most recent 30 from http://mathoverflow.net 2013-05-18T20:23:05Z http://mathoverflow.net/feeds/user/4985 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/19678/regular-functions-on-affine-schemes Albert N 2010-03-29T00:58:30Z 2010-03-29T07:04:04Z

<p>I'm just learning the language of schemes, so I'm sorry if this seems a little elementary. Consider an affine scheme $\text{Spec}(R)$. For an ideal $I$ of $R$, denote by $U(I)$ the open subset of $\text{Spec}(R)$ consisting of prime ideals $p$ that do not contain $I$.</p> <p>Is the ring of regular functions on $U(I)$ simply $R_I$ (the localization of $R$ with respect to $I$)? If $I$ is a principal ideal, then this is one of the earliest results in Hartschorne. Also, it is easy to see that $R_I$ injects into the ring of regular functions on $U(I)$. My guess is that this injection is not surjective, but I can't seem to come up with any examples. Thanks!</p>