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In answering a question on this site yesterday, I was led to introduce a non-separated scheme, which I said was normal.
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" is 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).

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 ?
[My guess is "yes", but I don't want to guess]

So, my question is:Do the EGA contain a definition of "normal (pre)scheme" ?
I'm interested because I have never used a definition at odds with EGA, and I have no intention to start now...

flag	asked Sep 27 2011 at 11:51 Georges Elencwajg 24.3k●65●132

Laurent Moret-Bailly · Answer 1 · 2011-09-27 13:39:28Z UTC

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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.

answered Sep 27 2011 at 13:39

Laurent Moret-Bailly
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Agree, see, e.g. Serre's normality criterion EGA IV$_2$ (5.8.6). – Leo Alonso Sep 27 2011 at 15:36

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@Leo Alonso: this is a very convincing reference. May I suggest that you transform your comment into an answer, so that I (and no doubt others) can upvote you? – Georges Elencwajg Sep 27 2011 at 20:17

@Laurent. Thank you very much: I wasn't aware of these very interesting and convincing references. (Readers: beware that Laurent's reference $0_I$ (6.5.1) is to the second edition of EGA I, the one edited by Springer .The first reference is common to both editions.) – Georges Elencwajg Sep 27 2011 at 20:37

Both you, Laurent, and Leo have given perfect answers to my question. Since, unfortunately, I can accept only one answer, I decided to encourage the more junior user. – Georges Elencwajg Sep 28 2011 at 16:51

Leo Alonso · Answer 2 · 2011-09-27 22:11:35Z UTC

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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$_1$ and S$_2$) is proved in EGA IV$_2$ (5.8.6). The definition is briefly recalled at the beginning of the proof.

answered Sep 27 2011 at 22:11

Leo Alonso
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It is astonishing how little is a role the notion of normality plays in Grothendieck's algebraic geometry, when we remember how central a notion it was for Zariski and his school, just one decade earlier. – Joël Sep 28 2011 at 3:24

Is “normal scheme” defined in EGA?

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