Exercise 10, § 5

Let A be a commutative ring,

an exact sequence of A-modules and $n$ an integer $> 0$. Let $L = \mathrm{Coker}(T^n(u))$ so that there is an exact sequence For every subset J of the set $\{1, 2, \ldots, n\}$, we write ($P_\varnothing$ is therefore equal to $T^n(M)$) and ($Q_\varnothing$ is therefore equal to $T^n(M'')$). For every integer $i$ such that $0 \leq i \leq n$, let $P^{(i)}$ denote the submodule of $T^n(M)$ generated by the union of the images of the $P_J$ for all the subsets $J$ such that $\mathrm{Card}(J) = i$ and let $L^{(i)}$ be the canonical image of $P^{(i)}$ in $L$.

(a) Show that there exists a canonical surjective homomorphism

(cf. II, § 5, no. 6, Proposition 6).

(b) If the sequence $0 \to M' \xrightarrow{u} M \xrightarrow{v} M'' \to 0$ is exact and splits, show that the homomorphism defined in (a) is bijective.