Talk:Tor functor

name
could someone give an indication about where the name comes from ? There seems to be no mathematician named "Tor". &mdash; MFH:Talk 12:56, 21 February 2006 (UTC)


 * Torsion in abstract algebra; see torsion (abstract algebra) Charles Matthews 13:43, 21 February 2006 (UTC)

Note that there is Tor functors too. nikita 17:26, 13 June 2006 (UTC)

When is this true?
Tor(A,B) is the tensor product of the torsion subgroups of A and B respectively. --Raijinili (talk) 05:39, 10 August 2008 (UTC)

Associativity of Tor
This has to be mentioned. -- Taku (talk) 16:10, 15 April 2012 (UTC)
 * Do you mean commutativity? If so, I agree... --Roentgenium111 (talk) 15:14, 23 November 2012 (UTC)
 * I meant associativity; Spanier, for example, has an exercise problem for the associativity of Tor_1. -- Taku (talk) 02:37, 4 November 2013 (UTC)

Non-exactness of Tor
For the record. Suppose M is an A-module such that $$\mathrm{Tor}^A_2 (M, N) \ne 0$$ for some N. Consider any short exact sequence $$0 \to K \to P \to M \to 0$$ with P projective. Then we get a long exact sequence
 * $$\cdots \to \mathrm{Tor}^A_2 (P, N) \to \mathrm{Tor}^A_2 (M, N) \to \mathrm{Tor}^A_1 (K, N) \to \mathrm{Tor}^A_1 (P, N) \to \mathrm{Tor}^A_1 (M, N) \to K \otimes_A N \to P \otimes_A N \to M \otimes_A N \to 0$$

but $$\mathrm{Tor}^A_i (P, N) = 0$$ for all $$i > 0$$, so in particular $$\mathrm{Tor}^A_1 (-, N)$$ is neither left exact nor right exact, let alone colimit-preserving. - 振霖T 01:08, 4 November 2013 (UTC)


 * I think I'm missing something. But, for example, an exercise in Atiyah-Macdonald asks you to show Tor commutes with colimit (direct limit). See also: http://mathoverflow.net/questions/97658/left-derived-functors-commute-with-filtered-colimits -- Taku (talk) 02:32, 4 November 2013 (UTC)


 * Not all colimits are filtered, obviously. ("Direct limit" should always be read as "filtered colimit" in old books.) Now would you please stop putting back false information and think before you revert? - 振霖T 08:33, 4 November 2013 (UTC)


 * I know that but "colimit" here was meant "direct limit" obviously (and I think that's a typical practice.) -- Taku (talk) 11:35, 4 November 2013 (UTC)


 * No, it is neither obvious nor typical. For instance, when we say left adjoints preserve colimits, we do in fact mean all colimits. - 振霖T 17:33, 4 November 2013 (UTC)


 * But, as you noticed (which is a good thing), it was not correct if colimit was not interpreted as direct limit; whence, "obvious". As for the "typical", a quick Google search with "colimit direct limit" shows they are frequently used synonymously. Perhaps, that's not a good practice, but then we have fixed that here. -- Taku (talk) 19:06, 5 November 2013 (UTC)

Note in the prove of the Symmetry of Tor
In the prove of Symmetry of Tor, I support that the resolution of Li(regard all the Li as R-mod) should be ····→Mi(f)→Ki→Li→0. In the way, Tor(Z,1)(L1,L2)=ker(f⊗i)/*, by ···→Mi⊗L2→(f⊗i)→Ki⊗L2→0. — Preceding unsigned comment added by Maozhou.Huang (talk • contribs) 04:53, 16 February 2016 (UTC)

Todo
Mention the following:
 * Tor in derived categories
 * the argument for computing $$P\otimes^\mathbf{L}Q$$ by resolving one, or the other, or both
 * Serre intersection formula

Have computations including the following:
 * finite abelian groups
 * koszul complexes => derived intersections of rings
 * sheafify computing tor to projecitve varieties and intersections
 * non-regular rings, e.g. $$k[x,y]/(xy)$$ pg. 7 https://webusers.imj-prg.fr/~yongqi.liang/files/mathjeunes/Javan_MJ.pdf
 * nilpotent artin local ring — Preceding unsigned comment added by 161.98.8.1 (talk) 00:58, 21 June 2017 (UTC)