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Demazure, lectures on p-divisible groups, I.9, the Frobenius morphism

This entry is about a section of the text

Reminder

Let s:RSs:R\to S be a morphism of rings. Then we have an adjunction

(s *s *):S.Mods *R.Mod(s^*\dashv s_*):S.Mod\stackrel{s_*}{\to} R.Mod

from the category of SS-modules to that of RR-modules where

s *:AA sSs^*:A\mapsto A\otimes_s S

is called scalar extension and s *s_* is called scalar restriction.

Idea

(Frobenius recognizes p-torsion)

Definition

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. 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}

end the linear map

α V:{TS pV pV aλλ(aa)\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 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: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 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

Note that the Frobenius F pF_p is an endomorphism of a field RR only if the characteristic of RR is pp. In this case it is automatically a monomorphism, since field homomorphisms always are.

However if we pass from rings to schemes, in general it is not true that Frobenius is a monomorphism. The following proposition gives necessary and sufficient conditions for the Frobenius to be a monomorphism in case of formal schemes.

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).

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 kk 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.

Revised on July 18, 2012 14:55:21 by Stephan Alexander Spahn (79.227.139.9)