Talk:Tor functor

Latest comment: 7 years ago by 161.98.8.1 in topic Todo

name

edit

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

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

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

When is this true?

edit

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

Associativity of Tor

edit

This has to be mentioned. -- Taku (talk) 16:10, 15 April 2012 (UTC)Reply

Do you mean commutativity? If so, I agree... --Roentgenium111 (talk) 15:14, 23 November 2012 (UTC)Reply
I meant associativity; Spanier, for example, has an exercise problem for the associativity of Tor_1. -- Taku (talk) 02:37, 4 November 2013 (UTC)Reply

Non-exactness of Tor

edit

For the record. Suppose M is an A-module such that   for some N. Consider any short exact sequence   with P projective. Then we get a long exact sequence

 

but   for all  , so in particular   is neither left exact nor right exact, let alone colimit-preserving. - 振霖T 01:08, 4 November 2013 (UTC)Reply

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)Reply
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)Reply
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)Reply
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)Reply
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)Reply

Note in the prove of the Symmetry of Tor

edit

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 (talkcontribs) 04:53, 16 February 2016 (UTC)Reply

Todo

edit

Mention the following:

  • Tor in derived categories
  • the argument for computing   by resolving one, or the other, or both
  • Serre intersection formula

Have computations including the following: