Contents

# Contents

## Idea

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

$x^p = \underbrace{x \cdot x \cdots x}_{p \; factors} \;.$

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 (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.

## Definition

this entry may need attention

## For fields

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

$F(a) \coloneqq a^p \,.$

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 $(a+b)^p=a^p+b^p$.

### Of schemes

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.

### For sheaves on $C Ring ^{op}$

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

$f_A: \begin{cases} A\to A \\ x\mapsto x^p \end{cases}$

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

$F_X: \begin{cases} X\to X^{(p)} \\ X(f_R):X(R)\to X(R_f) \end{cases}$

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.

### In terms of symmetric products

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

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

and the linear map

$\alpha_V \colon \begin{cases} V^{(p)} \to\otimes^p V \\ a\otimes \lambda\mapsto\lambda(a\otimes\cdots\otimes a) \end{cases}$

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

$\lambda_V \;\colon\; TS^p V\to V^{(p)}$

by

$\lambda_V\circ s=0$

and

$\lambda_V \circ \alpha_V= id$

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.

$\array{ X &\stackrel{F_X}{\to}& X^{(p)} \\ \downarrow&&\downarrow \\ X^p &\stackrel{can}{\to}& S^p X }$

### For $E_\infty$-Rings

###### Definition

Let $E$ be a E-infinity ring and $p$ a prime number. Then the Frobenius morphism on $R$ is the composite morphism of spectra

$R \overset{\Delta_p}{\longrightarrow} (R \wedge \cdots \wedge R)^{t C_p} \overset{prod^{t C_p}}{\longrightarrow} R^{C_p}$

where

1. $C_p = \mathbb{Z}/p\mathbb{Z}$ denotes the cyclic group of order $p$;

2. $(-)^{t C_p}$ denotes the Tate spectrum of a spectrum with $C_p$-action;

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

4. $\Delta_p$ dentotes the Tate diagonal map

5. $prod^{t C_p}$ 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.

###### Example

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.

$H A \longrightarrow H A^{t C_p}$

coincides on the 0th stable homotopy group with the ordinary Frobenius homomorphism

$A \longrightarrow A/p$

## Properties

### For fields

• The Frobenius morphism on algebras is always injective. Note that the Frobenius morphism of schemes (see below) is not always a monomorphism.

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

• The Frobenius morphism is surjective if and only if $k$ is perfect.

### As elements of the Galois group

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).

(..)

### For schemes

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 $W_2(k)$.

###### Proposition

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:

###### Remark

For any commutative affine group scheme $G$ the Frobenius- and the Verschiebung morphism correspond by “completed Cartier duality‘’; i.e. we have

$\hat D(V_G)=F_{\hat D(G)}$

For a more detailed account of the relationship of Frobenius-, Verschiebung- and homothety morphism? see Hazewinkel

## References

Lecture notes include

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)