Exercise 23, § 1

Let $A$ be a ring and $I$ the set of negative integers. Let $m$ be an integer $\geqslant 1$ and $[a_0, ..., a_{m-1}]$ an element of $W_m(A)$. Associate to it the element $(b_i)_{i \in I}$ of $A^I$ defined by

Thus $W_m(A)$ is identified with a subset of $A^1$. a) For every integer $n \geqslant 0$, the mapping $V_n : W_n(A) \to W_{n+1}(A)$ induced by the shift is a group homomorphism. The identifications of the groups $W_n(A)$ with subsets of $A^1$ are compatible with the mappings $V_n$ and make it possible to identify the group $\lim \overrightarrow{W_n(A)}$, the inductive limit of the groups $W_n(A)$ with respect to the $V_n$, with the subset $CW^u(A)$ of $A^1$ formed by the elements all but a finite number of whose components are zero, elements which shall be called unipotent Witt covectors. By transport of structure, one obtains on this subset a group structure. b) For $a = (a_i)_{i \in I}$ and $b = (b_i)_{i \in I}$ in $CW^u(A)$, one has

for every integer $m$ sufficiently large. c) For every ring homomorphism $\varphi : A \to B$, the mapping $CW^u(\varphi) : CW^u(A) \to CW^u(B)$ defined by $CW^u(\varphi)(a_i)_{i \in I} = (\varphi(a_i))_{i \in I}$ is a group homomorphism.