Well, my claims in this post were a little premature and may not be correct after all. It turns out that an argument in the lemma on etale localization of groupoids was wrong and has to be fixed. It doesn’t matter much for the overall picture since I can certainly use the arguments in the paper of Keel and Mori to fix it, but I wanted to prove it using only “etale localization” which may be hard.
The problem is the following: Suppose we have a groupoid scheme (U, R, s, t, c) with s, t flat, locally of finite presentation, and locally quasi-finite. Then we want to find many etale morphisms U’ –> U such that the restriction R’ of R to U’ is a disjoint union R’ = P \coprod Rest, with P a subgroupoid which is finite over U’ under both s’ and t’. The technique that I used (wrongly) in the current version of proof of the lemma mentioned above does prove that when we take U’ to be the spectrum of the henselization of the local ring of U at a point of U. But this doesn’t give you an etale morphism U’ —> U! What I am trying to see is if there is a kind of limit argument to descend P from the henselization to a smaller ring…
Let me just state here for the record that I think this means we can use this version of the lemma (with the henselization I mean) to define the (strictly) henselian local ring of the coarse moduli space without knowing that the coarse moduli space exists! Namely, since s, t : P —> Spec(O_{U,u}^h) are finite, flat and locally of finite presentation, we obtain that P is affine, and the “usual” arguments show that O_{U,u}^h is integral over the subring C of P-invariant elements of O_{U, u}^h. Presumably C (or its strict henselization) is what we are looking for. I haven’t thought this through completely, however.
I’ll post more here when I figure out how to repair the lemma.
I think that a non-flat version of the lemma could be interesting but I do not know if it is reasonable. An application in mind, requires the following conjecture:
Conjecture: Given an finite equivalence relation/groupoid R=>U in affine schemes, the spectrum of the invariant subring of O_U is a geometric quotient.
[As far as I know, this is very much an open question. Kollár has shown the conjecture in characteristic p for finite equivalence relations and under some strong semi-normality conditions in char. 0.]
Now, assume that the conjecture is true, then I think that it would follow from the lemma (if correct!) that quasi-finite equivalence relations/groupoids of arbitrary schemes have a geometric quotient.
Actually, the more I think about it, the more I am thinking: “What was I thinking?”. I mean the assumptions of the current version of Groupoids, Lemma Tag 03FM (see this link) are outrageously weak. Certainly at least some quasi-compactness has to be added and even in the case where everybody is of finite type over a field the thing isn’t clear. Soon I am going to replace it by the version with henselizations whilst adding some remarks on what more could be true. See also my post on the blog.
Lemma 03FM is actually correct! But you have to allow *non-separated* etale neighborhoods and this breaks the algorithm in Keel-Mori (we need the etale neighborhood to be affine or at least quasi-affine). Indeed, we have the following nice description of the Hilbert scheme trick in terms of etale sheaves:
Let f:X->Y be locally of finite type and separated. Consider Y’ = f_! (Z/2Z)_X. This is an etale sheaf on Y, i.e., an etale morphism Y’->Y (if f is of finite presentation, then Y’->Y is of finite presentation). Moreover, the inverse image sheaf X’=f^{-1}Y’ has a canonical open and closed subscheme X”, namely the support of the universal section. By the definition of f_!, we have that X”->Y’ is finite. We thus obtain a canonical way of making f finite etale-locally on X and Y.
Note that we do not need any flatness assumptions, however, if f is flat then X’=f_! (Z/2Z)_X is exactly the Hilbert-scheme of open and closed subschemes. Unfortunately, unless f is *flat*, X’->X is not necessarily separated.
In the above I am envision working on the stack level, i.e., my f is the quotient morphism U->X, but it is easy to lift this to the groupoid level as everything is canonical.
It is even possible to define f_! for non-separated morphisms (for etale sheaves of abelian groups, not for pointed sets!). But then there does not appear to be a canonical substitute for X” (the “support” is a space which is etale over X but only well-defined up to localizing with respect to trace maps).
Sorry for the bad typesetting. Is there any place where one can get information on valid html/mathml to insert in these blogs? (It also appears impossible to edit comments)
Do you mean by f_! sections with support finite over the base? Because if X = P^1_Y and f_! is the “usual” push forward with proper support then Y’ = f_!(Z/2Z)_X = (Z/2Z)_Y and X’ = (Z/2Z)_X and X” = X is the section 1. Hence in this case X” —> Y’ is proper but not finite. I understand the main case is when f is (locally) quasi-separated where the two notions agree. Just wanted to clarify what you were saying (hopefully correctly). In any case I’ll work out your suggestion a bit and see if I can add it to the stacks project without too much extra work. Thanks!
I of course forgot the assumption that f is locally quasi-finite (which supposedly is what you meant?). Without this assumption we do not get anything interesting with the (non-derived) f_!. For example, if f:X->Y is A^1->Spec(k) then Y’=Y and X’=\emptyset. Keel-Mori’s Hilbert scheme of clopen subschemes does agree with f_!Z/2Z for separated non-quasi-finite f but is as uninteresting.
For non-flat/open f, one get as I said Y’->Y non-separated which makes things difficult but at least we can reduce the question of existence of quotients of quasi-finite groupoids to that of finite groupoids (assuming the stabilizer is finite) where the space of objects is an algebraic space which is etale and non-separated over an affine scheme (these are not horribly difficult objects although perhaps inaccessible for commutative algebra methods).
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I could also add that it is enough that f is *universally open* to deduce that Y’->Y is separated.
Typo in previous comment, 3rd paragraph: Should be “Y’->Y separated” and not the weaker “X’->X separated”.