Let f : X —> B be a morphism of algebraic spaces. Let u : F —> G be a map of quasi-coherent O_X-modules. Consider the functor<\/p>\n
\nF<\/strong> : (Sch\/B) —> (Sets), T |—> singleton if u_T is zero and empty else\n<\/p><\/blockquote>\n
This functor always satisfies the sheaf property for the fpqc topology (Lemma 083H<\/a>). It turns out that if f is locally of finite presentation, G is locally of finite presentation, G is flat over B, and the support of G is proper over B, then F<\/strong> is an algebraic space and F<\/strong> —> B is a closed immersion. This is Lemma 083M<\/a> and the proof uses the Raynaud-Gruson techniques.<\/p>\n
Challenge:<\/strong> Give a simple proof of Lemma 083M.<\/p>\n
A while back I tried to do this. First, some reductions: you can reduce to the case where B is an affine scheme. You can reduce to the case where f is proper and locally of finite presentation (replace F by image of F in G and replace X by suitable closed subspace supporting G). I think you can also reduce to the case where F is of finite presentation (by a limit argument). Hence, if you like, you can reduce to the case where B is the spectrum of a Noetherian ring and everything is of finite type.<\/p>\n
In the case B is affine there is a simple argument that shows: if u_T = 0 for some quasi-compact T over B, then there is a closed subscheme Z ⊂ B such that u_Z = 0 and such that T —> B factors through Z (Lemma 083K<\/a>). The proof only uses that G is flat over B.<\/p>\n
The problem left over is somehow: What if we have infinitely many closed subschemes Z_1, Z_2, Z_3,… ⊂ B such that u_{Z_i} is zero. Why is it true that u_Z = 0 where Z is the scheme theoretic closure of ⋃ Z_n? E.g., what if B = Spec(Q[x, y]) and Z_n is cut out by the ideal (x^n, y – 1 – x – x^2\/2 – x^3\/6 – … – x^n\/n!).<\/p>\n
If F is globally generated then you can reduce to the case F = O_X and you can use that Rf_*G is (universally) computed by a perfect complex. This is related to Jack Hall’s paper “Coherence results for algebraic stacks”. Note that Lemma 083M is a consequence of the results there. Jack’s paper uses relative duality which we do not have available in the Stacks project.<\/p>\n
If f is projective, you can reduce to the case F = O_X and G such that Rf_*G is universally computed by a finite locally free sheaf, whence the result. This case is straightforward using only standard results.<\/p>\n
If the support of G is finite over B then the result is elementary. So you could try to argue by induction on the relative dimension. Alas, I’m having trouble producing enough quotients of G which are flat over B.<\/p>\n
I still think something simple might work in general. But I don’t see it. Do you?<\/p>\n","protected":false},"excerpt":{"rendered":"
Let f : X —> B be a morphism of algebraic spaces. Let u : F —> G be a map of quasi-coherent O_X-modules. Consider the functor F : (Sch\/B) —> (Sets), T |—> singleton if u_T is zero and … Continue reading