Timeline for Do $\infty$-categories make Grothendieck duality simpler?
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12 events
| when toggle format | what | by | license | comment | |
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| Oct 27, 2021 at 20:58 | comment | added | Exit path | @Gabriel I don't know most of the details, but my guess is that the formalism can be applied to any context in which one expects the six functors (see e.g. page $278$). In general, I don't think $QCoh$ is a subcategory of $IndCoh$, but there is a symmetric monoidal transformation of functors $QCoh \to IndCoh$. I believe this is an equivalence for nice enough (smooth?) schemes, as well as de Rham stacks. What they call $QCoh(X)$ may be thought of as the $DG$ enhancement of the derived category of quasicoherent sheaves on $X$ | |
| Oct 27, 2021 at 16:20 | comment | added | Gabriel | @leibnewtz that seems indeed very interesting. Do you know if this approach gives the desired information for QCoh, instead of IndCoh? (Is QCoh a "subcategory" of IndCoh, in some sense?) Also, just to confirm, what they denote by QCoh is actually D_qc, isn't it? | |
| Oct 27, 2021 at 15:04 | comment | added | Exit path | @Gabriel Let me direct your attention to page $273$ where they explain how the six functor formalism can be encoded in the data of a single functor. One can perhaps prove each property/compatibility in the language of derived or $1$-categories in a more elementary way, but I think being able to do it all in one step in the language of infinity categories is an advantage of the theory. | |
| Oct 27, 2021 at 11:41 | comment | added | Gabriel | @leibnewtz I surely don't understand well enough the book by Gaitsgory/Rozenblyum but it's not very clear to me that this approach simplifies the 3 parts. I mean... they spend most of the first book and even some of the second book to develop this theory. We can prove at least the first two parts with way less effort. (And I'm not sure that they prove part 2.) | |
| Oct 26, 2021 at 17:16 | comment | added | Exit path | @Gabriel If I understand correctly, it seems like the coherence conditions you are asking for just amount to the requirement that $X \mapsto D_{qc}(X)$ is functorial in $X$. In Gaitsgory/Rozenblyum they formulate the theory of ind-coherent sheaves as a functor out of the category of correspondences. This packages the six-functor formalism into a concise theory | |
| Oct 26, 2021 at 9:33 | comment | added | Gabriel | @DenisNardin some result similar to MacLane's coherence theorem, proving that all diagrams that should commute, commute. The preface to B. Conrad book "Grothendieck Duality and Base Change" explains exactly what I have in mind. This post is also in the same lines: mathoverflow.net/questions/404072/… | |
| Oct 26, 2021 at 8:54 | comment | added | Denis Nardin | What kind of coherence result do you mean in (3)? | |
| Oct 26, 2021 at 8:37 | comment | added | Gabriel | I took a look in Gaitsgory/Rozenblyum's work, and it seems very interesting indeed. I don't know if we gain anything in the context of schemes, though. We already have a functor $f^!$ for any separated map of finite-type. (Actually $\mathsf{R}f_*$ has a right adjoint in complete generality but, in some sense, this only gives the desired functor for proper $f$. So we define $f^!$ in general using a compactification.) | |
| Oct 26, 2021 at 8:31 | comment | added | Gabriel | @leibnewtz well... $X\mapsto \mathsf{D}_{\text{qc}}(X)$ forms a sheaf of stable $\infty$-categories. So this approach should work. But this appears to only solve part 1, which is already very simple (ever since Neeman's 1996 paper introducing Brown representability on triangulated categories) and works perfectly fine on triangulated categories! | |
| Oct 25, 2021 at 19:34 | comment | added | Exit path | Actually, you may be interested in the theory of ind-coherent sheaves developed by Gaitsgory/Rozenblyum. Part of the motivation there is to construct an exceptional inverse image functor for arbitrary morphisms | |
| Oct 25, 2021 at 19:24 | comment | added | Exit path | Just a guess: could it be that we’d like to construct $f^!$ first on some (etale) cover and then glue? If the the assignment $X \mapsto QCoh(X)$ formed a sheaf of categories then this would be a valid approach | |
| Oct 25, 2021 at 18:29 | history | asked | Gabriel | CC BY-SA 4.0 |