Is "normal scheme" defined in EGA? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T19:58:03Z http://mathoverflow.net/feeds/question/76498 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/76498/is-normal-scheme-defined-in-ega Is "normal scheme" defined in EGA? Georges Elencwajg 2011-09-27T11:51:55Z 2011-09-27T22:11:35Z <p>In answering a <a href="http://mathoverflow.net/questions/76349/is-there-a-connected-non-affine-scheme-s-such-that-it-is-the-union-of-rings-of/76365#76365" rel="nofollow">question</a> on this site yesterday, I was led to introduce a non-separated scheme, which I said was normal.<br> However when I tried to check the definition of "normal scheme" in EGA , in order to make sure that "normal" didn't have "separated" in its definition, I was surprised to find that I couldn't find the definition, although "normal prescheme" <em>is</em> used in many places (which seems to indicate that normal doesn't imply separated, else the authors would have said normal scheme: schemes were supposed to be separated in contradistinction to preschemes). </p> <p>Another ambiguity: it is not clear to me whether for Grothendieck-Dieudonné, "normal" only depends on the local rings of the scheme : do they consider that the spectrum of the product of two copies of a field $k$ is normal, even though $k\times k$ is not a domain, hence not an integrally closed domain ?<br> [My guess is "yes", but I don't want to guess] </p> <p>So, my question is:<strong>Do the EGA contain a definition of "normal (pre)scheme" ?</strong><br> I'm interested because I have never used a definition at odds with EGA, and I have no intention to start now...</p> http://mathoverflow.net/questions/76498/is-normal-scheme-defined-in-ega/76510#76510 Answer by Laurent Moret-Bailly for Is "normal scheme" defined in EGA? Laurent Moret-Bailly 2011-09-27T13:39:28Z 2011-09-27T13:39:28Z <p>Normal ringed spaces are defined in EGA $0_I$, (4.1.4), normal rings in $0_I$ (6.5.1) (with a strangely phrased comment which first led me to believe that the definition would be repeated for schemes in Chapter I, but it couldn't find this). In any case, a scheme is normal iff its local rings are integrally closed domains.</p> http://mathoverflow.net/questions/76498/is-normal-scheme-defined-in-ega/76571#76571 Answer by Leo Alonso for Is "normal scheme" defined in EGA? Leo Alonso 2011-09-27T22:11:35Z 2011-09-27T22:11:35Z <p>Normality for schemes (as opposed to general ringed spaces) seems not to be treated in EGA until $\S$5 in Chapter IV. It is discussed in the context of properties R$_n$ and S$_n$. Serre's normality criterion (Normal is equivalent to R$_1$ <em>and</em> S$_2$) is proved in EGA IV$_2$ (5.8.6). The definition is briefly recalled at the beginning of the proof.</p>