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X Tensor and Torsion Products

Elsevier B.V.
DOI: 10.1016/s0079-8169(08)62809-3


There is another fundamental bifunctor whose importance is comparable with that of Horn: this is the tensor product. The tensor products play a role that is—in a certain sense—dual to homomorphism groups, as will be apparent from the results of this chapter. Tensor products are defined in terms of generators and defining relations. They are universal for bilinear functions, and this property makes them of great importance. The exact sequence on tensor products which will be proved in 60 is just as useful as those on Homs. While Horn is left exact, the tensor product turns out to be right exact; exactness can be restored by making use of the functor Tor, the torsion product. If one of the groups is torsion, then their tensor product can be completely described. In particular, the tensor product of two torsion groups is always a direct sum of cyclic groups. The problem of determining the structure of the torsion product is more difficult, but recently a great deal of information has been obtained for Tor.

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