For a module $M$ over a commutative Noetherian ring $R$, let $pd_R (M)$ denote the projective dimension of $M$ as an $R$-module. Now let $R$ be a commutative Noetherian ring such that $\sup \{ pd_R (Q) : Q \in Spec (R) \} < \infty$, then definitely $R$ is regular.

My question is: Does $R$ have finite global dimension ?

  • $\begingroup$ what is your definition of global dimension? $\endgroup$ – Sunny Rathore Jun 8 at 17:37
  • $\begingroup$ @Sunny Rathore: For a commutative Noetherian ring $R$, the global dimension is $\sup \{pd_R(M) : M \in R$-$Mod \}=\sup \{pd_R(R/J) : J$ is an ideal of $R\}$ ... if this supremum doesn't exist, we say $R$ had infinite global dimension $\endgroup$ – user521337 Jul 3 at 15:43

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