Frobenius morphism (Rev #8) in nLab

Contents

Frobenius morphism of fields

Suppose kk is a field of positive characteristic pp. The Frobenius morphism is an endomorphism of the field F:k→kF:k\to k defined by F(a)=a pF(a)=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.

Properties

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

• Frobenius is surjective if and only if kk is perfect.

Frobenius morphism of schemes

In terms of schemes as locally ringed spaces

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.

Properties

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).

In terms of schemes as 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. 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}

end the linear map

α V:{TS pV→⊗ pV a⊗λ↦λ(a⊗⋯⊗a)\alpha_V: \begin{cases} TS^p V\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: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 a ring spectrum we abbreviate S pX=S k pX:=Sp k(TS pA)S^p X=S^p_k X:=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 }

Properties

Examples

If X=Sp kAX=Sp_k A is a kk-ring spectrum we have X (p)=Sp kA (p)X^{(p)}=Sp_k A^{(p)} and F X=Sp kF AF_X=Sp_k F_A.

If k=𝔽k=\mathbb{F} is a finite field we have X (p)=XX^{(p)}=X however F XF_X will not equal id Xid_X in general.

If k↪k ′k\hookrightarrow k^\prime is a field extension we have F X⊗ kk ′=F X⊗ kk ′F_{X\otimes_k k^\prime}=F_X\otimes_k k^\prime.

References

• Michel Demazure, lectures on p-divisible groups web

• Karen Smith, Brief Guide to Some of the Literature on F-singularities, American Institute of Mathematics