Timeline for Homological vs. cohomological dimension of a group/space
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 16, 2015 at 10:42 | comment | added | Yonatan Harpaz | -> The projective dimension of a complex $C_\bullet$ is the minimal $n$ such that $Ext^m(C_\bullet,M) = 0$ for every $m > n$ and every $G$-module $M$ (considered as a complex concentrated in degree $0$). Similarly, one can defined the flat dimension using $Tor$. These dimensions can be characterized as the minimal lengths of a projective/flat replacement of $C_\bullet$, respectively. The main point of the argument is that if $C_\bullet$ is a finite complex of finitely generated $G$-modules then it is enough to take the shortest finitely generated projective replacement. | |
| Jul 16, 2015 at 10:36 | comment | added | Yonatan Harpaz | @FernandoMuro. Maybe my terminology is not standard. Let us work with non-negatively graded complexes. By $Tor_n(C_\bullet,D_\bullet)$ I mean the $n$'th homology group of the derived tensor product of $C_\bullet$ and $D_\bullet$. If either $C_\bullet$ or $D_\bullet$ are flat than this will coincide with the $n$'th homology group of the ordinary tensor product. Similarly, by $Ext^n(C_\bullet,D_\bullet)$ I mean the $n$'th cohomology group of the derived hom complex. -> | |
| Jul 15, 2015 at 22:00 | comment | added | Fernando Muro | In don't understand this argument. What's the projective/flat dimension of a complex? How can Tor be non-trivial if the second argument is projective? | |
| Jul 15, 2015 at 6:27 | comment | added | KotelKanim | Nice :) And you are right that my argument has a problem. I'm not sure it's fixable... | |
| Jul 14, 2015 at 19:47 | history | answered | Yonatan Harpaz | CC BY-SA 3.0 |