Example 4

Tag 0CKR, permanent. Cite it as [Bourbaki, Tag 0CKR].

Commutative Algebra, chapter VIII, § 1. Dimension de Krull d’un anneau

Suppose A is an integral domain and finite-dimensional. Put $d = \dim(A)$ . Then we have $\mathscr{C}_{\leq d} = \mathscr{C}$ , $Z_d = \mathbf{Z}.[\mathrm{Spec}(A)] = Z^0$ , $z_d(M) = \mathrm{rg}(M).[Spec(A)] = z^0(M)$ , and the modules of type $\mathscr{C}_{\leq d-1}$ are the torsion modules.

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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