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…
Best Answer
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.