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Frobenius morphism

Contents

Idea

In number theory, Galois theory and arithmetic geometry in prime characteristic pp, the Frobenius morphism is the endomorphism acting on algebras, function algebras, structure sheaves etc., which takes each ring/algebra-element xx to its ppth power

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

It is precisely in positive characteristic pp 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 pp 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 kk be a field of positive characteristic pp. The Frobenius morphism is an endomorphism of the field F:kkF \colon k \to k defined by

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

Notice that this is indeed a homomorphism of fields: the identity (ab) p=a pb p(a b)^p=a^p b^p evidently holds for all a,bka,b\in k and the characteristic of the field is used to show (a+b) p=a p+b p(a+b)^p=a^p+b^p.

Of schemes

Suppose (X,𝒪 X)(X,\mathcal{O}_X) is an SS-scheme where SS is a scheme over kk. The absolute Frobenius is the map F ab:(X,𝒪 X)(X,𝒪 X)F^{ab}:(X,\mathcal{O}_X)\to (X,\mathcal{O}_X) which is the identity on the topological space XX and on the structure sheaves F *:𝒪 X𝒪 XF_*:\mathcal{O}_X\to \mathcal{O}_X is the pp-th power map. This is not a map of SS-schemes in general since it doesn’t respect the structure of XX as an SS-scheme, i.e. the diagram:

X F ab X S F ab S\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 SS-scheme morphism, F abF^{ab} must be the identity on SS, i.e. S=Spec(𝔽 p)S=Spec(\mathbb{F}_p).

Now we can form the fiber product using this square: X (p):=X× SSX^{(p)}:=X\times_{S} S. By the universal property of pullbacks there is a map F rel:XX (p)F^{rel}:X\to X^{(p)} so that the composition XX (p)XX\to X^{(p)}\to X is F abF^{ab}. This is called the relative Frobenius. By construction the relative Frobenius is a map of SS-schemes.

For sheaves on CRing opC Ring ^{op}

Let pp be a prime number, let kk be a field of characteristic pp. For a kk-ring AA we define

f A:{AA xx pf_A: \begin{cases} A\to A \\ x\mapsto x^p \end{cases}

The kk-ring obtained from AA by scalar restriction along f k:kkf_k:k\to k is denoted by A fA_{f}.

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

There are kk-ring morphisms f A:AA ff_A: A\to A_f and F A:{A (p)A xλx pλF_A:\begin{cases} A^{(p)}\to A \\ x\otimes \lambda\mapsto x^p \lambda \end{cases}.

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

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

If XX is a kk-scheme X (p)X^{(p)} is a kk-scheme, too.

Since the completion functor ^:Sch kfSch k{}^\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 pp be a prime number, let kk be a field of characteristic pp, let VV be a kk-vector space, let pV\otimes^p V denote the pp-fold tensor power of VV, let TS pVTS^p V denote the subspace of symmetric tensors, yielding the symmetric algebra. Then we have the symmetrization operator

s V:{ pVTS pV a 1a nΣ σS pa σ(1)a σ(n)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

α V:{V (p) pV aλλ(aa) \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)α VTS pVTS pV/s( pV)V^{(p)}\stackrel{\alpha_V}{\to}TS^p V\to TS^p V/s(\otimes^p V) is bijective and we define

λ V:TS pVV (p) \lambda_V \;\colon\; TS^p V\to V^{(p)}

by

λ Vs=0\lambda_V\circ s=0

and

λ Vα V=id\lambda_V \circ \alpha_V= id

If AA is a kk-ring we have that TS pATS^p A is a kk-ring and λ A\lambda_A is a kk-ring morphism.

If X=Sp kAX=Sp_k A is the spectrum of a commutative ring we abbreviate S pX0S k pXSp k(TS pA)S^p X 0 S^p_k X \coloneqq Sp_k (TS^p A) and the following diagram is commutative.

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

For E E_\infty-Rings

Definition

Let EE be a E-infinity ring and pp a prime number. Then the Frobenius morphism on RR is the composite morphism of spectra

RΔ p(RR) tC pprod tC pR C p 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=/pC_p = \mathbb{Z}/p\mathbb{Z} denotes the cyclic group of order pp;

  2. () tC p(-)^{t C_p} denotes the Tate spectrum of a spectrum with C pC_p-action;

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

  4. Δ p\Delta_p dentotes the Tate diagonal map

  5. prod tC pprod^{t C_p} is the image of the pp-fold product operation of the ring spectrum prod:RRRprod \;\colon\; R \wedge \cdots \wedge R \to R under the (infinity,1)-functor which forms Tate spectra.

(Nikolaus-Scholze 17, def. IV.1.1)

Example

Let ACRingA \in CRing be an ordinary commutative ring and write HAH A for its Eilenberg-MacLane spectrum. Then for pp a prime number the Frobenous homomorphsm from def. 1

HAHA tC p H A \longrightarrow H A^{t C_p}

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

AA/p A \longrightarrow A/p

(Nikolaus-Scholze 17, example IV.1.1)

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 kk with a pp-th root and is sometimes denoted k 1/pk^{1/p}.

  • The Frobenius morphism is surjective if and only if kk 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 𝔽 p n\mathbb{F}_{p^n} is a finite field,then Gal(𝔽 p n/𝔽 p)Gal(\mathbb{F}_{p^n}/\mathbb{F}_p) is generated by the Frobenius map xx px\mapsto x^p (e.g. Snyder 02, lemma 1.5.10).

(..)

See also at Artin L-function.

For schemes

For the purposes below kk will be a perfect field of characteristic pp>00.

  • XX is smooth over kk if and only if FF is a vector bundle, i.e. F *𝒪 XF_*\mathcal{O}_X is a free 𝒪 X\mathcal{O}_X-module of rank pp. One can study singularities of XX by studying properties of F *𝒪 XF_*\mathcal{O}_X.

  • If XX is smooth and proper over kk, the sequence 0𝒪 XF abF *𝒪 Xd𝒪 X00\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 XX has a lifting to W 2(k)W_2(k).

Proposition

Let XX be a kk-formal scheme (resp. a locally algebraic scheme) then XX is étale iff the Frobenius morphism F X:XX (p)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 GG the Frobenius- and the Verschiebung morphism correspond by ‘’completed Cartier duality’’; i.e. we have

D^(V G)=F D^(G)\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)

See also

Discussion in the context of power operations on E-infinity rings is in

Discussion for E-infinity rings via Tate spectra is due to

Revised on July 24, 2017 12:11:33 by Urs Schreiber (195.37.234.72)