Timeline for The minimal complexes and the direct limit
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 12, 2021 at 4:13 | comment | added | Aurora | If someone reads your counterexample he/she would understand what was the original question. I didn't delete it! I modified it! | |
| Jun 12, 2021 at 4:09 | comment | added | user1092847 | I think so. BTW, its considered bad manners to delete your original question after it is answered--people reading it later will get confused. You should indicate the edits/additional questions you make, by saying something like Edit 1, Edit 2, etc. | |
| Jun 12, 2021 at 4:04 | comment | added | Aurora | Yes, definitely that's the obstruction. Is the new one also easy? | |
| Jun 12, 2021 at 4:02 | comment | added | user1092847 | I'm not sure what a good restriction would be. One philosophical comment is that this counter-example is related to the fact that minimal iff $R/\mathfrak{m} \otimes_R F$ has zero differential breaks down when Nakayama's lemma fails. E.g. $k(x) \otimes_R R/(x) = 0$ even though $k(x) \neq 0$. | |
| Jun 12, 2021 at 3:47 | comment | added | Aurora | Yes, you are right again. What I am looking for is to put good restrictions to exclude simple counter-examples. Of course I have examples of such direct limits in mind which preserve the minimal property. | |
| Jun 12, 2021 at 3:38 | comment | added | user1092847 | @Aurora You can modify this example to answer your modified question. | |
| Jun 12, 2021 at 3:17 | comment | added | Aurora | Nice! That's correct, thanks. In this counterexample the direct limit morphisms induce zero map on the $0$-th homologies. So I change the question a lit bit to exclude this case. | |
| Jun 12, 2021 at 3:04 | history | answered | user1092847 | CC BY-SA 4.0 |