In number theory, Galois theory and arithmetic geometry in prime characteristic $p$, the Frobenius morphism is the endomorphism acting on algebras, function algebras, structure sheaves etc., which takes each ring/algebra-element $x$ to its $p$th power
It is precisely in positive characteristic $p$ that this operation is indeed an algebra homomorphism.
The Frobenius map is the shadow of the power operations in multiplicative cohomology theory/higher algebra (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 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 Borger’s absolute geometry – Motivation for more on this.
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Let $k$ be a field of positive characteristic $p$. The Frobenius morphism is an endomorphism of the field $F \colon k \to k$ defined by
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 $(a+b)^p=a^p+b^p$.
Suppose $(X,\mathcal{O}_X)$ is an $S$-scheme where $S$ is a scheme over $k$. The 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 pullbacks 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 relative Frobenius. By construction the relative Frobenius is a map of $S$-schemes.
Let $p$ be a prime number, let $k$ be a field of characteristic $p$. For a $k$-ring $A$ we define
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 Frobenius morphism for $X$ is the transformation of $k$-functors defined by
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 k-schemes, too.
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. Then we have the symmetrization operator
end the linear map
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 $\lambda_V:TS^p V\to V^{(p)}$ by
and
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 a ring spectrum we abbreviate $S^p X=S^p_k X:=Sp_k (TS^p A)$ and the following diagram is commutative.
Frobenius is always injective. Note that the Frobenius morphism of schemes (see below) is not always a monomorphism.
The image of Frobenius is the set of elements of $k$ with a $p$-th root and is sometimes denoted $k^{1/p}$.
Frobenius is surjective if and only if $k$ is perfect.
Some powers of the Frobenius morphism canonically induce elements in the Galois group (…)
Review of the standard story is for instance in (Snyder 02, section 1.5). Further developments include (Dokchitser-Dokchitser 10)
If $\mathbb{F}_{p^n}$ is a finite field,then $Gal(\mathbb{F}_{p^n}/\mathbb{F}_p)$ is generated by the Frobenius map $x\mapsto x^p$ (e.g. Snyder 02, lemma 1.5.10).
(..)
See also at Artin L-function.
For the purposes below $k$ will be a perfect field of characteristic $p$>$0$.
$X$ is 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 $F_*\mathcal{O}_X$.
If $X$ is smooth and 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 splits then $X$ has a lifting to $W_2(k)$.
Let $X$ be a $k$-formal scheme (resp. a locally algebraic scheme) then $X$ is étale iff the Frobenius morphism $F_X:X\to X^{(p)}$is a monomorphism (resp. an isomorphism).
The Frobenius as a morphism (natural transformation) of (affine) group schemes is one operation among other (related) operations of interest:
For any commutative affine group scheme $G$ the Frobenius- and the Verschiebung morphism? correspond by ‘’completed Cartier duality’’; i.e. we have
For a more detailed account of the relationship of Frobenius-, Verschiebung-? and homothety morphism? see Hazewinkel
Lecture notes include
Günter Tamme, section II 4.2 of Introduction to Étale Cohomology
James Milne, section 27 of Lectures on Étale Cohomology
Further discussion of the relation to the Galois group includes
Noah Snyder, section 1.5 of Artin L-Functions: A Historical Approach, 2002 (pdf)
Tim Dokchitser, Vladimir Dokchitser, Identifying Frobenius elements in Galois groups (arXiv:1009.5388)
See also
Michel Demazure, lectures on p-divisible groups web
Michiel Hazewinkel, witt vectors. part 1, arXiv:0804.3888v1
Karen Smith, Brief Guide to Some of the Literature on F-singularities, American Institute of Mathematics
Discussion in the context of power operations on E-infinity rings is in