Reduced scheme and closed points - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T21:05:27Z http://mathoverflow.net/feeds/question/12109 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/12109/reduced-scheme-and-closed-points Reduced scheme and closed points brunoh 2010-01-17T16:12:09Z 2011-07-14T18:51:21Z <p>In <i>The Geometry of Schemes</i> by Eisenbud and Harris, Exercise I-32 asks one to show that a scheme $X$ is reduced if and only if every local ring $\mathcal{O}_{X,p}$ is reduced for closed points $p \in X$. However, this does not seem to work in general, since $X$ may not have enough closed points. What additional hypotheses on $X$ do I need for such an assertion to hold?</p> http://mathoverflow.net/questions/12109/reduced-scheme-and-closed-points/12110#12110 Answer by brunoh for Reduced scheme and closed points brunoh 2010-01-17T16:33:39Z 2010-01-17T16:33:39Z <p>It seems to me that looking at closed points only is not sufficient since they are not always a dense set of X ...</p> http://mathoverflow.net/questions/12109/reduced-scheme-and-closed-points/12112#12112 Answer by Wanderer for Reduced scheme and closed points Wanderer 2010-01-17T16:42:22Z 2010-01-17T21:56:23Z <p>There do exist schemes without a closed point, yes. (Liu, exercises 3.3.26/27) </p> <p>But under some very reasonable additional conditions - I think quasi-compactness will be sufficient, if you are happy with using Zorn's lemma - the result holds. Use/prove the existence of a closed point, and the fact that localizing a reduced ring still gives you a reduced ring. </p> http://mathoverflow.net/questions/12109/reduced-scheme-and-closed-points/61549#61549 Answer by Y Y Zhu for Reduced scheme and closed points Y Y Zhu 2011-04-13T14:47:34Z 2011-04-13T14:47:34Z <p>I don't think quasi-compactness is enough,for Noether scheme it is true. in a noether scheme, every point P has a closed point in its closure, so ..... but i don't find a necessary and sufficient condition </p>