Rank of a projective module

Summary

The rank of a finitely generated projective module $P$ over a commutative ring $R$ is a function on prime ideals. It assigns to $\mathfrak{p}$ the rank of the free module $P_\mathfrak{p}$ over the local ring $R_\mathfrak{p}$. It is a locally constant function on the spectrum of $R$, so it is constant when the spectrum is connected. It extends the dimension of a vector space and gives a homomorphism from $K_0(R)$ to the group of locally constant integer-valued functions on $\mathrm{Spec}\, R$.