In Qing Liu's book Algebraic geometry and arithmetic curves I came across several confusing definitions. Several times he defines a notion only for a subclass of schemes/morphisms but later he is never explicitly mentioning these extra conditions again. Here are some examples:

  • Let $X$ be a locally Noetherian scheme, and let $x \in X$ be a point. We say that $X$ is regular at $x$ if [...]. We say that $X$ is regular if it is regular at all of its points. Question: If he later says "Let $X$ be a regular scheme", then is it implicit that $X$ is locally Noetherian? If so, then why doesn't he say "A scheme is called regular if it is locally Noetherian and [...]"?

  • Let $X$ be a reduced Noetherian scheme. Let $\xi_1,\ldots,\xi_n$ be the generic points of $X$. We say that a morphism of finite type $f:Z \rightarrow X$ is a birational morphism if [...]. Question: If he later says that a morphism $f:Z \rightarrow X$ of (arbitrary) schemes is birational, then is it implicit that $X$ is reduced Noetherian and that $f$ is of finite type? If so, then why doesn't he say "A morphism $f$ is called birational if it is of finite type, if $X$ is reduced Noetherian and if [...]"?

  • Now it gets really confusing: Let $X$ be a reduced locally Noetherian scheme. A proper birational morphism $\pi:Z \rightarrow X$ with $Z$ regular is called a desingularization of $X$. Question: He defined birational only for reduced Noetherian schemes. What is birational for reduced locally Noetherian schemes? Is his desingularization now automatically of finite type?


  1. In Liu's book I found the following definition now: We say that a morphism $f:X \rightarrow Y$ is proper if it is of finite type, separated and universally closed. So, first of all, I think that this definition is now given in the non-confusing style, and second, this implies that the desingularizations above are of finite type (although it doesn't answer the locally Noetherian/Noetherian question).

  2. I was asking "...then why doesn't he say that..." because I wasn't sure (and I'm still not sure) if there is some "higher truth" in this style of definition. Of course nobody except for Liu himself can answer this but perhaps someone else has more experience than I have and can give an explanation for this...

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    $\begingroup$ As to "why" only the author can provide a definite answer. Although he seems to be participating in MO, it would appear to be less argumentative if you asked him directly. I personally have always thought that "Let X have property P. We say that X is Q if ..." means that in talking about X that are Q, we only consider X with P (seems a bit tautological). It's a great devise for not dragging along monstrous sequences of standard assumptions (kind of like saying "in this book, all rings are commutative and have an identity"). $\endgroup$ – Victor Protsak Jul 18 '10 at 13:18
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    $\begingroup$ As for 1), the explanation is as given by Victor. As far as I know, regularity is defined only for locally noetherian schemes, so regular scheme are supposed to be locally noetherian, as least in my book. So yes, maybe it is better so say a scheme is regular if it is locally noetherian and etc. For 2), the reason is different. I only consider birational finite type morphism over a reduced noetherian scheme. Contrarily to 1), as pointed out by Georges, it can be defined in a much more general setting. Here I can not say a morphism if birational if it is finite type etc. $\endgroup$ – Qing Liu Jul 18 '10 at 14:14
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    $\begingroup$ Continued: I think in the book, the word birational is never used out of this situation. For 3), yes I should just restrict to noetherian and not locally noetherian schemes. It is already so hard to desingularize noetherian schemes :). Yes by definition (EGA IV), desingularization morphism are proper, so in particular are of finite type. $\endgroup$ – Qing Liu Jul 18 '10 at 14:17
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    $\begingroup$ Arminius's questions are of course completely legitimate and interesting, but I'd like to emphasize that Qing's book is very well-written and contains an amazing wealth of material. I bought it in 2003 but I am still very far, alas, from having mastered it... $\endgroup$ – Georges Elencwajg Jul 18 '10 at 14:52
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    $\begingroup$ Dear Georges, merci pour les compliments ! Dear Arminus, there is no problem critizing my book, and you are wellcome to do it. As I am keep writing an errata, it will help me to improve the book. $\endgroup$ – Qing Liu Jul 18 '10 at 22:26

Dear Arminius, I'm certainly not going to answer your questions "why doesn't he say...?": Qing is a frequent and friendly contributor to MO and he will answer himself if he wants to.

Here is what I think is the consensus about your questions.

1) For a scheme regular definitely implies locally noetherian: De Jong 19.8.2

2) Birational necessitates neither noetherian nor reducedness conditions on schemes nor finite type assumptions on morphisms: De Jong 20.7.1

3) Qing's definition now makes perfectly good sense in view of 1) and 2). Desingularization is automatically of finite type because a proper morphism is of finite type by definition : De Jong 20.36.1

Bibliographical note I didn't want to give a long list of references for the definitions you ask about. I have only quoted De Jong and collaborators' monumental Stacks Project which is the most up-to-date reference and which is incredibly well thought-out. Also De Jong is arguably the mathematician who has made the greatest progress on the resolution of singularities for schemes since Hironaka in 1964 .

  • $\begingroup$ Thank you for the link to Stacks Project! Another 2054 pages to digest (on the plus side: in English; on the minus side: Publ Math IHES are typographically superior). $\endgroup$ – Victor Protsak Jul 18 '10 at 14:05
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    $\begingroup$ Hurry up, Victor, the number of pages increases almost daily :-) I completely agree with you that the typography of the Publications de l'IHES is exceptionally beautiful: some modification of Baskerville maybe? [I mean John Baskerville, the type designer, not Conan Doyle's poor Charles, the target of a notorious hound] $\endgroup$ – Georges Elencwajg Jul 18 '10 at 14:27
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    $\begingroup$ I've been admiring the font of the Publications de l'IHES as well. It is indeed a variation of Baskerville: tex.stackexchange.com/questions/16027/… $\endgroup$ – Eivind Dahl Nov 9 '11 at 0:13

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