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I am in the condition where I have a noetherian ring $R$ of finite global dimension. Consider the category of finitely generated (right) modules over $R$. Then I want to show that every module admits a finite projective resolution of finitely generated projective modules.

The definition of being having finite global dimension only means that all modules have a finite projective resolution, I do not know how to prove that the resolution is finitely generated.

Any hint?

Thanks!

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    $\begingroup$ Every f.g. module over a noetherian ring is noetherian. $\endgroup$ – user26857 Jan 3 '16 at 15:25
  • $\begingroup$ My bad for that! I now did. About your answer, I agree that my modules are noetherian, but I do not see how it gives me the existence of a finite projective resolution of finitely generated modules. In fact in this way I know that every submodule is also finitely generated but what gives me that the only finite projective resolution is only made of not finitely generated projective modules? Can you be more specific? $\endgroup$ – Ale Jan 3 '16 at 16:01
  • $\begingroup$ In fact you can find a finite free resolution. If $M$ is your module then there is a finite free module $F_0$ such that $F_0\to M\to 0$. The kernel of this map is also f.g. and can do the same for this, etc. $\endgroup$ – user26857 Jan 3 '16 at 16:05
  • $\begingroup$ Oh I see. So it eventually stops because the generators of the kernels are less than the generators of the free modules they are injected in. So this concludes right? $\endgroup$ – Ale Jan 3 '16 at 16:09
  • $\begingroup$ No, it stops since the projective dimension of the module is finite. $\endgroup$ – user26857 Jan 3 '16 at 16:13

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