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Any R-module admits a projective resolution.

When the ring is Noetherian, can we show the existence of a projective resolution of any finitely generated R-module by finite rank free R-modules?

In particular, what would prevent one these free R-modules from having a infinite basis?

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Since $R$ is noetherian (as a ring), every finitely generated $R$-module $M$ is noetherian (as an $R$-module), i.e. every submodule of $M$ is also finitely generated.

In particular, we can construct a resolution of $M$ by finitely generated free $R$-modules. Start with a finite generating set for $M$ of size $r_0$, take $F_0 = R^{r_0}$ and consider the kernel of the projection $F_0 \to M$: it is finitely generated, so pick a finite generating set for the kernel of size $r_1$, take $F_1 = R^{r_1}$, etc.

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