suppose $R$ is a non-commutative ring of finite (left) global dimension and $M$ is a finitely generated (left) $R$-module.
So we know that there is a projective resolution of $M$ of finite length. The first term $P_0$ of the standard resolution will be finitely generated free. However, the next step would take into account the kernel $P_0\to M$, which need not be finitely generated.
So what about the general case? Is there always a resolution by finitely generated projective modules (allowing that specific resolution to be infinite)? If the answer is negative, which I would expect, what conditions on $R$ would make it true?