Timeline for On construction of Hilbert and Quot schemes
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16 events
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| Feb 20, 2022 at 9:50 | comment | added | Lao-tzu | To record, see also here mathoverflow.net/questions/44516/… for (1) on locally noetherian assumption. | |
| Feb 17, 2022 at 13:54 | comment | added | Lao-tzu | ...to confirm that the Yoneda embedding followed by sheafification is fully faithful. | |
| Feb 17, 2022 at 12:30 | comment | added | Lao-tzu | Great, thanks Marc! I can be sure (2) is not a problem for me now: we first sheafify the coproduct decomposition of functor in that book to a coproduct decomposition in the category of sheaves for the topology whose covers are given by coproduct decompositions (and use that representable functors are still sheaves). | |
| Feb 17, 2022 at 12:23 | comment | added | Marc Hoyois | Yes, exactly. This sheafification does not change the values on connected schemes, so it's also clear what it does on disjoint unions of connected schemes (e.g. locally noetherian schemes). | |
| Feb 17, 2022 at 9:25 | comment | added | Lao-tzu | @Marc Hoyois Thanks! To "make the latter (presheaf) product-preserving", can I understand as the sheafification of the presheaf $Q$ in question w.r.t. the topology whose covers (of an object X) are given by coproduct decompositions $\coprod_i U_i\cong X$? | |
| Feb 17, 2022 at 8:09 | comment | added | Marc Hoyois | By the way it is well-known that the noetherian assumptions in Grothendieck's presentation of the Quot scheme are not essential, but I do not know a reference. It might be in the Stacks project. | |
| Feb 17, 2022 at 8:01 | comment | added | Marc Hoyois | Indeed, and correspondingly the value of the functor Quot on a non-connected scheme is not simply the coproduct of the values of the functors Quot^Phi, one has to make the latter product-preserving. | |
| Feb 16, 2022 at 8:24 | history | edited | Lao-tzu | CC BY-SA 4.0 |
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| Feb 15, 2022 at 21:56 | history | edited | Lao-tzu | CC BY-SA 4.0 |
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| Feb 15, 2022 at 21:50 | history | edited | Lao-tzu | CC BY-SA 4.0 |
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| Feb 15, 2022 at 20:26 | comment | added | Lao-tzu | @Marc Hoyois Moreover, I think in the category of product-preserving presheaves, the value of the coproduct of (say two) product-preserving presheaves on an object is not simply the coproduct (i.e. disjoint union) of the values of these presheaves. | |
| Feb 15, 2022 at 20:09 | comment | added | Lao-tzu | @ Marc Hoyois Thanks Marc! By "product-preserving presheaves", do you mean those presheaves taking (arbitrary) coproducts to products? (Namely, you are using that products in the opposite category are coproducts in the original one?) | |
| Feb 15, 2022 at 19:28 | comment | added | Jason Starr | You can use limit theorems as in EGA III to reduce representability to the Noetherian case, cf. the comments to the following MO question: mathoverflow.net/questions/259423/… | |
| Feb 15, 2022 at 18:48 | comment | added | Marc Hoyois | The first coproduct decomposition does not actually take place in the category of presheaves. It takes place in the subcategory of product-preserving presheaves. The Yoneda embedding into that subcategory preserves coproducts. | |
| Feb 15, 2022 at 18:28 | history | edited | Lao-tzu | CC BY-SA 4.0 |
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| Feb 15, 2022 at 17:52 | history | asked | Lao-tzu | CC BY-SA 4.0 |