Exercise 10, § 1

Let k be a field of characteristic $p$, having a finite $p$-base. Let A be a ring and $\varphi$ a homomorphism of k into A, making A into a finitely generated $k$-algebra. a) For every integer $n \geqslant 1$, A is a finitely generated module over $A^{p^n}$. b) For every integer $n \geqslant 1$, $W_n(A)$ is a finitely generated $W_n(k)$-algebra. (If $a_1, ..., a_N$ generate A as a $k$-algebra and as an $A^{p^{n-1}}$-module, prove that the elements $V^j \tau(a_i), \ 1 \leqslant i \leqslant N, \ 0 \leqslant j \leqslant n-1$, generate the $W_n(k)$-algebra $W_n(A)$.)