Timeline for Is there a sheaf of categories $\text{QCoh}_X(1)$ analogous to $\mathcal{O}_X(1)$?
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| Jul 28, 2023 at 5:12 | comment | added | Marc Hoyois | I think these results are originally due to Toën and Antieau-Gepner, but a convenient reference is section 11.5 in Spectral Algebraic Geometry. There is a bijection between invertible quasi-coherent sheaves of $\infty$-categories on a qcqs scheme $X$ and (derived) Azumaya algebras modulo Morita equivalence. You may also be interested in Remark 11.5.5.5: if $X$ is normal and noetherian, these are further equivalent to $B\mathbb G_m$-torsors. | |
| Jul 28, 2023 at 5:09 | comment | added | David Ben-Zvi | I don’t think there’s a canonical gerbe associated to a divisor (which I think is what your question is asking for), rather there are many - the root stacks (gerbe of n-th roots of O(D) for given n, and the corresponding n-torsion invertible sheaf of categories), cf eg papers of Talpo and Vistoli | |
| Jul 27, 2023 at 22:03 | comment | added | მამუკა ჯიბლაძე | Concerning $\operatorname{QCoh}_\eta$ - maybe my question about meromorphic bundles is related? There is an answer there... | |
| Jul 27, 2023 at 20:52 | comment | added | Pulcinella | @MarcHoyois Thanks - if you happen to know a reference for that I'd be happy to type up an answer here. | |
| Jul 27, 2023 at 20:51 | comment | added | Pulcinella | @TimCampion (3/4) Understanding what $\text{QCoh}_X(1)$ is in the affine case should be the hard part of the question ($\text{QCoh}_X(-1)$ should presumably be the sheaves vanishing along $D$). (1) I have no idea how to generalise (G_m bundles <-> invertible sheaves) to (BG_m gerbes <-> "invertible sheaves of categories") but if you know how I'd love to hear! | |
| Jul 27, 2023 at 20:43 | history | edited | Pulcinella | CC BY-SA 4.0 |
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| Jul 27, 2023 at 18:34 | comment | added | Marc Hoyois | I'd say the analogue is modules over Azumaya algebras (which are the invertible objects among quasi-coherent sheaves of categories). | |
| Jul 27, 2023 at 17:49 | comment | added | Tim Campion | Let me check if I understand a few things right: (1) You want to complete the analogy Picard group : Divisors :: Brauer group : (something), right? (2) $\eta$ denotes the localization of $X$ away from the generic point and $j : \eta \to X$ is the inclusion, right? Some suggestions: (3) Is it clear what should happen in the affine case? (4) I might suggest editing the question title to make the question clearer. | |
| Jul 27, 2023 at 14:35 | history | edited | Pulcinella | CC BY-SA 4.0 |
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| Jul 27, 2023 at 14:25 | history | edited | Pulcinella | CC BY-SA 4.0 |
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| Jul 27, 2023 at 12:31 | history | asked | Pulcinella | CC BY-SA 4.0 |