# nLab Demazure, lectures on p-divisible groups, I.10, Frobenius morphism and symmetric products

This entry is about a section of the text

Let $p$ be a prime number, let $k$ be a filed of characteristic $p$, let $V$ be a $k$-vector space, let ${\otimes }^{p}V$ denote the $p$-fold tensor power of $V$, let ${\mathrm{TS}}^{p}V$ denote the subspace of symmetric tensors. Then we have the symmetrization operator

${s}_{V}:\left\{\begin{array}{l}{\otimes }^{p}V\to {\mathrm{TS}}^{p}V\\ {a}_{1}\otimes \cdots \otimes {a}_{n}↦{\Sigma }_{\sigma \in {S}_{p}}{a}_{\sigma \left(1\right)}\otimes \cdots \otimes {a}_{\sigma \left(n\right)}\end{array}$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

${\alpha }_{V}:\left\{\begin{array}{l}{\mathrm{TS}}^{p}V\to {\otimes }^{p}V\\ a\otimes \lambda ↦\lambda \left(a\otimes \cdots \otimes a\right)\end{array}$\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 ${V}^{\left(p\right)}\stackrel{{\alpha }_{V}}{\to }{\mathrm{TS}}^{p}V\to {\mathrm{TS}}^{p}V/s\left({\otimes }^{p}V\right)$ is bijective and we define ${\lambda }_{V}:{\mathrm{TS}}^{p}V\to {V}^{\left(p\right)}$ by

${\lambda }_{V}\circ s=0$\lambda_V\circ s=0

and

${\lambda }_{V}\circ {\alpha }_{V}=\mathrm{id}$\lambda_V \circ \alpha_V= id

If $A$ is a $k$-ring we have that ${\mathrm{TS}}^{p}A$ is a $k$-ring and ${\lambda }_{A}$ is a $k$-ring morphism.

If $X={\mathrm{Sp}}_{k}A$ is a ring spectrum we abbreviate ${S}^{p}X={S}_{k}^{p}X:={\mathrm{Sp}}_{k}\left({\mathrm{TS}}^{p}A\right)$ and the following diagram is commutative.

$\begin{array}{ccc}X& \stackrel{{F}_{X}}{\to }& {X}^{\left(p\right)}\\ ↓& & ↓\\ {X}^{p}& \stackrel{\mathrm{can}}{\to }& {S}^{p}X\end{array}$\array{ X &\stackrel{F_X}{\to}& X^{(p)} \\ \downarrow&&\downarrow \\ X^p &\stackrel{can}{\to}& S^p X }
Revised on May 27, 2012 13:23:13 by Stephan Alexander Spahn (79.227.168.80)