# Finitely generated projective resolution of a module over a regular local ring

by Anonymous   Last Updated May 16, 2019 04:20 AM

Let $$R$$ be a regular local ring and let $$M$$ be an $$R$$-module. Then there exists a finite projective resolution $$P_\bullet\to M\to 0$$. However, need there exist a finite projective resolution consisting of finitely generated projective modules? What if we require that $$M$$ be finitely generated?

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A regular local ring $$R$$ is Noetherian by hypothesis. If $$M$$ is a finitely generated $$R$$-module, then there is a finite rank free module $$F$$ and a surjection $$F\to M$$ fitting into a short exact sequence $$0\to N\to F\to M\to0.$$ As $$N$$ is a submodule of the finitely generated free module $$F$$ over the Noetherian ring $$R$$ then $$N$$ is finitely generated. Iterating this, gives a resolution of $$N$$ by finitely generated free modules. As $$R$$ is regular, it has finite global dimension, and the iterated kernels are eventually projective (so free) and the resolution can be brought to an end.

Lord Shark the Unknown
May 16, 2019 04:09 AM

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