Definition 1

Let P be a finitely generated projective A-module. For every prime ideal $p$ of A, the rank of the free $A_p$-module $P_p$ is called the rank of P at $p$ and is denoted by $\operatorname{rg}_p(P)$.

By Theorem 1 the integer-valued function $p \mapsto \mathrm{rg}_p(P)$ is locally constant on $X = \mathrm{Spec}(A)$; it is therefore constant if $X$ is connected and in particular if the ring $A$ is an integral domain (§ 4, no. 3, Corollary 2 to Proposition 15).