Timeline for Rigidity of the category of schemes
Current License: CC BY-SA 2.5
22 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 1, 2011 at 19:33 | comment | added | Ben Webster♦ | Gil- Yes, look at the reputation graph under the user page and select a range including the life of the question. | |
| Mar 1, 2011 at 19:16 | comment | added | Gil Kalai | How do you know? Is there a way to tell how many upvotes and down votse are there? | |
| Mar 1, 2011 at 18:19 | comment | added | Harry Gindi | @Steven: 4 upvotes and 4 downvotes. | |
| Mar 1, 2011 at 16:52 | comment | added | Steven Gubkin | I wonder if this is a "reputation pump", as descibed on meta at some point. Just wondering how many upvotes this has... | |
| Mar 1, 2011 at 15:16 | comment | added | Harry Gindi | @Scott: Ah, but that's very different from a "categorical definition", as I'm sure you know. | |
| Mar 1, 2011 at 9:33 | comment | added | Martin Brandenburg | @Scott: 1+, this is exactly what I'm trying to say. | |
| Mar 1, 2011 at 8:12 | comment | added | S. Carnahan♦ | As far as I can tell, the problem is that Martin wants a characterization of open immersion and étale morphism that is invariant under autoequivalences of the category of schemes, and you have provided a characterization of open immersion that depends on the Zariski topos (and the corresponding geometry), which is not obviously invariant under such autoequivalences. You claim that we know how to construct everything ring-theoretically, but it is not obvious that affine schemes are taken to affine schemes under all autoequivalences of schemes (and in fact, if they were, he would be done). | |
| Feb 28, 2011 at 20:19 | comment | added | Harry Gindi | Anyway, since we actually do know how to construct everything ring-theoretically first (that's the point of a geometry!). I don't understand what is meant by the idea that associated Zariski-toposes of schemes are not preserved by the auto-equivalences of $Sch$ and why, in particular that is a problem. | |
| Feb 28, 2011 at 19:53 | comment | added | Harry Gindi | Anyway, a geometry is an essentially small finitely-complete, idempotent-complete category $\mathcal{G}$ with a distinguished wide subcategory $\mathcal{G}^{ad}$ that is closed under pullbacks along morphisms of $\mathcal{G}$, is closed under retracts, and given any $g\in \mathcal{G}^{ad}$ and any map $f\in \mathcal{G}$ composable with $g$, then $f$ is admissible if and only if $g\circ f$ is admissible. Further $\mathcal{G}^{ad}$ is equipped with a Grothendieck topology such that its covering sieves are stable under pullback by any morphism in $\mathcal{G}$. | |
| Feb 28, 2011 at 19:39 | comment | added | Harry Gindi | @YBL: This was my attempt to un-infinity-ize Lurie's approach from DAG 5, not Toen-Vezzosi's (from HAG II), which was un-infinity-ized in Toen's notes for the Master course on stacks he did a few years ago. Toen-Vezzosi's approach is somehow substantially more complicated and involves a recursion procedure that I think is somewhat problematic in terms of how "canonical" and "categorical" it is. | |
| Feb 28, 2011 at 19:26 | comment | added | Qfwfq | I agree with YBL: +1 | |
| Feb 28, 2011 at 19:17 | comment | added | AFK | Some people seem to have skipped the first sentence: "This is a just a comment". As such it is very interesting (even though it could have quickly recalled what Toen means by "a geometry"). It doesn't bother me that it doesn't solve the problem (because we don't know how to construct the Zariski topoi without defining open immersions in ring theoretic terms first. The down votes seem unecesssary. | |
| Feb 28, 2011 at 16:34 | comment | added | Martin Brandenburg | @Saul: Yes, actually my problem is equivalent to this one. | |
| Feb 28, 2011 at 16:19 | comment | added | Saul Glasman | I guess the problem is that the Zariski topoi may not be preserved by automorphisms of the category of schemes? | |
| Feb 28, 2011 at 16:15 | comment | added | Steven Gubkin | Not sure why this is getting down votes... | |
| Feb 28, 2011 at 16:14 | comment | added | Harry Gindi | @Martin: Are you sure about that? | |
| Feb 28, 2011 at 15:49 | comment | added | Martin Brandenburg | Yes but this char. is not preserved by equivalences of the category of schemes. | |
| Feb 28, 2011 at 15:42 | comment | added | Harry Gindi | @Martin: I disagree. You were asking about categorical ways to characterize open and closed embeddings of schemes. If you don't want to hear about such ways, then don't ask such questions. You said in your question that if you had a way to characterize open immersions, you would be done. I just gave you an alternative characterization. | |
| Feb 28, 2011 at 15:09 | comment | added | Martin Brandenburg | @Harry: This is offtopic and just again showing off your knowledge about topos theory etc. ... please read the question more carefully. Your answer is not connected with my question. | |
| Feb 28, 2011 at 14:33 | comment | added | Harry Gindi | The part about a categorical definition of an open morphism or a proper morphism between schemes! | |
| Feb 28, 2011 at 14:25 | comment | added | Mariano Suárez-Álvarez | I cannot tell what is the relation between this and Martin's question! | |
| Feb 28, 2011 at 14:19 | history | answered | Harry Gindi | CC BY-SA 2.5 |