Proposition 10

The sequences of $\mathbf{Z}$-modules and homomorphisms

are exact.

One has $\zeta_d \circ i_d = 0$ by Lemma 1. For every $p \in \mathrm{Spec}(A)$ such that $\dim(A/p) = d$, one has $\zeta_d([A/p]_{\leq d}) = z_d(A/p) = [V(p)]$, hence the homomorphism $\zeta_d$ is surjective. By IV, § 1, No. 4, Theorem 1, $K(\mathscr{C}_{\leq d})$ is generated by the $[A/p]_{\leq d}$, where $p \in \mathrm{Spec}(A)$ and $\dim(A/p) \leq d$; consequently, every element $\xi$ of $K(\mathscr{C}_{\leq d})$ can be written $\xi = i_d(\eta) + \sum_{i=1}^k n_i[A/p_i]_{\leq d}$, with $\eta \in K(\mathscr{C}_{\leq d-1})$, $n_i \in \mathbf{Z}$ and $\dim(A/p_i) = d$ for $1 \leq i \leq k$; one has $\zeta_d(\xi) = \sum_{i=1}^k n_i[V(p_i)]$ and consequently $\zeta_d(\xi) = 0$ implies $\xi = i_d(\eta) \in \mathrm{Im}(i_d)$, whence $\mathrm{Ker}(\zeta_d) = \mathrm{Im}(i_d)$.

One reasons analogously for the second sequence.