What is an excellent algebraic space?

Question

What does it mean to say that an algebraic space $S$ is excellent? One knows that excellence of a Noetherian ring is not a property that is etale local (that is, excellence cannot be checked over an etale cover; there are counterexamples in EGA). Thus I wonder: what does excellence actually mean in the context of algebraic spaces?

The question comes from looking at this paper http://imrn.oxfordjournals.org/content/2006/75273 of Max Lieblich. He uses the phrase "excellent algebraic space" five times in the paper without discussing its meaning (as far as I can tell), so I presume the notion is standard. I would be very grateful if someone could explain what it means.

Answer 1

Let $X$ be a Noetherian algebraic space.

We say $X$ is quasi-excellent if the following equivalent conditions hold: (1) for every scheme $U$ and etale morphism $U \to X$ the scheme $U$ is quasi-excellent, and (2) for some scheme $U$ and surjective etale morphism $U \to X$ the scheme $U$ is quasi-excellent.

We say $X$ is universally catenary if for every quasi-separated morphism $Y \to X$ locally of finite type, the topological space $|Y|$ of $Y$ is catenary. (Note that the space $|Y|$ is a sober locally Noetherian topological space.)

We say $X$ is excellent if it is quasi-excellent and universally catenary.

Discussion: The problem with this definition is that it is hard to check (so it is not clear that there exist excellent algebraic spaces, besides the ones we know about, namely algebraic spaces of finite type over an excellent base scheme). On the other hand, everywhere in Max's paper you can replace excellent by quasi-excellent. In fact, in order to use Artin's results on representability all you need is to work over a base where the local rings are G-rings. Moreover, to establish a stack is algebraic, you may work etale locally on the base (if I understand correctly, this is how the algebraic space you are talking about arises in his paper), hence you can always assume the base is a scheme.