Exercise 13, § 1

Let A be a ring of characteristic $p$. The filtration of $W(A)$ by the ideals $V_n(A)$ is compatible with the ring structure of $W(A)$, and $\mathrm{gr}_V(W(A))$ denotes the associated graded ring. For every integer $n \geq 0$, let $\varphi^n_*A$ denote the ring A endowed with the structure of an A-algebra given by the homomorphism $\varphi^n : A \to A$, where $\varphi$ denotes raising to the $p$-th power. Prove that, for every integer $n \geq 0$, the mapping of A into $V_n(A)/V_{n+1}(A)$ which to $x \in A$ associates the class of $V^n \tau(x)$ is an isomorphism of the A-module $\varphi^n_*A$ onto the A-module $\mathrm{gr}_V^n(W(A))$. Endow the A-module $\bigoplus_{n \in \mathbf{N}} \varphi^n_*A$ with the structure of a graded ring given by the mappings $(x, y) \mapsto \varphi^n x \cdot \varphi^m y$ of $\varphi^m_*A \times \varphi^n_*A$ into $\varphi^{m+n}_*A$ (for every pair of positive integers $(m, n)$). Now prove that the preceding isomorphisms define an isomorphism of graded A-algebras of $\bigoplus_{n \in \mathbf{N}} \varphi^n_*A$ onto $\mathrm{gr}_V(W(A))$.