Abstract
Given 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.
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© 2013 Springer-Verlag London
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Bosch, S. (2013). Homological Methods: Ext and Tor. In: Algebraic Geometry and Commutative Algebra. Universitext. Springer, London. https://doi.org/10.1007/978-1-4471-4829-6_5
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DOI: https://doi.org/10.1007/978-1-4471-4829-6_5
Publisher Name: Springer, London
Print ISBN: 978-1-4471-4828-9
Online ISBN: 978-1-4471-4829-6
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