Timeline for Morphism on schemes induced by continuous morphism of sites
Current License: CC BY-SA 3.0
13 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 11, 2014 at 14:57 | vote | accept | user46578 | ||
| Dec 9, 2014 at 9:25 | comment | added | David Carchedi | @ZhenLin: Ah ha! Mystery solved. | |
| Dec 9, 2014 at 8:52 | comment | added | Zhen Lin | @DavidCarchedi I am not familiar with this paper. I suppose you refer to Theorem 47? It appears to me that the structure sheaf is being carried around, albeit disguised as a geometric morphism to the base topos, which is the big étale topos (i.e. the classifying topos for strictly henselian rings). | |
| Dec 9, 2014 at 6:58 | comment | added | David Carchedi | @ZhenLin, how does this tie in with your counterexample? | |
| Dec 9, 2014 at 6:57 | comment | added | David Carchedi | @user46578: See this paper of Pronk: eudml.org/doc/90454. Apparently you do not have carry along the structure sheaf. | |
| Dec 9, 2014 at 6:53 | answer | added | David Carchedi | timeline score: 3 | |
| Dec 8, 2014 at 2:52 | comment | added | user74230 | @user46578: I think the question is not worth dwelling about until you have acquired a better understanding of etale morphisms and descent theory (prior to such knowledge it doesn't make sense to think about the etale topology), after which the affirmative answer is an easy exercise. You don't give a reason for posing the question in the first place; if it is idle curiosity then I recommend learning more about etale morphisms and descent. (A good reference is Chapter 2 and early parts of Chapter 6 of the book "Neron Models".) | |
| Dec 7, 2014 at 19:26 | comment | added | user46578 | @user74230: Thank you for the comment. This is very helpful. I do not know where to look for to get an affirmative answer to my question. The question is vague partly because of my ignorance and partly I am hoping that it would help me learn more about the different possible directions to look at. | |
| Dec 7, 2014 at 19:03 | comment | added | user74230 | @user46578: A topos is like a topological space, ignorant of a choice of structure sheaf, so your question is analogous to asking when a continuous map between smooth manifolds is infinitely differentiable. Perhaps what you mean to ask is whether morphisms of the associated ringed topoi are the "same" as scheme morphisms, subject to an appropriate "locality" condition (as for locally ringed spaces, but adjusted for the etale topology)? This is addressed affirmatively in SGA4. You have to pay attention to the structure sheaf. | |
| Dec 7, 2014 at 19:01 | comment | added | user74230 | @ZhenLin: There is no such morphism "over $X$"; in general there could of course be such morphisms of abstract schemes (but I agree that the question is ill-posed). | |
| Dec 7, 2014 at 18:50 | answer | added | anon | timeline score: 1 | |
| Dec 7, 2014 at 16:56 | comment | added | Zhen Lin | You will have to be more specific. For instance, let $X$ be $\operatorname{Spec} k$ and let $Y$ be $\operatorname{Spec} K$ for some algebraically closed transcendental extension of $k$. Then the corresponding small étale toposes are equivalent, but there is no morphism $X \to Y$. | |
| Dec 7, 2014 at 14:33 | history | asked | user46578 | CC BY-SA 3.0 |