Exercise 31, § 1

Let $A$ be a ring. Let $n \in J$. Let $(a_d)_{d \in J_n - \{n\}}$ be a family of elements of $A$. Put $u_d = \Phi_d((a_e)_{e \in J_d})$ for $d \in J_n - \{n\}$, and let $u_n$ be an element of $A$. For every prime number $p \in J_n$, suppose moreover given an endomorphism $\sigma_p$ of $A$ such that $\sigma_p(a) \equiv a^p \mod. pA$ for every $a \in A$. Then the following conditions are equivalent : a) There exists an element $a_n$ of $A$ such that $u_n = \Phi_n((a_d)_{d \in J_n})$. b) For every prime number $p \in J_n$, one has $u_n \equiv \sigma_p(u_{n/p}) \mod. p^{w_p(n)}A$.