I noticed that many results in positive characteristic assumes that the object of the theorem is *excellent*. I have looked up the definition of excellent and have tried to get a feeling for it, but all I could really surmise was that it is nearly harmless to assume that a scheme is excellent as most schemes one naturally encounters are excellent. This is OK and I am happy to assume that every scheme I ever work with is excellent, but I still wonder (and here comes the first question):


**Question 1:** In what situations do we really need excellent schemes?


I discovered that excellence implies for example that the singular set is a closed set. For what else is it important?


I also wonder what other assumptions imply excellence. For example 


**Question 2:** Are there some other assumptions that imply excellence? For instance, "noetherian", "regular", "Cohen-Macaulay", etc.



Excellent schemes
An excellent reference for questions as yours is Grothendieck's EGA IV.7.8.


