User kripton - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T00:25:55Z http://mathoverflow.net/feeds/user/11414 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/48694/how-to-calculate-torr-i-r-j How to calculate Tor(R/I, R/J) ?? Kripton 2010-12-08T23:48:57Z 2012-04-28T20:55:38Z <p>How can I prove that $\text{Tor}_1(R/I,R/J) = (I \cap J)/IJ$, where $R$ is a ring and $I, J$ ideals. </p> <p>Moreover, if we suppose $R=I+J$, how do I prove that $\text{Tor}_1(R/I,R/J)=0$?</p> <p>Ps: No, this is not a homework question. </p> http://mathoverflow.net/questions/48694/how-to-calculate-torr-i-r-j/48696#48696 Comment by Kripton Kripton 2010-12-09T03:16:16Z 2010-12-09T03:16:16Z yes, that I know since the basic. lolol What I don't know is to prove that indirectly, I mean, using the fact that Tor(R/I,R/J)=(I∩J)/IJ. I must prove that tor vanishes to conclude that I∩J = IJ, see? http://mathoverflow.net/questions/48694/how-to-calculate-torr-i-r-j Comment by Kripton Kripton 2010-12-09T03:10:13Z 2010-12-09T03:10:13Z yes, I've already proved the first part, but now I can't see the second. Anyway thank you for your comment. http://mathoverflow.net/questions/48694/how-to-calculate-torr-i-r-j Comment by Kripton Kripton 2010-12-09T02:33:54Z 2010-12-09T02:33:54Z Karl, thank you for the answer. It with the one below allowed me to have the solution. I really apreciate that. http://mathoverflow.net/questions/48694/how-to-calculate-torr-i-r-j/48696#48696 Comment by Kripton Kripton 2010-12-09T02:25:56Z 2010-12-09T02:25:56Z Thank you very much for your help. It was very useful. Concerning to 2), I think you don't understand what I asked. I need to prove that (I∩J) = IJ, using that Tor1(R/I,R/J)=(I∩J)/IJ and the fact that R=I+J. So I think if we prove that Tor vanishes in this case, we have the problem solved. Once again thank you. http://mathoverflow.net/questions/48694/how-to-calculate-torr-i-r-j Comment by Kripton Kripton 2010-12-09T00:06:34Z 2010-12-09T00:06:34Z I mean Tor_1 over the ring R. That is exactly what I did, but I get Tor(R/I x R/J) = Ker(I x R/I ----&gt; A x A/J), where x is the tensor product, but then I don't know how to prove the equality.