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

This entry is about a section of the text

• Michel Demazure, lectures on p-divisible groups, web We give here another characterization of the Frobenius morphism in terms of symmetric products.

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

$$s_V:\{\begin{array}{l}{\otimes }^{p}V\to {\mathrm{TS}}^{p}V\\ a_1\otimes \cdots \otimes a_n\mapsto {\Sigma }_{\sigma \in S_p}{a}_{\sigma (1)}\otimes \cdots \otimes a_{\sigma (n)}\end{array}$$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

$${\alpha }_V:\{\begin{array}{l}{\mathrm{TS}}^{p}V\to {\otimes }^{p}V\\ a\otimes \lambda \mapsto \lambda (a\otimes \cdots \otimes a)\end{array}$$\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 $V^{(p)}\stackrel{{\alpha }_V}{\to }{\mathrm{TS}}^{p}V\to {\mathrm{TS}}^{p}V/s({\otimes }^{p}V)$ is bijective and we define ${\lambda }_V:{\mathrm{TS}}^{p}V\to V^{(p)}$ 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 $S^{p}X=S_k^{p}X:={\mathrm{Sp}}_k({\mathrm{TS}}^{p}A)$ and the following diagram is commutative.

$$\begin{array}{ccc}X& \stackrel{F_X}{\to }& X^{(p)}\\ \downarrow & & \downarrow \\ X^p& \stackrel{\mathrm{can}}{\to }& S^{p}X\end{array}$$\array{ X &\stackrel{F_X}{\to}& X^{(p)} \\ \downarrow&&\downarrow \\ X^p &\stackrel{can}{\to}& S^p X }