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?

Answers 1

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

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