Timeline for Understanding sheaves on lisse-etale site of an algebraic stack
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 31, 2017 at 20:30 | comment | added | Marc Hoyois | @Sam I'm not sure what your confusion is. You seem to be asking why descent for a pretopology implies descent for the associated topology? The smooth and etale pretopologies on this category are different, but the existence of $Z$ tells you they generate the same topology. | |
| May 31, 2017 at 14:58 | comment | added | Sam | What I mean is when you evaluate the sequence on $V \rightarrow \mathcal{X}$, you have to deal with $V \times_{\mathcal{X}}U \rightarrow \mathcal{X}$. How does the existence of $Z$ helps? Also, if small smooth site is same as lisse-etale site, why is this difference in terminology? | |
| May 31, 2017 at 14:53 | history | edited | Sam | CC BY-SA 3.0 |
deleted 3 characters in body
|
| May 31, 2017 at 14:51 | comment | added | Sam | @MarcHoyois: I do understand that for the smooth surjection $V\times_{\mathcal{X}} U\xrightarrow{smooth} V$ there exists some scheme $Z$ such that there is a map from $Z \rightarrow V\times_{\mathcal{X}} U$ and an etale surjection $Z \rightarrow V$ such that the diagram commutes. But I don't see how that makes the sequence mentioned in my question exact! Could you explain a bit? | |
| May 31, 2017 at 8:02 | comment | added | David Roberts♦ | @MarcHoyois that's the kind of useful information that people never seem to say in public. Thanks! | |
| May 31, 2017 at 1:15 | comment | added | Marc Hoyois | Put another way, the small smooth site and the lisse-etale site of a stack are actually the same. | |
| May 31, 2017 at 1:13 | comment | added | Marc Hoyois | Smooth surjections between algebraic spaces are in fact covering for the etale topology, because they have sections after a surjective etale base change. | |
| May 30, 2017 at 20:20 | history | asked | Sam | CC BY-SA 3.0 |