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?


Edit:


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...



Confusing definitions in Liu's Algebraic geometry and arithmetic curves?
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](http://math.columbia.edu/algebraic_geometry/stacks-git/) 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 .


