Timeline for Tot and colimits
Current License: CC BY-SA 2.5
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 16, 2010 at 8:40 | vote | accept | Agustí Roig | ||
| Jun 16, 2010 at 8:36 | comment | added | Leonid Positselski | It is a standard fact (mentioned in Grothendieck's Tohoku paper) that in this case the object A must be zero. Basically, the isomorphism of the finite coproduct and the finite product in an additive category allows to add morphisms, and an isomorphism between the countable coproduct and product would allow to take countable sums of morphisms. In particular, there is a well-defined countable sum of copies of the identity endomorphism of A. This can be only non-contradictory when the identity endomorphism is zero. | |
| Jun 16, 2010 at 8:32 | comment | added | Leonid Positselski | Consider the set of diagrams D_n = (0->0->...->A->A->...) -- 0 on the first n positions and A on the subsequent ones, where A is a certain fixed object in our abelian category, and the maps between copies of A are the identity maps. If the countable filtered colimit commutes with the countable product for this set of diagrams, it simply means that the natural map from the coproduct of a countable set of copies of A to their product is an isomorphism. | |
| Jun 16, 2010 at 0:56 | comment | added | Agustí Roig | Sorry, but why are necessarily all the objects zero? | |
| Jun 15, 2010 at 16:37 | history | answered | Leonid Positselski | CC BY-SA 2.5 |