Is every regular (excellent) scheme separated? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T08:54:30Z http://mathoverflow.net/feeds/question/63592 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/63592/is-every-regular-excellent-scheme-separated Is every regular (excellent) scheme separated? Mikhail Bondarko 2011-05-01T08:58:05Z 2011-05-01T09:36:49Z <p>Sorry for one more stupid AG question. I need schemes that are regular, excellent and separated. Are these three conditions independent?</p> http://mathoverflow.net/questions/63592/is-every-regular-excellent-scheme-separated/63594#63594 Answer by Kevin Ventullo for Is every regular (excellent) scheme separated? Kevin Ventullo 2011-05-01T09:36:49Z 2011-05-01T09:36:49Z <p><strong>-</strong> Separated, excellent, regular: Spec$(k)$.</p> <p><strong>-</strong> Separated, excellent, not regular: Spec$(k[\epsilon]/\epsilon^2)$.</p> <p><strong>-</strong> Separated, not excellent, regular: See <a href="http://en.wikipedia.org/wiki/Excellent_ring" rel="nofollow">http://en.wikipedia.org/wiki/Excellent_ring</a></p> <p><strong>-</strong> Separated, not excellent, not regular: Spec$(k[\epsilon_1,\epsilon_2,\ldots]/\langle\epsilon_1^2,\epsilon_2^2,\ldots\rangle$.</p> <p><strong>-</strong> Not separated, excellent, regular: Glue Spec$(\mathbb{Z})$ to itself along the complement of a closed point.</p> <p>To get the other three, take the disjoint union of the fifth example with any of the second, third, or fourth examples.</p>