Let $f:A\rightarrow B$ be a ring morphism, $P$ a prime ideal of $B$. If $M$ is an $A$-module, then can we express $(M\otimes_A B)_P=M\otimes_A B_P$ as a tensor product over $A_{f^{-1}(P)}$? I would expect something like $M_{f^{-1}(P)}\otimes_{A_{f^{-1}(P)}} B_P$?
2026-03-28 03:27:30.1774668450
Localization of an extension of scalars
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