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\section*{Frobenius morphism}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{arithmetic_geometry}{}\paragraph*{{Arithmetic geometry}}\label{arithmetic_geometry}

[[!include arithmetic geometry - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{for_fields}{For fields}\dotfill \pageref*{for_fields} \linebreak
\noindent\hyperlink{of_schemes}{Of schemes}\dotfill \pageref*{of_schemes} \linebreak
\noindent\hyperlink{for_sheaves_on_}{For sheaves on $C Ring ^{op}$}\dotfill \pageref*{for_sheaves_on_} \linebreak
\noindent\hyperlink{InTermsOfSymmetricProducts}{In terms of symmetric products}\dotfill \pageref*{InTermsOfSymmetricProducts} \linebreak
\noindent\hyperlink{ForEInfinityRings}{For $E_\infty$-Rings}\dotfill \pageref*{ForEInfinityRings} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{for_fields_2}{For fields}\dotfill \pageref*{for_fields_2} \linebreak
\noindent\hyperlink{AsElementsOfGaloisGroup}{As elements of the Galois group}\dotfill \pageref*{AsElementsOfGaloisGroup} \linebreak
\noindent\hyperlink{for_schemes}{For schemes}\dotfill \pageref*{for_schemes} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

In [[number theory]], [[Galois theory]] and [[arithmetic geometry]] in [[prime number|prime]] [[characteristic]] $p$, the \emph{Frobenius morphism} is the [[endomorphism]] acting on [[algebras]], [[function algebras]], [[structure sheaves]] etc., which takes each [[ring]]/[[associative algebra|algebra]]-element $x$ to its $p$th power

\begin{displaymath}
x^p = \underbrace{x \cdot x \cdots x}_{p \; factors}
  \;.
\end{displaymath}
It is precisely in [[positive characteristic]] $p$ that this operation is indeed an algebra [[homomorphism]] (``[[freshman dream arithmetic]]'').

The Frobenius map is the shadow of the [[power operations]] in [[multiplicative cohomology theory]]/[[higher algebra]] (\hyperlink{Lurie}{Lurie, remark 2.2.7}).

The presence of the Frobenius endomorphism in characteristic $p$ is a fundamental property in [[arithmetic geometry]] that controls many of its deep aspects. Notably [[zeta functions]] are typically expressed in terms of the [[action]] of the Frobenius endomorphisms on [[cohomology groups]] and so it features prominently for instance in the [[Weil conjectures]].

In [[Borger's absolute geometry]] lifts of Frobenius endomorphisms through [[base change]] for all primes at once -- in the sense of \emph{\href{Lambda-ring#LambdaRingByFrobeniusLifts}{Lambda-ring structure}} -- is interpreted as encoding [[descent]] data from traditional [[arithmetic geometry]] over [[Spec(Z)]] down to the ``absolute'' geometry over ``[[F1]]''. See at \emph{\href{Borger%27s+absolute+geometry#Motivation}{Borger's absolute geometry -- Motivation}} for more on this.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\begin{quote}%
this entry may need attention


\end{quote}
\hypertarget{for_fields}{}\subsection*{{For fields}}\label{for_fields}

Let $k$ be a [[field]] of positive [[characteristic]] $p$. The \textbf{Frobenius morphism} is an [[endomorphism]] of the field $F \colon k \to k$ defined by

\begin{displaymath}
F(a) \coloneqq  a^p
  \,.
\end{displaymath}
Notice that this is indeed a [[homomorphism]] of fields: the identity $(a b)^p=a^p b^p$ evidently holds for all $a,b\in k$ and the [[characteristic]] of the field is used to show $(a+b)^p=a^p+b^p$.

\hypertarget{of_schemes}{}\subsubsection*{{Of schemes}}\label{of_schemes}

Suppose $(X,\mathcal{O}_X)$ is an $S$-[[scheme]] where $S$ is a [[scheme]] over $k$. The \textbf{absolute Frobenius} is the map $F^{ab}:(X,\mathcal{O}_X)\to (X,\mathcal{O}_X)$ which is the identity on the [[topological space]] $X$ and on the structure sheaves $F_*:\mathcal{O}_X\to \mathcal{O}_X$ is the $p$-th power map. This is not a map of $S$-schemes in general since it doesn't respect the structure of $X$ as an $S$-scheme, i.e. the diagram:

$\displaystyle \begin{matrix} X & \stackrel{F^{ab}}{\to} & X \\
\downarrow &  & \downarrow \\
S & \stackrel{F^{ab}}{\to} & S \end{matrix}$,

so in order for the map to be an $S$-scheme morphism, $F^{ab}$ must be the identity on $S$, i.e. $S=Spec(\mathbb{F}_p)$.

Now we can form the [[fiber product]] using this square: $X^{(p)}:=X\times_{S} S$. By the universal property of [[pullback]]s there is a map $F^{rel}:X\to X^{(p)}$ so that the composition $X\to X^{(p)}\to X$ is $F^{ab}$. This is called the \textbf{relative Frobenius}. By construction the relative Frobenius is a map of $S$-schemes.

\hypertarget{for_sheaves_on_}{}\subsubsection*{{For sheaves on $C Ring ^{op}$}}\label{for_sheaves_on_}

Let $p$ be a prime number, let $k$ be a field of characteristic $p$. For a $k$-ring $A$ we define

\begin{displaymath}
f_A:
\begin{cases}
A\to A
\\
x\mapsto x^p
\end{cases}
\end{displaymath}
The $k$-ring obtained from $A$ by scalar restriction along $f_k:k\to k$ is denoted by $A_{f}$.

The $k$-ring obtained from $A$ by scalar extension along $f_k:k\to k$ is denoted by $A^{(p)}:=A\otimes_{k,f} k$.

There are $k$-ring morphisms $f_A: A\to A_f$ and $F_A:\begin{cases}
A^{(p)}\to A
\\
x\otimes \lambda\mapsto x^p \lambda
\end{cases}$.

For a $k$-functor $X$ we define $X^{(p)}:X\otimes_{k,f_k} k$ which satisfies $X^{(p)}(R)=X(R_f)$. The \emph{Frobenius morphism} for $X$ is the transformation of $k$-functors defined by

\begin{displaymath}
F_X:
\begin{cases}

X\to X^{(p)}
\\
X(f_R):X(R)\to X(R_f)
\end{cases}
\end{displaymath}
If $X$ is a $k$-scheme $X^{(p)}$ is a $k$-scheme, too.

Since the completion functor ${}^\hat\;:Sch_k\to fSch_k$ commutes with the above constructions the Frobenius morphism can be defined for [[formal scheme|formal k-schemes]], too.

\hypertarget{InTermsOfSymmetricProducts}{}\subsubsection*{{In terms of symmetric products}}\label{InTermsOfSymmetricProducts}

We give here another characterization of the [[Frobenius morphism]] in terms of symmetric products.

Let $p$ be a prime number, let $k$ be a field of [[characteristic]] $p$, let $V$ be a $k$-vector space, let $\otimes^p V$ denote the $p$-fold tensor power of $V$, let $TS^p V$ denote the subspace of symmetric tensors, yielding the [[symmetric algebra]]. Then we have the symmetrization operator

\begin{displaymath}
s_V:
\begin{cases}
\otimes^p V\to TS^p V
\\
a_1\otimes\cdots\otimes a_n\mapsto \Sigma_{\sigma\in S_p}a_{\sigma(1)}\otimes\cdots\otimes a_{\sigma(n)}
\end{cases}
\end{displaymath}
and the linear map

\begin{displaymath}
\alpha_V 
  \colon
\begin{cases}
  V^{(p)} \to\otimes^p V
  \\
  a\otimes \lambda\mapsto\lambda(a\otimes\cdots\otimes a)
\end{cases}
\end{displaymath}
then the map $V^{(p)}\stackrel{\alpha_V}{\to}TS^p V\to TS^p V/s(\otimes^p V)$ is [[bijective]] and we define

\begin{displaymath}
\lambda_V
   \;\colon\;
  TS^p V\to V^{(p)}
\end{displaymath}
by

\begin{displaymath}
\lambda_V\circ s=0
\end{displaymath}
and

\begin{displaymath}
\lambda_V \circ \alpha_V= id
\end{displaymath}
If $A$ is a $k$-ring we have that $TS^p A$ is a $k$-ring and $\lambda_A$ is a $k$-ring morphism.

If $X=Sp_k A$ is the [[spectrum of a commutative ring]] we abbreviate $S^p X 0 S^p_k X \coloneqq Sp_k (TS^p A)$ and the following diagram is commutative.

\begin{displaymath}
\itexarray{
X
&\stackrel{F_X}{\to}&
X^{(p)}
\\
\downarrow&&\downarrow
\\
X^p
&\stackrel{can}{\to}& S^p X
}
\end{displaymath}
\hypertarget{ForEInfinityRings}{}\subsubsection*{{For $E_\infty$-Rings}}\label{ForEInfinityRings}

\begin{defn}
\label{EInfinityFrobenius}\hypertarget{EInfinityFrobenius}{}
Let $E$ be a [[E-infinity ring]] and $p$ a [[prime number]]. Then the \emph{Frobenius morphism} on $R$ is the composite morphism of [[spectra]]

\begin{displaymath}
R
    \overset{\Delta_p}{\longrightarrow}
  (R \wedge \cdots \wedge R)^{t C_p}
    \overset{prod^{t C_p}}{\longrightarrow}
  R^{C_p}
\end{displaymath}
where

\begin{enumerate}%
\item $C_p = \mathbb{Z}/p\mathbb{Z}$ denotes the [[cyclic group]] of order $p$;


\item $(-)^{t C_p}$ denotes the [[Tate spectrum]] of a spectrum with $C_p$-[[action]];


\item the [[smash product of spectra]] $R \wedge \cdots \wedge R$ is regarded with the $C_p$-action given by [[permutation]] of smash factors;


\item $\Delta_p$ dentotes the \emph{[[Tate diagonal]]} map


\item $prod^{t C_p}$ is the image of the $p$-fold product operation of the ring spectrum $prod \;\colon\; R \wedge \cdots \wedge R \to R$ under the [[(infinity,1)-functor]] which forms Tate spectra.



\end{enumerate}
\end{defn}
(\hyperlink{NikolausScholze17}{Nikolaus-Scholze 17, def. IV.1.1})

\begin{example}
\label{EInfinityFrobeniusReducesToOrdinaryFrobenius}\hypertarget{EInfinityFrobeniusReducesToOrdinaryFrobenius}{}
Let $A \in CRing$ be an ordinary [[commutative ring]] and write $H A$ for its [[Eilenberg-MacLane spectrum]]. Then for $p$ a [[prime number]] the Frobenius homomorphism from def. \ref{EInfinityFrobenius}

\begin{displaymath}
H A \longrightarrow H A^{t C_p}
\end{displaymath}
coincides on the 0th [[stable homotopy group]] with the ordinary Frobenius homomorphism

\begin{displaymath}
A \longrightarrow A/p
\end{displaymath}
\end{example}
(\hyperlink{NikolausScholze17}{Nikolaus-Scholze 17, example IV.1.1})

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{for_fields_2}{}\subsubsection*{{For fields}}\label{for_fields_2}

\begin{itemize}%
\item The Frobenius morphism on algebras is always [[injective]]. Note that the Frobenius morphism of schemes (see below) is \emph{not} always a monomorphism.


\item The [[image]] of the Frobenius morphism is the set of elements of $k$ with a $p$-th root and is sometimes denoted $k^{1/p}$.


\item The Frobenius morphism is [[surjective]] if and only if $k$ is [[perfect field|perfect]].



\end{itemize}
\hypertarget{AsElementsOfGaloisGroup}{}\subsubsection*{{As elements of the Galois group}}\label{AsElementsOfGaloisGroup}

Some powers of the Frobenius morphism canonically induce elements in the [[Galois group]] (\ldots{})

Review of the standard story is for instance in (\hyperlink{Snyder02}{Snyder 02, section 1.5}). Further developments include (\hyperlink{DokchitserDokchitser10}{Dokchitser-Dokchitser 10})

If $\mathbb{F}_{p^n}$ is a [[finite field]],then $Gal(\mathbb{F}_{p^n}/\mathbb{F}_p)$ is [[generators and relations|generated]] by the Frobenius map $x\mapsto x^p$ (e.g. \hyperlink{Snyder02}{Snyder 02, lemma 1.5.10}).

(..)

See also at \emph{[[Artin L-function]]}.

\hypertarget{for_schemes}{}\subsubsection*{{For schemes}}\label{for_schemes}

For the purposes below $k$ will be a [[perfect field]] of [[characteristic]] $p${\tt \symbol{62}}$0$.

\begin{itemize}%
\item $X$ is [[smooth scheme|smooth]] over $k$ if and only if $F$ is a [[vector bundle]], i.e. $F_*\mathcal{O}_X$ is a free $\mathcal{O}_X$-module of rank $p$. One can study singularities of $X$ by studying properties of $F_*\mathcal{O}_X$.


\item If $X$ is [[smooth scheme|smooth]] and [[proper scheme| proper]] over $k$, the sequence $0\to \mathcal{O}_X\stackrel{F^{ab}}{\to} F_*\mathcal{O}_X \to d\mathcal{O}_X\to 0$ is exact and if it [[split exact sequence|splits]] then $X$ has a lifting to $W_2(k)$.



\end{itemize}
\begin{prop}
\label{}\hypertarget{}{}
Let $X$ be a $k$-formal scheme (resp. a [[locally algebraic scheme]]) then $X$ is [[etale scheme|étale]] iff the [[Frobenius morphism]] $F_X:X\to X^{(p)}$is a monomorphism (resp. an isomorphism).

\end{prop}
The Frobenius as a morphism (natural transformation) of (affine) group schemes is one operation among other (related) operations of interest:

\begin{remark}
\label{}\hypertarget{}{}
For any commutative affine [[group scheme]] $G$ the Frobenius- and the [[Verschiebung morphism]] correspond by ``completed [[Cartier duality|Cartier duality]]''; i.e. we have

\begin{displaymath}
\hat D(V_G)=F_{\hat D(G)}
\end{displaymath}
\end{remark}
For a more detailed account of the relationship of Frobenius-, [[Verschiebung morphism|Verschiebung-]] and [[homothety morphism]] see \hyperlink{Hazewinkel}{Hazewinkel}

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[Artin-Schreier sequence]]


\item [[Weil conjecture]]


\item [[restricted Lie algebra]]


\item [[shtuka]]


\item [[cyclotomic spectrum]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item [[Ferdinand Georg Frobenius]], Vol 2, around p. 719 of \emph{Gesammelte Abhandlungen}, Springer-Verlag, Berlin, 1968.

\end{itemize}
Lecture notes include

\begin{itemize}%
\item [[Günter Tamme]], section II 4.2 of \emph{[[Introduction to Étale Cohomology]]}


\item [[James Milne]], section 27 of \emph{[[Lectures on Étale Cohomology]]}



\end{itemize}
Further discussion of the relation to the [[Galois group]] includes

\begin{itemize}%
\item [[Noah Snyder]], section 1.5 of \emph{Artin L-Functions: A Historical Approach}, 2002 (\href{http://www.math.columbia.edu/~nsnyder/thesismain.pdf}{pdf})


\item Tim Dokchitser, Vladimir Dokchitser, \emph{Identifying Frobenius elements in Galois groups} (\href{http://arxiv.org/abs/1009.5388}{arXiv:1009.5388})



\end{itemize}
See also

\begin{itemize}%
\item Michel Demazure, \emph{[[lectures on p-divisible groups]]} \href{http://sites.google.com/site/mtnpdivisblegroupsworkshop/lecture-notes-on-p-divisible-groups}{web}


\item [[Michiel Hazewinkel]], witt vectors. part 1, \href{http://arxiv.org/abs/0804.3888v1}{arXiv:0804.3888v1}


\item Karen Smith, \emph{Brief Guide to Some of the Literature on F-singularities}, 



\end{itemize}
Discussion in the context of [[power operations]] on [[E-infinity rings]] is in

\begin{itemize}%
\item [[Jacob Lurie]], \emph{[[Rational and p-adic Homotopy Theory]]}

\end{itemize}
Discussion for [[E-infinity rings]] via [[Tate spectra]] is due to

\begin{itemize}%
\item [[Thomas Nikolaus]], [[Peter Scholze]], \emph{On topological cyclic homology} (\href{https://arxiv.org/abs/1707.01799}{arXiv:1707.01799})

\end{itemize}
A brief discussion in the context of algebraic geometry over rigs is in

\begin{itemize}%
\item [[William Lawvere]], \emph{Core Varieties, Extensivity, and Rig Geometry} , TAC \textbf{20} no.14 (2008) pp.497-503. (\href{http://www.tac.mta.ca/tac/volumes/20/14/20-14abs.html}{abstract})

\end{itemize}
[[!redirects Frobenius morphisms]]

[[!redirects Frobenius homomorphism]] [[!redirects Frobenius homomorphisms]]

[[!redirects Frobenius endomorphism]] [[!redirects Frobenius endomorphisms]]

[[!redirects Frobenius automorphism]] [[!redirects Frobenius automorphisms]]

[[!redirects Frobenius map]] [[!redirects Frobenius maps]]



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