# When can one localize Ext?

Let $R\to S$ be a ring map such that $S$ is projective over $R$ (I am willing to assume $S=R[X_1,...,X_n]$). Let $M,N$ be finite $S$-modules. Let $P\in Spec R$ such that $M_P$ is $R_P$-flat. Under what condition can one say that $Ext^1_R(M,N)_P=0$?

This is trivial if $M$ is finite over $R$, but in general $Ext$ does not commute with localization. I would appreciate any reference on this matter.

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I would suggest having a look at the article "Compactifying the Picard Scheme" by Altman-Kleiman. They discuss base change issues for Ext. I'm not sure if this will be applicable in your particular context, but it may be a start.

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Thanks! Unfortunately I could not find a copy online, but I will check it out. –  Hailong Dao Dec 3 '09 at 16:00
Hmmm... did my link (click on the title) not work? –  B. Cais Dec 3 '09 at 19:40
This exchange is obviously quite old, but I think the issue might that many pages on sciencedirect look like paywalls, but aren't. There is an "Open Preview" link, which makes it sound like the full article isn't available, but there is also a "Download PDF" link. This has tripped me up an embarrassing number of times. –  Andrew Dudzik Sep 27 at 4:17