Corollary

Suppose $A$ left noetherian and let $\mathscr{C}_0$ (resp. $\mathscr{C}$) be the set of classes of $A$-modules projective of finite type (resp. of $A$-modules of finite projective dimension and of finite type). Then the homomorphism of Grothendieck groups $K(\mathscr{C}_0) \to K(\mathscr{C})$ is bijective.

This follows from X, p. 58, th. 1 (note that $\mathscr{C}_0$ and $\mathscr{C}$ are left exact by cor. 2).