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- W179677972 abstract "AbstractGiven a module E over a ring R, the process ⋅⊗ R E of taking tensor products with E over R may be viewed as a functor on the category of R-modules. Unless E is flat over R, this functor fails to be exact, i.e. there will exist injective morphisms of R-modules φ: M′→M such that the kernel of the morphism φ⊗id E : M′⊗ R E→M⊗ R E is non-trivial. A natural description of such a kernel is possible via the so-called long exact Tor sequence involving Tor functors as left-derived functors of the tensor product. In the chapter the construction of derived functors is explained, including the associated long exact (co-) homology sequences. As an application, Tor and Ext functors are studied as the derived functors of the tensor product and of the Hom functor.KeywordsExact SequenceShort Exact SequenceProjective ResolutionCovariant FunctorCochain ComplexThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves." @default.
- W179677972 created "2016-06-24" @default.
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- W179677972 date "2013-01-01" @default.
- W179677972 modified "2023-09-27" @default.
- W179677972 title "Homological Methods: Ext and Tor" @default.
- W179677972 doi "https://doi.org/10.1007/978-1-4471-4829-6_5" @default.
- W179677972 hasPublicationYear "2013" @default.
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