Let $R$ be a Noetherian ring. For any nonzero $R$-module $M$, and any prime $P\in \operatorname{Ass}M$, do we have $\operatorname{pd}M\geq\operatorname{depth}P$?
When $M$ is finitely generated, this can be shown by first reducing to the local case, and then using Auslander-Buchsbaum Formula. However, I have no idea how to deal with the infinitely generated case. Any suggestions?
