The characteristic-p tag has no usage guidance.

**4**

votes

**2**answers

407 views

### Adem-Wu relations from Bullett-Macdonald identities

Question. Let $p$ be a prime. Let $q$ be a power of $p$. Let $P^0$, $P^1$, $P^2$, ... be elements of some associative $\mathbb F_q$-algebra $A$. (Here, $P^i$ does not mean $P$ to the $i$-th power; ...

**5**

votes

**1**answer

384 views

### Given a branched cover with branch cycle description $(g_1,…,g_r)$, does $g_i$ generate some decomposition group?

Classically:
Let $a_1,...,a_r$ be points in $\mathbb{P}^1_{\mathbb{C}}$, and let $\alpha_1,...,\alpha_r$ be simple loops around the $a_i$, all counterclockwise, and none touching (so $\alpha_1...\...

**10**

votes

**1**answer

1k views

### State of resolution in positive characteristic?

Heisuke Hironaka's coming talk makes me wonder how the state of the work on that theme is. So far, I noticed (but didn't read) these programs: 1, 2. It would be great if someone who listenes Hironaka'...

**3**

votes

**2**answers

511 views

### Ample line bundle and Frobenius morphism on smooth toric variety

Let $k$ be an algebraically closed field of $\mathrm{char}(k)=p>0$, $X$ a smooth toric projective variety of $\dim X=n$, $F_X:X\rightarrow X$ the absolute Frobenius morphism of $X$. Then for any $\...

**1**

vote

**1**answer

247 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 $X$. ...

**0**

votes

**0**answers

309 views

### Liftability in positive characteristic

What clsses of algebraic varieties over field of positive characteristic can be lift to $W_2(k)$?

**7**

votes

**2**answers

407 views

### Tameness for the Galois closure of a map of curves

Say we are working over some $K=\overline{K}$, of characteristic $p>0$. Let $\phi: Y\rightarrow X$ be a nonconstant map of smooth projective curves. To this map we can associate a map $\psi: Z\...

**0**

votes

**0**answers

462 views

### DeRham cohomology

The Poincare lemma fails in positive characteristic, since pth powers vanish under differention. My question is : is there still some kind of resolution of the local system k by considering some ...

**14**

votes

**2**answers

1k views

### Frobenius splitting and derived Cartier isomorphism

Let $X$ be a smooth algebraic variety over an algebraically closed field $k$ of characteristic $p>\dim X$. The motivation for my question comes from the following results.
1. If $X$ is ...

**2**

votes

**1**answer

439 views

### More on universal homeomorphisms

I would like to understand this notion better; where could I find some examples? In particular, I am interested in the following questions (and references for the answers).
Is a universal ...

**4**

votes

**2**answers

246 views

### Is the pre-image of a regular subscheme with respect to a universal homeomorphism of regular schemes regular?

Let $f:X\to Y$ be a universal homeomorphism of regular (excellent finite-dimensional) schemes, $Z\subset Y$ be a regular subscheme. Is $f^{-1}(Z)$ necessarily regular?

**4**

votes

**1**answer

301 views

### finite quotients of fundamental groups in positive characteristic

For affine smooth curves over $k=\bar{k}$ of char. $p,$ Abhyankar's conjecture (proved by Raynaud and Harbater) tells us exactly which finite groups can be realized as quotients of their fundamental ...

**10**

votes

**2**answers

549 views

### Valuations and separable extensions

Let $R$ be a valuation ring containing a field $k$, with residue field $F$ and quotient field $K$. Assume $F/k$ is separable. Is $K/k$ separable?
I have convinced myself that (for a positive answer) ...

**2**

votes

**1**answer

396 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 ...

**-2**

votes

**2**answers

1k views

### non discrete valuation ring [closed]

Hi,
I am looking for examples of non-discrete valuation rings. Could you help me?
Thanks

**1**

vote

**2**answers

305 views

### Could the Kunneth decomposition of a motif depend on the choice of $l$?

Suppose that over some (algebraically closed) field $K$ of characteristic $p>0$ we have: numerical equivalence of cycles coincides with homological one with respect to ${\mathbb{Q}}_{l}$ and ${\...

**11**

votes

**0**answers

523 views

### What's known about the mod 2 reduction of the level l Jacobi modular equation?

Motivation:
Let $\ell$ be an odd prime. Let $A$ in ${\mathbb Z}/2[[x]]$ be $x+x^9+x^{25}+x^{49}+...$, and $B=A(x^\ell)$. One can use the level $\ell$ Jacobi modular equation to get a polynomial ...

**3**

votes

**1**answer

256 views

### Do $Q_l$-etale Euler characteristics of Chow motives coincide for all $l$?

I am interested in (Chow) motives over (algebraically closed) characteristic $p>0$ fields. For $H$ being $\mathbb{Q}_l$-adic cohomology, one can consider $Ch_l(M)=\sum (-1)^i\dim_{\mathbb{Q}_l} H^i(...

**0**

votes

**1**answer

149 views

### An inseparable lift of a regular variety.

Let $X$ be a variety over an (imperfect) field $k$, that is regular as a scheme. Let $k'/k$ be an algebraic inseparable extension (I am interested in $k'$ being the perfection or the algebraic closure ...

**22**

votes

**4**answers

3k views

### The Frobenius morphism

I found the following list on the "Frobenius Page" by David Ben-Zvi, described by the author as "an outdated collection of intuitive ways to think about raising to the p-th power".
Generates a ...

**2**

votes

**1**answer

454 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[\...

**9**

votes

**0**answers

991 views

### Do the Standard Conjectures imply parts of the “Weil II” Riemann Hypothesis?

It is known that Grothendieck's Standard Conjectures on algebraic cycles imply the Riemann Hypothesis of the original Weil Conjectures. However, do they also say something about the version of the ...

**5**

votes

**0**answers

350 views

### Given two linear operators A and B over a finite field, is there a third operator C whose kernel is the intersection of kernels of A and B?

Let $V$ be a finite dimensional linear space over a finite field $k$. Let $A$ and $B$ be two endomorphisms of $V$.
Question 1. Is there an endomorphism $C$ of $V$, which is expressed in terms of ...

**7**

votes

**2**answers

511 views

### What are the polynomial relations between these characteristic 2 “thetas” ?

Suppose $\ell=2m+1$, $m>0$. Define $[i]$ in $\mathbb{Z}/2\mathbb{Z}[[x]]$ to be $$\sum_{n\equiv i\mod l} x^{n^2}.$$ Note that $[0]=1$, and that $[i]=[j]$ whenever $\ell$ divides $i+j$ or $i-j$.
...

**2**

votes

**0**answers

278 views

### Dimension of fibres of moment maps in characteristic $p$

Suppose $G$ is a connected semisimple linear algebraic group with Lie algebra $\mathfrak{g}$ and $X$ is a homogeneous $G$-space with isotropy subgroup $H$ (associated Lie algebra $\mathfrak{h}$) that ...

**16**

votes

**1**answer

865 views

### Is there a connected $k$-group scheme $G$ such that $G_{red}$ is not a subgroup?

I've been trying a learn a little more about group schemes by working through a set of exercises on Brian Conrad's website. Exercise 8.3 of http://math.stanford.edu/~conrad/papers/gpschemehw1.pdf ...

**13**

votes

**2**answers

887 views

### Can a reductive group act non-linearly on a vector group?

Let $k$ be a field; I'm going to discuss linear algebraic groups over $k$. The question I'll pose is only interesting when the characteristic is $p>0$.
1. Some motivation
A vector group is an ...

**8**

votes

**3**answers

542 views

### Variations on a theme of O'Bryant, Cooper and Eichhorn concerning power series over $\mathbb Z/2\mathbb Z$

Define 2 power series over the field $\mathbb Z/2\mathbb Z$ by $f=1+x+x^3+x^6+\dots$, the exponents being the triangular numbers, and $g=1+x+x^4+x^9+\dots$, the exponents being the squares. Write $f/g$...

**2**

votes

**1**answer

444 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 ...

**7**

votes

**1**answer

514 views

### Geometric (or intuitive) interpretation of additional derivatives in characteristic p > 0

In characteristic $p > 0$ there are "extra" differential operators, i.e., ones that are outside the algebra generated by first-order derivations.
Is there any interpretation of these operators in ...

**2**

votes

**1**answer

250 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 $f \equiv 1$, ...

**6**

votes

**0**answers

338 views

### Alterations of regular varieties

Let $X$ be a regular quasi-projective variety over a perfect field $k$. The existence of a "good compactification" of $X$, i.e. a regular projective variety $\bar{X}$ with an embedding $X\...

**13**

votes

**3**answers

1k views

### Lifting varieties to characteristic zero.

If you want to compute crystalline cohomology of a smooth proper variety $X$ over a perfect field $k$ of characteristic $p$, the first thing you might want to try is to lift $X$ to the Witt ring $W_k$ ...

**26**

votes

**3**answers

2k views

### Is there a nice proof of the fact that there are (p-1)/24 supersingular elliptic curves in characteristic p?

If $k$ is a characteristic $p$ field containing a subfield with $p^2$ elements (e.g., an algebraic closure of $\mathbb{F}_p$), then the number of isomorphism classes of supersingular elliptic curves ...

**10**

votes

**1**answer

851 views

### Are there “reasonable” criteria for existence/non-existence of Levi factors or their conjugacy in prime characteristic?

Classical theorems attributed to Levi, Mal'cev, Harish-Chandra for a finite
dimensional Lie algebra over a field of characteristic 0 state that it has a Levi decomposition (semisimple subalgebra plus ...

**23**

votes

**1**answer

2k views

### Wanted: Quadratic Space in Characteristic 2 as a Counterexample to a Theorem of Arf

Hi. Peter Roquette sent me an email asking for an example of a quadratic space in characteristic 2 having certain features. I have no idea on this, but maybe someone reading this does.
He would ...

**7**

votes

**1**answer

378 views

### Liftability of Enriques Surfaces (from char. p to zero)

Let $k$ be an algebraically closed field of characteristic $p > 0$, $X$ a variety over $k$.
We say $X$ lifts to characteristic zero, if there exists a local ring $R$ containing $\mathbb Z$ with ...

**11**

votes

**0**answers

498 views

### Lifting abelian varieties in (the closed fiber of) a fixed Neron model

Suppose that $R$ is a dvr with field of fractions $K$ and residue field $k$ and that $A_K$ is an abelian variety over $K$ with Neron model $A$ over $R$. Then the closed fiber $A_k$ is a smooth ...

**4**

votes

**2**answers

355 views

### lower bound for torsion of abelian varieties

Let $A$ be an abelian variety defined over a field $K$ of characteristic $p>0$. Let $A[\ell]$ be the group of $\ell$-torsion points, $\ell\neq p$ a prime. Are there positive constants $C, \eta$ ...

**3**

votes

**2**answers

496 views

### If the morphism of root data induced by an isogeny of a reductive group is a Frobenius, is then the isogeny itself a Frobenius?

Let $G$ be a reductive (or just semisimple) algebraic group over an algebraically closed field $k$ of characteristic $p > 0$, let $T$ be a maximal Torus and let $f:G \rightarrow G$ be an isogeny. ...

**4**

votes

**0**answers

401 views

### Hochschild Kostant Rosenberg theorem for varieties in positive characteristic?

Is there is a known version of the HKR theorem as proved in say Swan's paper "Hochschild Cohomology of Quasiprojective Varieties" in positive characteristic? I assume something is known about this as ...

**5**

votes

**3**answers

392 views

### Smoothness of hyperplane sections

Suppose $X\subset \mathbb{P}^n$ is a smooth hypersurface defined over $\mathbb{Q}$. For a "generic" prime $p$, what can be said about the set of hyperplanes $H$ in $\mathbb{P}^n(\mathbb{F}_p)$ for ...

**6**

votes

**4**answers

511 views

### Simply connected quasi-projective varieties in positive characteristic

I am looking for examples of non-projective (quasi-projective) varieties $X$ defined over a field of positive characteristic, which have trivial étale fundamental group.
It is well known that the ...

**21**

votes

**3**answers

2k views

### One dimensional (phi,Gamma)-modules in char p

I would like to better understand the simplest case of the correspondence between Galois representations and (phi,Gamma)-modules. Namely, consider 1-dimensional Galois representations of $G_{Q_p}$ ...

**8**

votes

**1**answer

602 views

### Restriction theorems over finite fields

A short while ago, Dvir proved the Kakeya conjecture over finite fields. Does this have any implications for restriction theorems over finite fields? I am aware only of implications going in the ...

**11**

votes

**1**answer

1k views

### The Infinitesimal topos in positive characteristic

This question was inspired by and is somewhat related to this question.
In his article "Crystals and the de Rham cohomology of schemes" in the collection "Dix exposes sur la cohomologie des schemas" ...

**6**

votes

**2**answers

417 views

### Concerning the dimension of a complex variety modulo a prime

Let V be a complex affine variety given as the vanishing set of a set of polynomials with integral coefficients. I have 3 questions.
1)
Under what assumption will the dimension of V over C remain ...

**3**

votes

**2**answers

427 views

### What is the correct formulation of the CDE triangle?

The CDE triangle with C Cartan matrix and D decomposition matrix is well-known for finite groups. I have not seen a full account of this for finite dimensional algebras which I find surprising as ...

**8**

votes

**3**answers

1k views

### Is $Sym^n (V^*) \cong Sym^n (V)^\ast$ naturally in positive characteristic?

Background/motivation
It is a classical fact that we have a natural isomorphism $Sym^n (V^*) \cong Sym^n (V) ^\ast$ for vector spaces $V$ over a field $k$ of characteristic 0. One way to see this is ...

**25**

votes

**3**answers

2k views

### product of all F_p, p prime

Let $R$ be the ring $$R = \prod_{p\ \text{prime}} \mathbb{F}_p$$ where $\mathbb{F}_p$ is the field having $p$ elements.
Is it true that $R$ has a quotient by a maximal ideal which is a field of ...