nLab I.10, Frobenius morphism and symmetric products

This entry is about a section of the text

We give here another characterization of the Frobenius morphism in terms of symmetric products.

Let pp be a prime number, let kk be a filed 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


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 }

Created on May 27, 2012 at 13:23:15. See the history of this page for a list of all contributions to it.