6
$\begingroup$

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...

Improve this question
$\endgroup$

2 Answers 2

Reset to default
12
$\begingroup$

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.

Improve this answer
$\endgroup$
4
  • 4
    $\begingroup$ Agree, see, e.g. Serre's normality criterion EGA IV$_2$ (5.8.6). $\endgroup$
    – Leo Alonso
    Sep 27, 2011 at 15:36
  • 1
    $\begingroup$ @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? $\endgroup$
    – Georges Elencwajg
    Sep 27, 2011 at 20:17
  • $\begingroup$ @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.) $\endgroup$
    – Georges Elencwajg
    Sep 27, 2011 at 20:37
  • $\begingroup$ 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. $\endgroup$
    – Georges Elencwajg
    Sep 28, 2011 at 16:51
11
$\begingroup$

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.

Improve this answer
$\endgroup$
1
  • 3
    $\begingroup$ 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. $\endgroup$
    – Joël
    Sep 28, 2011 at 3:24

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.