Can you please help me to know: How "the description of localization of $D$ at a prime ideal of it" is?
Thank you.
2026-03-27 18:14:53.1774635293
description of localization of pullback at a prime ideal
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You can consider $A$, $B$, and $C$ as $D$-modules via the maps from $D$, and then the square is also a pullback square of $D$-modules. Since localization is exact, it preserves pullbacks, so if $p\subset D$ is a prime ideal, then $D_p$ can be described as the pullback of $A_p$ and $B_p$ over $C_p$. Here $A_p=A\otimes_D D_p$ means the localization of $A$ at $p$ as a $D$-module, and similarly for the others. Note that these can be seen as localizations of $A$, $B$, and $C$ with respect to multiplicatively closed sets (namely, the image of $D\setminus p$ in each of them), but not necessarily with respect to prime ideals.