# Noetherian rings whose prime ideals have projective dimension bounded above

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 ?

• what is your definition of global dimension? – Sunny Rathore Jun 8 at 17:37
• @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 – user521337 Jul 3 at 15:43