Prove that the functor T=Tor1^Z(G, □) is left exact for every abelian group G, and compute its r

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Prove that the functor T=Tor1^Z(G, □) is left exact for...

Key Concepts

Derived Functors

Derived functors are a fundamental tool in homological algebra used to extend a (usually non-exact) functor to a sequence of functors that measure the exactness failures. They are constructed by applying the original functor to a resolution (typically a projective or injective resolution) of an object and taking the homology of the resulting complex. This process reveals hidden homological information that the original functor, when applied directly, might miss.

Tor Functor

The Tor functors, denoted as Tor?, are specific derived functors of the tensor product. They arise when one tensors a projective resolution with a module and studies the homology of the resulting complex. In particular, Tor? captures obstructions to the flatness of a module and measures how tensor product fails to be exact. They play a crucial role in understanding extensions and relate to various obstructions in module theory.

Left Exact Functor

A functor is left exact if it preserves all finite limits, notably kernels and monomorphisms. This property means that when a left exact functor is applied to a short exact sequence, the resulting sequence remains exact at the initial terms. Establishing that a functor is left exact often involves proving that it preserves injectivity and the structure of the initial segments of exact sequences.

Projective Resolutions

Projective resolutions are sequences of projective objects (such as free modules in the category of abelian groups) that provide a way to study modules via homological methods. By resolving an object with projective modules, one can apply right exact functors (like the tensor product) term by term, and then compute the derived functors (such as the Tor functors) by taking the homology of the resulting complex. This device is essential for verifying properties like left exactness and for computing higher derived functors.

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