Exercise 34, § 1

Let $R = \mathbf{Z}[X, Y]$ be the $\mathbf{Z}$-algebra of polynomials in two families of indeterminates $X = (X_n)_{n \in J}$ and $Y = (Y_n)_{n \in J}$. For every prime number $p$, let $\sigma_p$ be the endomorphism of $R$ defined by $\sigma_p(X_n) = X_n^p$ and $\sigma_p(Y_n) = Y_n^p$ for every $n \in J$; one then has $\sigma_p(a) \equiv a^p \mod. pR$ for every $a \in R$. Moreover, $R$ is $\mathbf{Z}$-torsion-free. a) There exist in $R^J$ elements $S = (S_n)_{n \in J}$, $P = (P_n)_{n \in J}$, $I = (I_n)_{n \in J}$, and, for every $q \in J$, $F_q = (F_{q,n})_{n \in J}$ and $V_q = (V_{q,n})_{n \in J}$, characterized respectively by the equalities

(On embedding R in Q $\otimes$ R, we have, with the notation of Exercise 33,

) b) For every $n \in J$, assign weight $n$ to $X_n$ and $Y_n$. Then $S_n, P_n$ and $I_n$ depend only on the families $(X_d)_{d \in J_n}$ and $(Y_d)_{d \in J_n}$. Moreover: $\alpha)$ $S_n$ is isobaric of weight $n$. $\beta)$ $P_n$ is isobaric of weight $2n$, and isobaric of weight $n$ in each of the families $(X_d)_{d \in J_n}$ and $(Y_d)_{d \in J_n}$. $\gamma)$ $I_n$ is isobaric of weight $n$. c) $F_{q,n}$ is isobaric of weight $qn$, and depends only on the family $(X_d)_{d \in J_{qn}}$. d) One has $V_{q,n}(X) = X_{n/q}$, where by convention $X_{n/q} = 0$ if $n/q \notin J$. (It is enough to verify that with this definition of $V_q$, one has $v_q(\Phi(X)) = \Phi(V_q(X))$.)