Schemes with isomorphic stalks - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T19:41:07Z http://mathoverflow.net/feeds/question/110550 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/110550/schemes-with-isomorphic-stalks Schemes with isomorphic stalks Daniel Barter 2012-10-24T15:49:11Z 2012-10-24T15:49:11Z <blockquote> <blockquote> <p><strong>Fact:</strong> If $X$ and $Y$ are varieties and we have <code>$\mathcal{O}_{X,q} \cong \mathcal{O}_{Y,q}$</code> then there are neighborhoods $U$ of $p$ and $V$ of $q$ which are isomorphic. </p> </blockquote> </blockquote> <p>I understand the intuition behind this result: Since the open sets are so big, the scheme $$\varprojlim_{p \in U\subseteq X \; \text{open}} U$$ captures a lot of infomation about $X$ at the point $p$. </p> <p>I can imagine a proof which goes like this: We can assume without loss of generality that the varieties in question are affine. Therefore we are reduced to proving that an isomorphism $R_{\mathfrak{p}} \to S_{\mathfrak{q}}$ of rings induces an isomorphism $R_f \to S_g$ for suitable elements $f \notin \mathfrak{p}$ and $g \notin \mathfrak{q}$. </p> <p>This approach is a bit disappointing in my opinion because the result is so geometric and we reduce it to a formal proof about localization.</p> <blockquote> <blockquote> <p><strong>Question:</strong> Can anyone explain a geometric proof of this result to me?</p> </blockquote> </blockquote>