 votes
1 answer
1k views

Gelfand's trick (Gelfand's lemma) in positive characteristic?

I came across this preprint that claims in Lemma 1.1 that Gelfand's trick (also known as Gelfand's lemma) only works in characteristic zero: Let $H < G$ be finite groups. Suppose we have an anti-...
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  • 121
3 votes
1 answer
224 views

Is this unipotent group, over characteristic 2, connected?

Let $E_{ij}(x)\in \mathrm{Mat}_{7\times7}(\overline{\mathbb{F}}_2)$ be the matrix with zeros everywhere, except for the value $x$ at $ij$. Set $$a(x)=1+E_{12}(x)+E_{34}(x)+E_{56}(x),\quad b(y)=1+E_{23}...
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  • 4,262
9 votes
2 answers
597 views

Nakano vanishing in positive characteristic

Let $X$ be a smooth projective variety defined over a field $k$. In characteristic zero, the following is a special case of the (Kodaira-Akizuki-)Nakano vanishing theorem: $(\ast) \quad$ $\mathrm H^...
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  • 992
16 votes
2 answers
1k views

Biggest Field Of Characteristic $p$

The Surreal nummbers, $\boldsymbol{No}$, are according to Wikipidia the biggest ordered field, and the Surrcomplex numbers are the biggest field of characteristic 0. Biggest in the sense that every ...
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15 votes
2 answers
536 views

When is the etale cohomology of $\mathrm{Sym}^n(X)$ isomorphic to the $\Sigma_n$-invariants in the étale cohomology of $X^n$?

Suppose $X$ is a smooth projective variety defined over an arbitrary algebraically closed field $k$, and consider the action of $\Sigma_n$ on the $n$-fold product $X^n$. Is it true that $H_{\acute{e}t}...
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4 votes
0 answers
154 views

Symmetric power contained in tensor power?

Let $V$ be an $R$-module. Traditionally the symmetric algebra $S(V)$ is defined as a quotient of the tensor algebra $T(V)$, by the ideal generated by all $a\otimes b-b\otimes a$. Can $S^n(V)$ also be ...
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  • 2,287
2 votes
1 answer
312 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 ...
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4 votes
0 answers
173 views

Semisimplicity of the p-adic étale Tate module over $F_p(t)$

Let $k$ be a finitely generated field of positive characteristic p. Let $A$ be an abelian variety over $k$ and write $T_p(A)$ for the $p$-adic étale Tate module of $A$. Is it known if the natural ...
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2 votes
1 answer
178 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 $...
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  • 21
9 votes
1 answer
635 views

Endomorphism ring of simple ordinary abelian variety

Is there an example of an ordinary and simple abelian variety $A$ over an algebraically closed field $K$ (of characteristic $p>0$) such that ${\rm End}(A)$ is not commutative? Note that the answer ...
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11 votes
2 answers
797 views

Cotangent complex of perfect algebra over a perfect field

Let $A$ be a perfect $\kappa$-algebra over a perfect field $\kappa$ of positive characteristic $p$. Then the algebraic (= classical) cotangent complex $L_{A/\kappa}^{\operatorname{alg}}$ is known to ...
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5 votes
0 answers
188 views

Map associated to linear system onto curve is morphism

In Mumford's first paper on Surfaces in char $p$ [1], part 2 Step (II), he wants to show that, given an indecomposable curve of canonical type $D$ on a smooth projective surface $F$ with $p_g(F)=0, ...
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7 votes
1 answer
484 views

absolute Galois group of the function field of a curve over $\mathbb{F}_p^{alg}$

In their 2008 paper "Torelli theorem for curves over finite fields" Bogomolov, Korotiaev and Tschinkel mention in the beginning of Section 9 that absolute Galois groups of curves over $\mathbb{F}_p^{...
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1 vote
1 answer
250 views

Norms of elements in Artin-Schreier extensions

The following is claimed in the proof of Theorem 7.5 of Auslander, Goldman, "The Brauer group of a commutative ring": Let $k$ be a nonperfect field of positive characteristic $p$, let $K := k(x)$ ...
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2 votes
0 answers
260 views

Surjectivity of map of Picard schemes implies abelian

Note: This question was asked on MSE first, but got zero reactions. So I deleted it there, and am now posting it here. I am looking for a reference or explanation of the fact that is used in Mumford'...
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  • 193
2 votes
0 answers
115 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. ...
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