Exercise 24, § 1

Let $I$ be the set of negative integers. For every ring $A$, every nilpotent two-sided ideal $\mathfrak{n}$ of $A$, and every integer $r \geqslant 0$, let $CW(A, \mathfrak{n}, r)$ be the subset of $A^1$ formed by the elements $(a_i)_{i \in I}$ such that $a_{-n} \in \mathfrak{n}$ if $n \geqslant r$. In other words, one has

Endow $CW(A, \mathfrak{n}, r)$ with the product topology, each factor $A$ or $\mathfrak{n}$ being endowed with the discrete topology. Let $CW(A)$ denote the union of the $CW(A, \mathfrak{n}, r)$, and endow $CW(A)$ with the inductive limit topology. For this topology, $CW(A)$ is separated and $CW^u(A)$ (cf. exercise 23) is dense in $CW(A)$. The elements of $CW(A)$ are called Witt covectors of A. a) Let $t$ be a positive integer, and let $\omega_0, ..., \omega_t$ be positive integers such that $\omega_0$ be nonzero and that $p^{t+1}$ divide $\sum_{i=0}^t p^i \omega_i$. Then one has $\sum_{i=0}^t \omega_i \geqslant t(p-1) + p$. b) Let $\mathscr{A}$ be the ring $\mathbf{Z}[X, Y]$ of polynomials in two families of indeterminates $X = (X_i)_{i \in I}$ and $Y = (Y_i)_{i \in I}$. For every integer $r \geqslant 0$, let $\mathfrak{n}_r$ be the ideal of $\mathscr{A}$ generated by the $X_{-n}$ and $Y_{-n}$ for $n \geqslant r$. Let $r$ and $s$ be two integers $\geqslant 1$. Then one has modulo $\mathfrak{n}_r^s$ for every integer $m \geqslant r-1$ if $s < p$ and for every integer $m \geqslant r-1 + (s-p)/(p-1)$ if $s \geqslant p$. (By weight arguments, now prove that the difference of the two members is a linear combination with integer coefficients of terms of the form $X_{-m-1}^{u_0} Y_{-m-1}^{v_0} X_{-m}^{u_1} Y_{-m}^{v_1} ... X_0^{u_{m+1}} Y_0^{v_{m+1}}$) where, if one puts $\omega_i = u_i + v_i$ for $0 \leqslant i \leqslant m+1$, one has $\omega_0 \neq 0$ and $\sum_{i=0}^t p^i \omega_i = p^{m+1}$. Then use a). c) Let $A$ be a ring, $\mathfrak{n}$ a nilpotent two-sided ideal of $A$, $r$ a positive integer, and $a, b$ two elements of $CW(A, \mathfrak{n}, r)$. Then: (i) For every integer $n \geqslant 0$, the sequence of elements of $A$ is stationary. (ii) For every integer $n \geqslant 0$, let $c_{-n}$ be the limit of the preceding sequence. Then the element $c = (c_i)_{i \in I}$ belong to $CW(A, \mathfrak{n}, r)$. One will put $a + b = c$. (iii) The preceding law of addition endows $CW(A)$ with a commutative group structure, compatible with its topology. For every nilpotent two-sided ideal $\mathfrak{n}$ of $A$ and every integer $r \geqslant 0$, the subset $CW(A, \mathfrak{n}, r)$ of $CW(A)$ is a topological subgroup of it. The same holds for $CW^u(A)$. d) Let $A$ and $B$ be two rings, $\varphi$ a ring homomorphism from $A$ into $B$. Let us denote by $CW(\varphi)$ the mapping of $CW(A)$ into $CW(B)$ which to $(a_i)_{i \in I}$ associates $(\varphi(a_i))_{i \in I}$. Then $CW(\varphi)$ is a group homomorphism and a continuous mapping.