When can one localize Ext? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T08:25:43Z http://mathoverflow.net/feeds/question/7567 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/7567/when-can-one-localize-ext When can one localize Ext? Hailong Dao 2009-12-02T07:33:47Z 2009-12-03T09:38:38Z <p>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 $Ext^1_R(M,N)_P=0$? </p> <p>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. </p> http://mathoverflow.net/questions/7567/when-can-one-localize-ext/7651#7651 Answer by B. Cais for When can one localize Ext? B. Cais 2009-12-03T09:38:38Z 2009-12-03T09:38:38Z <p>I would suggest having a look at the article <a href="http://www.sciencedirect.com/science?%5Fob=ArticleURL&amp;%5Fudi=B6W9F-4CRY60R-1H3&amp;%5Fuser=2459438&amp;%5Frdoc=1&amp;%5Ffmt=&amp;%5Forig=search&amp;%5Fsort=d&amp;%5Fdocanchor=&amp;view=c&amp;%5Facct=C000057302&amp;%5Fversion=1&amp;%5FurlVersion=0&amp;%5Fuserid=2459438&amp;md5=deca64ad5b5501727ed918a10bc1ffd9" rel="nofollow">"Compactifying the Picard Scheme"</a> 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.</p>