Timeline for Finite generatation of Ext
Current License: CC BY-SA 2.5
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 7, 2010 at 19:39 | comment | added | Hailong Dao | I think the question has a lot of potential. As Robin and Karl pointed out, there are a lot of examples of complexes of "big" objects with "small" cohomology. It would be nice if someone can give some underlying reasons. | |
| Jun 7, 2010 at 19:20 | comment | added | Karl Schwede | Heck, a slight variant of the same example, if you have a projective variety $X$ over $\mathbb{C}$ and a coherent sheaf $F$ on $X$, then the cohomology $H^i(X, F)$ are finitely dimensional vector spaces! They are also computed using an injective resolution of $F$. | |
| Jun 7, 2010 at 15:57 | comment | added | Robin Chapman | You get finitely generated abelian groups from taking the homology of the singular chain complex of a CW-complex with finitely many cells. This chain complex is likely to be even more monstrously huge :-) | |
| Jun 7, 2010 at 15:46 | comment | added | ashpool | Getting a finitely generated module from a possibly monstrously huge module is a bit hard for me to swallow. I was hoping there might be some straight forward explanation without translating it to the projective world. | |
| Jun 7, 2010 at 15:14 | comment | added | Robin Chapman | You'll get the $\mathrm{Ext}_A^k(M,N)$ which will be finitely generated; whether that is magical or not isn't really a mathematical question. | |
| Jun 7, 2010 at 15:13 | comment | added | Simon Wadsley | Yes. It does mean that. | |
| Jun 7, 2010 at 15:09 | history | asked | ashpool | CC BY-SA 2.5 |