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. 



When can one localize Ext? 
I would suggest having a look at the article ["Compactifying the Picard Scheme"](http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6W9F-4CRY60R-1H3&_user=2459438&_rdoc=1&_fmt=&_orig=search&_sort=d&_docanchor=&view=c&_acct=C000057302&_version=1&_urlVersion=0&_userid=2459438&md5=deca64ad5b5501727ed918a10bc1ffd9)
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.


