So in the near future I want to write a bit more about Hilbert schemes in the stacks project. Now it feels a bit wrong to say “Hilbert space” for the… uh… Hilbert space of an algebraic space. My promethean colleague Davesh Maulik suggests using “Hilbert skeem” so that typographically at least we make the reader aware that the… uh… Hilbert skeem may not be a scheme. What do you think?
Hilbert sheaf? ‘Hilbert algebraic space’ seems too clunky.
I do not object at all to the innovative Hilbert Skeem, but personally I would just stick with the *Hilbert functor* and say things like “the Hilbert functor (of X/S) is an algebraic space when X->S is a separated algebraic space/stack etc.” When the Hilbert functor is *represented* by a scheme, we call that scheme the Hilbert scheme. I dislike saying–although I certainly have done so quite frequently–that a functor is represented by an algebraic space.
Hi David, I agree with everything you said!
Well, just to be pedantic, if X —> S is a proper finitely presented morphism of algebraic spaces then its own Hilbert functor is a functor on algebraic spaces and hence to say that it is represented by an algebraic space isn’t “wrong” in the way it is to say that a functor defined just on the category of schemes is represented by an algebraic space (rather than simply that to say that the functor literally is an algebraic space, which I assume is what David was getting at in the end of his comment, and with which I completely agree). Apart from this quibbling, I also agree with everything David wrote except that I am not too keen on the “skeem” idea (sorry, Davesh).
Instead of “promethean” I might have said “devilish”!
The problem with “skeem” is that in speech it sounds the same, so in a talk one would have to keep specifying.
The deeper problem is that “algebraic space” is too awkward. One encounters this awkwardness in many settings (e.g., “every group algebraic space is a group scheme”). Let’s rename them! How about “stage”?