5
$\begingroup$

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.

Improve this question
$\endgroup$
7
  • 6
    $\begingroup$ Dear R3D3, have you read EGA IV.7.8 yet? $\endgroup$
    – Fred Rohrer
    Commented Jan 17, 2015 at 7:25
  • 4
    $\begingroup$ There are non-excellent dvr's in positive characteristic (though every complete local noetherian ring and Dedekind domain of char. 0 is excellent). It is unwise to "assume that every scheme I ever work with is excellent". It is better to understand how to reduce problems to the excellent case (or when excellence is truly a necessary assumption), and more importantly from the proofs of many results you have read you can learn the answer to Question 1 (e.g., passage to various kinds of completion is well-behaved for properties like reducedness, normality, etc. under excellence assumptions). $\endgroup$
    – user74230
    Commented Jan 17, 2015 at 14:02
  • $\begingroup$ @FredRohrer: Thanks for that suggestion. If you would add it as an answer, I am happy to accept it. $\endgroup$
    – R3D3
    Commented Jan 21, 2015 at 7:14
  • $\begingroup$ @user74230: Thank you, but this isn't really helpful. You are saying I should do what I am asking help to do. I rarely work with situations where this matters and I asked the question so I could get some guidance in where to look. Suggesting to look at many results is not helpful at all. $\endgroup$
    – R3D3
    Commented Jan 21, 2015 at 7:18
  • $\begingroup$ @R3D3: You say in the question that you have "noticed that many results" assume excellence, so I assumed you have a big supply of results using excellence that you already care about. The only way I ever grokked this topic was to see how it is used in proofs of things I already cared about. So my advice is not flippant and is based on exactly what I did myself. Pick at random among the "many" results you say you have noticed, and as you read the proofs using excellence you will understand what its purpose is (especially to address Question 1). $\endgroup$
    – user74230
    Commented Jan 21, 2015 at 8:40

1 Answer 1

Reset to default
1
$\begingroup$

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

Improve this answer
$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .