Timeline for Construction of smooth projective space in Spectral Algebraic Geometry
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
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| Oct 15 at 21:58 | history | edited | Stahl | CC BY-SA 4.0 |
Added clarification that the functor of points description may be the dual of the classical description, depending on conventions
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| Oct 15 at 17:10 | comment | added | Stahl | As for the comment of quotients vs. subs, I'm aware of the distinction in convention (i.e., do we use $\operatorname{Sym}\mathcal{E}$ or $\operatorname{Sym}\mathcal{E}^{\vee}$ when defining $\mathbb{P}(\mathcal{E})$)-- good to mention just in case someone isn't familiar, of course. | |
| Oct 15 at 17:08 | comment | added | Stahl | @Z.M oh, of course! That makes a lot of sense -- then we just need to check that the functor $A\mapsto (\mathsf{Mod}_A)_{/A^{n+1}}$ preserves the subcategories described. I suppose it's also fine to use the slice category over $A^{n+1}$ here since the pullback/base change along $A\to B$ will send $A^{n+1}$ to $B^{n+1}.$ However, I don't see why we need point 3. | |
| Oct 15 at 10:17 | comment | added | Z. M | I think that it decomposes into the followings: 1. The association $A\mapsto(\operatorname{Mod}_A)_{/A^{n+1}}$ upgrades to a functor $\operatorname{CAlg}\to\operatorname{Cat}_\infty$; 2. Composing with the maximal groupoid $\operatorname{Cat}_\infty\to\operatorname{An}$; 3. Selecting connected components which satisfy these two properties. By the way, Lurie's definition does not seem to literally coincide with usual functor of points, since in usual description, one considers quotients of $A^{n+1}$ being projective of rank 1 (equivalent up to taking dual, but not the same). | |
| Oct 15 at 2:11 | history | asked | Stahl | CC BY-SA 4.0 |