Frobenius morphism (Rev #23) in nLab

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=x⋅x⋯x⏟ 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:k→kF \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,b∈ka,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 X→𝒪 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:X→X (p)F^{rel}:X\to X^{(p)} so that the composition X→X (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:{A→A x↦x 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:k→kf_k:k\to k is denoted by A fA_{f}.

The kk-ring obtained from AA by scalar extension along f k:k→kf_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:A→A ff_A: A\to A_f and F A:{A (p)→A x⊗λ↦x 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:{X→X (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 k→fSch 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:{⊗ pV→TS pV a 1⊗⋯⊗a n↦Σ σ∈S 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 a⊗λ↦λ(a⊗⋯⊗a) \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 pV→TS 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 pV→V (p) \lambda_V \;\colon\; TS^p V\to V^{(p)}

by

λ V∘s=0\lambda_V\circ s=0

and

λ V∘α 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 pX≔Sp 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

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

(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 x↦x 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 0→𝒪 X→F abF *𝒪 X→d𝒪 X→00\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).

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

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

• Artin-Schreier sequence

• Weil conjecture

• restricted Lie algebra

• shtuka

• cyclotomic spectrum

References

• Ferdinand Georg Frobenius, Vol 2, around p. 719 of Gesammelte Abhandlungen, Springer-Verlag, Berlin, 1968.

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

• Jacob Lurie, Rational and p-adic Homotopy Theory

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

• Thomas Nikolaus, Peter Scholze, On topological cyclic homology (arXiv:1707.01799)