Timeline for What exactly goes wrong with $f_!$ outside of locally compact spaces?
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| Aug 17, 2021 at 10:38 | comment | added | Dan Petersen | In the setting of étale constructible sheaves, Laszlo-Olsson define functors $Rf_!$ and $f^!$ for any finite type morphism between algebraic stacks locally of finite type. No separation hypotheses are needed. | |
| Aug 16, 2021 at 22:54 | comment | added | Tyrone | It is a property of the map $f$ which is needed, not a property of the spaces. Reasonably $f$ must be both separated and locally proper. Any mapping between locally compact Hausdorff spaces has these properties. | |
| Aug 16, 2021 at 21:30 | comment | added | Gabriel | Dear @Z.M, in his article Resolutions of Unbounded Complexes, N. Spaltenstein proves a version of it (supposing some extra condition). In mathoverflow.net/questions/400948/…, I proposed a possible proof of it without this condition but I'm not entirely sure if it's correct. | |
| Aug 16, 2021 at 21:11 | comment | added | Z. M | Could you please give a reference of the Verdier duality for ringed spaces (or topoi)? On the other hand, I don't see any obvious way to relate the solid lower shriek in Clausen-Scholze to the classical one when the map is not proper, since in this case the solid lower shriek is in general not discrete, so at least, it does not "coincide" with the classical one in the naive sense. | |
| Aug 16, 2021 at 20:17 | comment | added | Gabriel | @MarcHoyois What confuses me is that the definition of $f_!$ is SGA4 is the one for the étale topology and it doesn't coincide with the one I've written in the post. | |
| Aug 16, 2021 at 20:15 | comment | added | Gabriel | @Z.M Well... Verdier duality holds just as well for module sheaves. (I'm not sure about the étale case though.) I indeed looked at this lecture and Dustin Clausen's masterclass and this seems very interesting. Do you happen to know if the $f_!$ constructed using condensed mathematics coincides with the one I've defined for nonproper $f$? (Perhaps it is not the derived functor of anything?) | |
| Aug 16, 2021 at 16:34 | comment | added | Marc Hoyois | In short, the issue for schemes is that $R(p_*\circ j_!)\neq R(p_*)\circ j_!$, and it's the right-hand side we're interested in. | |
| Aug 16, 2021 at 16:32 | comment | added | Marc Hoyois | At the level of derived categories, we know what we want $f_!$ to be when $f$ is either proper or an open immersion, which forces the definition for compactifiable morphisms. Now in the proper case we have a right derived functor, and in the open immersion case we have a left derived functor, so we shouldn't expect the composite to be a derived functor in general. For nice enough topological spaces it turns out to still be the right derived functor of underived $f_!$, but this uses partitions of unity and does not work for schemes: see SGA4 XVII Contre-exemple 6.1.6. | |
| Aug 16, 2021 at 16:03 | comment | added | Z. M | Under what circumstances do you study the lower shriek of module sheaves? In the topological / étale case, one considers abelian sheaves. There is a version of lower shriek of "quasi-coherent sheaves", but it is a very recent work: Lecture XI of Lectures on Condensed Mathematics. | |
| Aug 16, 2021 at 12:28 | comment | added | Gabriel | A very practical problem that I see with the functor $f_!$ when dealing with schemes is that it doesn't preserve quasi-coherence (I think). But I wonder if there are other reasons and, in particular, what's the matter with $\mathsf{R}f_!$ that Deligne was hinting. | |
| Aug 16, 2021 at 12:26 | comment | added | Gabriel | What I mean is that a fiber product of schemes (for example) does not coincide with the fiber product of the underlying topological spaces. Also, I see the tag 09V5 on the Stacks Project as morally a special case of proper base change, and it holds for arbitrary proper separated morphisms on ringed spaces. | |
| Aug 16, 2021 at 11:43 | comment | added | Vivek Shende | Consider their Proposition 2.5.1, in which they do not assume local compactness. It is morally a special case of proper base change, e.g. imagine Z and neighborhoods of Z are compact; push forward along a distance function to Z. As they write, it may not be an isomorphism unless also the ambient space is Hausdorff; if you want proper base change in some other category this is one thing you have to correct. | |
| Aug 16, 2021 at 9:56 | comment | added | Gabriel | @VivekShende I did, in fact, read it. The chapter begins with the definition of $f_!$ without any supposition on the spaces and then the authors say that "Until the end of this section, all spaces will again be assumed to be locally compact spaces." This supposition is used in the proof of the proper base change theorem, but I don't see why the same result can't be true in other categories of ringed spaces. (Since fibered products are different in other categories.) | |
| Aug 16, 2021 at 9:16 | comment | added | Vivek Shende | You may find some ideas about this by looking in Chapter 2.5 in the book "Sheaves on Manifolds" by Kashiwara and Schapira. | |
| Aug 16, 2021 at 1:33 | history | edited | LSpice | CC BY-SA 4.0 |
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| Aug 15, 2021 at 20:29 | history | asked | Gabriel | CC BY-SA 4.0 |