Examples of computing Ext and Tor functors? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T10:59:53Z http://mathoverflow.net/feeds/question/25062 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/25062/examples-of-computing-ext-and-tor-functors Examples of computing Ext and Tor functors? Adeel 2010-05-17T21:49:29Z 2010-07-06T03:51:34Z <p>So I understand in theory the definition of Ext and Tor, but when it comes to actually computing them, I'm stuck. For example, could someone show me how to compute $\text{Ext}(\mathbb{Z}/m\mathbb{Z}, \mathbb{Z}/n\mathbb{Z})$? I tried this by taking an injective resolution ($0 \to \mathbb{Z}/m\mathbb{Z} \to \mathbb{Q}/\mathbb{Z} \to \mathbb{Q}/\mathbb{Z} \to 0$?) and I got an exact sequence with $\text{Ext}^1(\mathbb{Z}/m\mathbb{Z}, \mathbb{Q}/\mathbb{Z}) = 0$, but I don't see what to do next?</p> <p>P.S. No, this is not a homework question.</p> http://mathoverflow.net/questions/25062/examples-of-computing-ext-and-tor-functors/30714#30714 Answer by Sean Tilson for Examples of computing Ext and Tor functors? Sean Tilson 2010-07-06T03:26:59Z 2010-07-06T03:51:34Z <p>$\mathbb{Q}/\mathbb{Z}$ is a pretty terrible abelian group, or a rather hard one, there may be better injective resolutions to work with. It would certainly be easier to do the projective resolution, use $0 \to \mathbb{Z} \to \mathbb{Z} \to \mathbb{Z}/n \to 0$. this will surely be easier to work through than the one involving $\mathbb{Q}/\mathbb{Z}$. Then compute the appropriate tensor product or hom group.</p> <p>I started learning this stuff on more interesting modules as Schmidt suggests. For example, modules over the group ring of some cyclic group, or maybe an exterior algebra on two generators (if you make the generators of different gradings, in particular 1 and 3). This happens to be the category of modules you need to understand in order to compute complex connective k-theory! </p> <p>this should help get you going. These computations are very fun!</p>