I'm reading Bruns-Herzog's book Cohen Macaulay rings and have a probably elementary question. Why we may consider both modules as modules over $R_{(p)}$ in this theorem?
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i know that both modules are modules over $R_{(p)}$. but i dont know if depth and $r$ change with changing ring?
The key observation is that $R_p$ and $R_{p^*}$ can be obtained as localizations of $R_{(p)}$. Similarly we can obtain $M_p, M_{p^*}$ as localizations of $M_{(p)}$. So what B&H effectively do in their proof, is to replace $R$ by $R_{(p)}$ and $M$ by $M_{(p)}$. Hence, there is no computation of the depth with respect to a different ring, just a replacement of the underlying ring.