Exercise 57, § 1

Let $m, n$ be two integers $\geq 1$. For every ring A of characteristic $p$, let us denote by $_m W_n(A)$ the kernel of the endomorphism $F^m$ of $W_n(A)$. a) For $m \geq 2$ and $n \geq 1$, the following diagram is commutative

where the mappings V, F are induced by the shift and Frobenius homomorphisms respectively, I is the natural injection and R the natural projection. b) For $n \geq 1$ and $a = [a_0, ..., a_{n-1}]$ in $W_n(A)$, we denote by $\tilde{a}$ the element $(a_0, ..., a_{n-1}, 0, ...)$ of $W(A)$.

Let $a \in {}_m W_n(A)$, $b \in {}_n W_m(A)$. We then put

(cf. p. 57, Exercise 43 for the notation E, and p. 54, Exercise 40 for the notation $\omega_A$). Prove that the mapping $(a, b) \mapsto \langle a, b \rangle$ of ${}_m W_n(A) \times {}_n W_m(A)$ into $A^*$ is $\mathbf{Z}$-bilinear. c) With the notation of $a$, one has and