Yesterday, David Rydh in this comment found a wonderful approach to proving Lemma Tag 03FM. In fact I had already more or less given up hope that the lemma could be fixed! I am so exited about David’s idea that I am going to try to explain it in this post. Of course any mistakes are mine.

First we are going to discuss a completely general construction. Namely, suppose that f : X —> Y is a morphism of schemes which is separated and locally of finite type. On the small etale site Y_{etale} of Y consider the following presheaf

(*) V/Y |—> {Z \subset X_V open, Z —> V is finite}

Note that as we assumed X —> Y is separated such a Z is also closed in X_V. Also note that the empty set is closed and finite over any scheme. Finite morphisms satisfies descent for the etale topology. This implies that the presheaf above is a sheaf on Y_{etale}. Now, it is a general fact that any sheaf on Y_{etale} is representable by an algebraic space etale over Y. Hence we obtain an algebraic space, which I am going to denote f_!X which comes equipped with an etale morphism f_!X —> Y and we get a universal open and closed subspace Z_{univ} \subset f_!X \times_Y X which is finite over f_!X.

Before we say more about this construction let us point out that f_!X is *really* an algebraic space and not a scheme. For example suppose that X = Spec(C) and Y = Spec(R[t]) and the morphism f : X —> Y maps the unique point of X to (t = 0) inside Y. In that case f_!X is a copy of Y with {0} replaced by {0_1, 0_2} with residue fields R, respectively C (but of course f_!X is still etale over Y). The complement of {0_1} in f_!X is an open subspace which is a copy of Example Tag 03FN. And this is the simplest example of an algebraic space etale over a scheme which is not a scheme.

Now a very important property of the construction above is that it commutes with arbitrary base change. This is the analogue of the fact that cohomology with proper supports commutes with any base change. Namely, suppose that Y’ —> Y is any morphism of schemes, and denote f’ : X’ —> Y’ the base change of f. By universality of f’_!X’ we obtain a morphism f_!X \times_Y Y’ —> f’_!X’ over Y’. To prove this map is an isomorphism it suffices to show that the stalks of the sheaf (*) can be computed from the geometric fibres of the morphism X —> Y. And this is almost exactly the content of Lemma Tag 02LN.

As in Keel-Mori we are going to use a slight modification of f_! in the case that the morphism f : X —> Y comes equipped with a section e : Y —> X. Namely, in that case we consider the sub presheaf of (*) consisting of those Z \subset X_V with e(V) contained in Z. In the same manner as above you show that this is representable by an algebraic space fe_!X etale over Y whose formation commutes with base change.

OK, so how do we apply this construction to a groupoid scheme (U, R, s, t, c, e, i) with s, t separated and locally of finite type? Well, consider U’ = se_!R and the universal closed sub space Z \subset R \times_{U, s} U’. The inverse map of the groupoid scheme shows that U’ is canonically isomorphic to te_!R which gives that i(Z) \subset U’ \times_{U, t} R is universal. Compatibility with base change and the fact that R \times_{s, U, t} R is isomorphic to R \times_{s, U, s} R over (the second) R shows that there is a canonical isomorphism U’ \times_{U, t} R = R \times_{s, U} U’ over R. Then you check by chasing some diagrams that this isomorphism maps the universal closed subscheme Z into i(Z). Hence if

R’ = U’ \times_{U, t} R \times_{s, U} U’

is the restriction of R to U’, then we conclude that Z is an open sub space of R’ which is finite under both s’ : R’ \to U’ and t’ : R’ \to U’. (Here we use that U’ \times_{U, t} R = R \times_{s, U} U’ can be identified with an open in R’ as these spaces are etale over R.) Because we have e’ inside of Z by construction some more diagram chasing shows that Z is a sub groupoid space! Awesome.

Of course it is possible that U’ is empty. But, as in the statement of Lemma Tag 03FM if we have a point u of U where the set of points {r_1, …, r_n} of R mapping to u under both s and t is finite, and if s, t are quasi-finite at each of those points, then we get a canonical point u’ of U’ whose residue field is the same as the residue field of u, and which corresponds to an open and closed sub scheme of the fibre of R —> U whose points are exactly the set {r_1, …, r_n}.

OK, so this utterly general construction allows us to find an open subgroupoid which is a finite groupoid, at the cost of replacing U by an algebraic space etale over U. And, as pointed out by David Rydh in his further comments we can then use further properties of the original groupoid (U, R, s, t, c, e, i) to prove that U’ has additional properties. For example, in case the morphisms s, t are also of finite presentation and flat, then (I think) U’ agrees with an open of the Hilbert scheme used by Keel and Mori and it is a quasi-projective scheme over U. Wonderful.

I am curious why you insist on defining f_! as open closed subschemes with FINITE support and not PROPER support (so that f_!X = f_! (Z/2Z)_X is the usual zeroth part of cohomology with proper support). Of course, it doesn’t matter when f is quasi-finite which is the only really interesting case for this construction anyway.

Another curious fact (unless I am mistaken): If f:X->Y is flat (and perhaps we need quasi-finite), then f_! Z_X (a non-constructible etale sheaf) is the coarse moduli space of the Hilbert stack H of proper flat families mapping etale to X. Note that H->Y is etale so that the coarse moduli space of H is simply the sheafification of \pi_0 H. Also, f_! Z_X (resp. H) is an abelian group (resp. abelian group stack) and this structure is preserved. This also works if f is non-separated and gives a construction of f_!Z in this case.

I really intended to use finite support instead of proper support in the post, because it gives me exactly what I want without changing the hypotheses of Lemma Tag 03FM. However, I should have used another symbol (not f_!X) to denote the algebraic space representing the sheaf (*). Sorry for the confusion.

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