(i) For a ring R and a fixed k ≥0, prove that Tork^R(□, □) is a bifunctor. (ii) For a ring R and

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(i) For a ring R and a fixed k ≥0, prove that Tork^R(□, □)...

(i) For a ring $R$ and a fixed $k \geq 0$, prove that $\operatorname{Tor}_{k}^{R}(\square, \square)$ is a bifunctor.
(ii) For a ring $R$ and a fixed $k \geq 0$, prove that $\operatorname{Ext}_{R}^{k}(\square, \square$ ) is a bifunctor.

Key Concepts

Projective and Injective Resolutions

Projective resolutions (for Tor) and injective resolutions (for Ext) are key tools in homological algebra that allow one to compute derived functors. A projective resolution of a module is an exact sequence where each module is projective, which makes tensoring with another module well-behaved. Similarly, an injective resolution allows for the computation of the Hom functor in a way that leads to the definition of Ext. The functorial nature of these resolutions underlies the bifunctor properties of Tor and Ext.

Hom Functor

The Hom functor, which assigns to a pair of modules the group of module homomorphisms, is left exact but fails to be exact in general. Ext groups, defined as the right derived functors of Hom, quantify the extent to which Hom fails to preserve exactness. Establishing that Extk is a bifunctor requires demonstrating that it is functorial in each argument, a property derived from the naturality of the constructions involved in taking Hom and forming injective (or projective) resolutions.

Tensor Product

The tensor product is a bilinear operation on modules that is right-exact in each variable. As a bifunctor from the product of two module categories to the category of abelian groups, it does not in general preserve exact sequences. Tor emerges as the derived functor of the tensor product, measuring the failure of exactness. Proving that Tork is a bifunctor involves showing that the construction is natural in both arguments through the use of projective resolutions.

Derived Functor

Derived functors generalize classical functors by providing homological information when the classical functor fails to be exact. In the case of Tor and Ext, these are the left and right derived functors, respectively, of the tensor product and Hom functors. The process involves replacing modules by resolutions (projective or injective) to account for the lack of exactness, ensuring that these constructions yield invariants independent of the chosen resolution.

Bifunctor

A bifunctor is a construction that assigns an object to every pair of objects from two categories, in such a way that it is functorial in each variable separately. This means that if you have morphisms in either argument, the bifunctor provides a corresponding morphism in the target category, and these assignments respect compositions and identities. Showing that Tork and Extk are bifunctors involves demonstrating that they act compatibly with morphisms in both module arguments.

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