Example

$*$ Let A be a Noetherian commutative ring, and let Σ be its spectrum. For any integer $n\geqslant 0$, denote by $\mathscr{C}^{\geqslant n}$ the set of classes of finitely generated A-modules with support of codimension $\geqslant n$ in Σ. Set $K(n,A) =$ $K(\mathscr{C}^{\geqslant n})$ and $\gamma_n=\gamma_{\mathscr{C}^{\geqslant n},\mathscr{C}^{\geqslant n+1}}$. We have a sequence of homomorphisms

We can prove (AC, VIII, §1, n$^o5$, p. 13, proposition 10) that in $K(n,A)$, the elements $[A/\mathfrak{p}]_{\mathscr{C}_n}$, where $\mathfrak{p}$ runs through the set of elements of Σ of height $n$, form a basis of a $\mathbf{Z}$-module supplementary to the image of $\gamma_n$. More precisely, for every module E of type $\mathscr{C}^{\geqslant n}$, we have

$[E]_{\mathscr{C}^{\geqslant n}}\equiv \sum$ long$_{A_{\mathfrak{p}}}(E_{\mathfrak{p}})\cdot [A/\mathfrak{p}]_{\mathscr{C}^{\geqslant n}}$ (mod Im $\gamma_n$)$.*$

$\{\mathfrak{p}\in \Sigma |$ht($\mathfrak{p}$)$=n\}$

The group $K(\mathscr{C})$ is generated by the elements of the form $[E]_{\mathscr{C}}$ with $E\in \mathscr{C}$, and we have $[E\oplus E']_{\mathscr{C}}= [E]_{\mathscr{C}}+ [E']_{\mathscr{C}}$ by relation (1) (VIII, p. 184). Every element of $K(\mathscr{C})$ is therefore of the form $[E]_{\mathscr{C}}-[F]_{\mathscr{C}}$, where E and F belong to $\mathscr{C}$.

An element of $K(\mathscr{C})$ is called effective if it is of the form $[E]_{\mathscr{C}}$ for an A-module E of type $\mathscr{C}$. The set of effective elements of $K(\mathscr{C})$ is denoted by $K(\mathscr{C})^+$; it is a submonoid of $K(\mathscr{C})$, and $K(\mathscr{C})$ can be identified with the group of differences of $K(\mathscr{C})^+$ (I, §2, No. 4, p. 20).