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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
667:{\displaystyle \cdots \to \mathrm {Tor} _{2}^{A}(P,N)\to \mathrm {Tor} _{2}^{A}(M,N)\to \mathrm {Tor} _{1}^{A}(K,N)\to \mathrm {Tor} _{1}^{A}(P,N)\to \mathrm {Tor} _{1}^{A}(M,N)\to K\otimes _{A}N\to P\otimes _{A}N\to M\otimes _{A}N\to 0}
970:
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.
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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
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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:
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No, it is neither obvious nor typical. For instance, when we say left adjoints preserve colimits, we do in fact mean all colimits. -
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could someone give an indication about where the name comes from ? There seems to be no mathematician named "Tor". —
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I know that but "colimit" here was meant "direct limit" obviously (and I think that's a typical practice.) --
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I meant associativity; Spanier, for example, has an exercise problem for the associativity of Tor_1. --
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on
Knowledge. If you would like to participate, please visit the project page, where you can join
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http://mathoverflow.net/questions/97658/left-derived-functors-commute-with-filtered-colimits
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Tor(A,B) is the tensor product of the torsion subgroups of A and B respectively. --
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https://webusers.imj-prg.fr/~yongqi.liang/files/mathjeunes/Javan_MJ.pdf
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is neither left exact nor right exact, let alone colimit-preserving. -
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sheafify computing tor to projecitve varieties and intersections
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with P projective. Then we get a long exact sequence
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109:and see a list of open tasks.
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801:−
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