Timeline for Describe a topic in one sentence.
Current License: CC BY-SA 2.5
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 8, 2012 at 22:12 | comment | added | Greg Muller | In an abelian category, a chain complex can be made into a cochain complex by flipping the sign of the indices. So, there's no concrete difference between homology groups and cohomology groups. (Though, in specific examples, usually one numbering is better and so a group feels more homological or cohomological) | |
| Oct 8, 2012 at 15:32 | comment | added | Camilo Sarmiento | What would be the difference measured by cohomology groups? | |
| Feb 28, 2011 at 2:05 | comment | added | Greg Muller | For example, once I was comparing $\overline{I\cap J}$ to $\overline{I}\cap \overline{J}$, where the bar denotes taking the associated graded module with respect to some filtration of $R$-ideals $I$ and $J$. I suspected that there was some homology group which vanished exactly when those coincided, and I was correct (it was a rather complicated $Tor$). | |
| Feb 28, 2011 at 2:03 | comment | added | Greg Muller | Those are certainly the examples that give the most classical derived functors, but I think this platitude is more helpful when applied to a question which isn't obviously measured by a homology group. | |
| Feb 28, 2011 at 0:26 | comment | added | J.C. Ottem | @Colin: One wants certain functors to be exact, e.g., the Hom-functor gives Exts, tensoring with a module gives Tor. | |
| Feb 27, 2011 at 13:11 | comment | added | user2529 | May I ask, what is it that one wishes to be true (in an abelian category)? | |
| Oct 24, 2009 at 14:40 | history | answered | Greg Muller | CC BY-SA 2.5 |