# Questions tagged [frobenius-map]

The frobenius-map tag has no usage guidance.

23
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### Endomorphisms of a Finite Field [migrated]

Let $\mathbb{F}_q$ be the finite field with $q$ elements, where $q$ is a power of a prime $p$. Define the dual of $\mathbb{F}_q$, denoted $\mathbb{F}_q^*$, to be the set of endomorphisms of $\mathbb{F}...

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### How can I prove this stronger version of Fedder's Criterion?

I was reading Fedder's original paper which proved what is now known as "Fedder's criterion". I noticed that the abstract stated something which is a priori stronger than what is proved in ...

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### Frobenius action on the trivial connection

Let $F$ denote the absolute Frobenius acting on a smooth quasiprojective scheme $X$ over a finite field $k$.
Denote the trivial connection on $\mathcal{O}_X$ by $d$. Denote its pullback by Frobenius ...

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### $F$-structure implies regular singularities + unipotent local monodromy?

Let $(\mathcal{E},\nabla)$ be a vector bundle with an integrable connection on a smooth quasi-projective $K$ scheme $X$, with $K$ a $p$-adic number field of characteristic $0$. Let $F$ denote a semi-...

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### Frobenius and regular scheme

Let $X$ be a noetherian regular scheme over $\mathbb{F}_{p}$. Then by Kunz's theorem, the absolute Frobenius $F: X\rightarrow X$ is flat and integral. Can it be written as a projective limit of finite ...

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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 ...

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### Relative 1 form of Frobenius morphism

Let $X$ be a connected smooth algebraic variety over a field $k$ of characteristic $p > 0$. We can consider the Frobenius morphism associated to $X \mapsto Spec(k)$. I'd like to show $\Omega^{1}_{X/...

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229
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### Group homomorphism from $\mathrm{GL}_p$ to $\mathrm{SL}_p$ in characteristic $p$

If $k$ is a commutative field of characteristic $p>0$, then the map
$$ \theta \colon \mathrm{GL}_p(k) \to \mathrm{SL}_p(k) \colon A = (a_{ij}) \mapsto (\det A)^{-1} (a_{ij}^p) $$
is a group ...

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163
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### Supersingular points on the modular curve $X_0(p)$ over $\mathbb{F}_p$ and their Frobenius action

I am trying to understand certain points on the modular curve $X_0(p)$ over $\mathbb{F}_p$ (where $p$ is a rational prime) via their moduli description.
A point $P \in X_0(p)$ can be viewed as $P:=(E, ...

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### Is there a separable isogeny between any two isogenous abelian varieties?

Question: Let $k$ be an algebraically closed field, and $A,B$ abelian varieties over $k$. Suppose there exists an isogeny $A\to B$. Does this imply existence of a separable isogeny $A\to B$?
Known ...

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### If $ n \in \mathbb{N} $, then does the Reynolds operator of $ \mathbb{G}_{m}^{n} $ commute with the Frobenius endomorphism?

If $ n \in \mathbb{N} $, then $ \mathbb{G}_{m}^{n} $ is linearly reductive. Let $ \beta: \mathbb{G}_{m}^{n} \to \operatorname{GL}(\mathbf{V}) $ where $ \mathbf{V} $ is a vector space over an ...

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### When is a smooth point of a projective, simplicial, toric variety $ X_{\Sigma} $ compatibly $ F $-split?

A variety $ X $ is $ F $-split if there exists an $ \mathcal{O}_{X} $-linear map $ \phi: F_{\ast}(\mathcal{O}_{X}) \to \mathcal{O}_{X} $ such that $ \phi \circ F^{\sharp} = \operatorname{id}_{\mathcal{...

4
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377
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### Frobenius pushforward of an equivariant tautological bundle on the flag variety

Let $m,n$ be nonnegative integers. Let $k$ be a field of positive characteristic. Let the vector space $V$ over the field $k$ have basis $e_1,...,e_{m+n}, f_1,..., f_n$. Let $k^*$ act on $V$ by $t \...

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145
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### 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 ...

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### Cycle type in Galois group from ramified primes

Let $P \in \mathbb Z[X]$ be monic, separable, of degree $d$, $K$ its splitting field over $\mathbb Q$ and $G$ the Galois group of $K$ over $\mathbb Q$.
Now, let $p$ be a prime number unramified in $K$....

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147
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### Uniqueness of Galois descent

Let $Y,Z$ be $\mathbb{F}_p$-schemes, they are both models of scheme $X$ over $\overline{\mathbb{F}_p}$ . Let $F$ be the absolute Frobenius of $\overline{\mathbb{F}_p}$, If $1_Y\times F$ and $1_Z\times ...

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### Does anyone know an example of a non-singular, globally $ F $-regular variety $ X $ for which generic smoothness does not hold?

Let us denote the Frobenius endomorphism of a variety $ X $ by $ F $. A variety $ X $ over a field $ k $ of positive characteristic is globally $ F $-regular if for every effective Weil divisor $ D $,...

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### $F_q×1$-stable affine subspace

Let $A^n$ be an affine space over $\mathbb{F}_q$. Let $F_q$ be the absolutely Frobenius of $A^n$. Let $\bar{A^n}$ be the base change to $\bar{\mathbb{F}_q}$ and $F_q×1$ be $F_q\times_{\mathbb{F}_q}id_{...

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### Fixed space of absolutely Frobenius

Let $A$ be an affine space over $\bar{\mathbb{F}}_q$, $F$ be the absolutely Frobenius. Let $B$ be an $F-$ invariant affine subspace contained in $A$, $B^F$ be the fixed points of $F$ in $B$.
My ...

3
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508
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### The cycle class map with values in crystalline cohomology

Let $ k = \mathbb{F}_q $ be a finite field of characteristic $ p > 0 $.
Let $ X $ be a smooth proper scheme of dimension $ d $ over $ k $.
Consider the associated $ K $ - linear cycle class map ...

4
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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 ...

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### Automorphisms of Frobenius liftings and degeneration of the Hodge-de Rham spectral sequence

I am still studying Deligne and Illusie's paper (https://eudml.org/doc/143480), and I am again stuck, this time on pages 262/263.
Assume $X\longrightarrow S$ is a smooth morphism of $\mathbb{F}_{p}$-...

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### Length of a module and Frobenius map

Let $(R,m)$ be a regular local ring of dimension $d$ and char $p>0.$ Let $F^e:R\longrightarrow R$ defined by $r\longrightarrow r^{p^e}$be the Frobenius map.
How to compute $l(R/m^{[p^e]})?.$
I ...