Timeline for "Sums-compact" objects = f.g. objects in categories of modules?
Current License: CC BY-SA 2.5
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 30, 2014 at 11:20 | vote | accept | Sasha | ||
| Jun 6, 2021 at 10:42 | |||||
| Nov 22, 2011 at 22:53 | comment | added | Tom Leinster | ...Take a category in which monos, epis and coproducts behave "well"; I'm too lazy to figure out the exact hypotheses, but something like "regular category" should do. Take a connected object X and an epi e: X -> Y. I claim Y is connected. For take f: Y -> A+B. WLOG, fe factors through the inclusion A -> A+B, which I'll assume is mono. So we get a commutative square with an epi down the left-hand edge and a mono down the right. In the usual fashion, we can fill it in with a diagonal, i.e. a map Y -> A. This gives the required factorization. (I hope this can be made respectable.) | |
| Nov 22, 2011 at 22:49 | comment | added | Tom Leinster | As I said in my comment on the main question, the standard term for "sumpact" is "connected" (modulo the question of whether we're dealing with finite coproducts or all of them). So Fernando's result says: a quotient of a connected set is connected. That makes sense! And it's true much more generally than Fernando states it. Sketch proof follows in next comment... | |
| Mar 23, 2011 at 20:41 | comment | added | Fernando Muro | @Sasha: the relation between perfect and noetherian rings is not that close, but for instance, perfect + noetherian is the same as artinian, so my argument really applies to many more rings than just perfect rings. The fact that noetherianity is equivalent to the fact that direct sums of injectives is injetive is a very classical result. | |
| Mar 23, 2011 at 17:33 | comment | added | Sasha | @Martin: $g$ exists because the sum of $E_i$ is injective, hence we can extend a map to it. | |
| Mar 23, 2011 at 17:28 | comment | added | Martin Brandenburg | Why does $g$ exist? | |
| Mar 23, 2011 at 17:15 | comment | added | Sasha | Thank you! Those are very interesting comments. I have just two questions: 1) I don't know really what is perfect; you claim left perfect implies left Noetherian? 2) Is the fact that sum of injectives is injective for Noetherian ring a hard fact? Thanks. | |
| Mar 23, 2011 at 16:50 | history | answered | Fernando Muro | CC BY-SA 2.5 |