Remark

For every $p \in \mathrm{Spec}(A)$, one has $z(A/p) = [V(p)]$. More generally, let $M$ be a finitely generated $A$-module, and let $(M_i)_{0 \leq i \leq n}$ be a composition sequence of $M$ such that for $0 \leq i \leq n-1$, the module $M_i/M_{i+1}$ is isomorphic to $A/p_i$, where $p_i$ is a prime ideal of $A$ (cf. IV, § 1, No. 4, Theorem 1); then one has $z(M) = \sum_{i \in J} [V(p_i)]$, where $J$ is the subset of $I$ consisting of the $i$ such that $p_i$ is a minimal element of $\{ p_0, ..., p_{n-1} \}$ (IV, § 1, No. 4, Theorem 2 and § 2, No. 5, Remark 1).

For every integer $d$, let us denote by $Z_{\leq d}$ (resp. $Z_d$, resp. $Z^{>d}$, resp. $Z^d$) the $\mathbf{Z}$-submodule of $Z(A)$ generated by the elements $[V(p)]$ where $p$ is a prime ideal of $A$ such that $\dim(A/p) \leq d$ (resp. $\dim(A/p) = d$, resp. $\mathrm{ht}(p) \geq d$, resp. $\mathrm{ht}(p) = d$). The elements of $Z_d$ (resp. $Z^d$) are called cycles of dimension $d$ (resp. of codimension $d$). One obviously has

Let moreover $C$ be the set of classes of finitely generated $A$-modules (A, VIII, § 3, No. 5), and for each integer $d$, let $C_{\leq d}$ (resp. $C^{>d}$) be the subset of $C$ consisting of the classes of finitely generated $A$-modules of dimension $\leq d$ (resp. whose support is of codimension $\geq d$ in $\mathrm{Spec}(A)$).