Talk:Tor functor
This article is rated Start-class on Wikipedia's content assessment scale. It is of interest to the following WikiProjects: | |||||||||||
|
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)
- 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?[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)
Associativity of Tor[edit]
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[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)
- 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)
- 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[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 (talk • contribs) 04:53, 16 February 2016 (UTC)
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:
- finite abelian groups
- koszul complexes => derived intersections of rings
- sheafify computing tor to projecitve varieties and intersections
- non-regular rings, e.g. 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)