Exercise 7, § 10

Let $A$ be a ring, $M$ an $A$-module, $x = (x_i)_{i \in I}$ a finite family of elements of $A$. For every integer $n \geq 1$, we denote by $x^n$ the family $(x_i^n)_{i \in I}$. Let $\Delta(x)$ be the endomorphism of $A^I$ defined by $\Delta(x)(e_i) = x_i e_i$.

a) For every couple of integers $(n, m)$ such that $n \geq m$, now prove that the mappings

define an inductive system of complexes; we denote by $K^\bullet(x^\infty, M)$ the limit of this system, and by $H^\bullet(x^\infty, M)$ its cohomology.

b) We choose a total ordering on $I$. For every subset $J$ of $I$, we put $x_J = \prod_{j \in J} x_j$.

For $p \in \mathbf{Z}$, we put $\mathscr{C}^p = \bigoplus_{|J|=p} A_{x_J}$; for $a \in A_{x_J}$, with $\operatorname{Card}(J) = p$, we put $d^p(a) = \sum_{i \notin J} (-1)^{\varepsilon(J, i)} \frac{a}{1}$ in $A_{x_{J \cup \{i\}}}$, where $\varepsilon(J, i)$ is the number of elements of $J$ strictly inferior to $i$. Exercise prove that $(\mathscr{C}, d)$ is a complex, and define a canonical isomorphism of $K^\bullet(x^\infty, M)$ onto the complex $\mathscr{C} \otimes_A M$.