# Questions tagged [positive-characteristic]

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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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### Sheaf whose singular support is not Lagrangian

For constructible sheaves $\mathcal F$ on real analytic manifolds $X$, there is a notion of the singular support $SS(\mathcal F)$ which is a radially invariant singular Lagrangian subset of the ...