Tensor-hom adjunction


In mathematics, the tensor-hom adjunction is that the tensor product and hom-functor form an adjoint pair:
This is made more precise below. The order of terms in the phrase "tensor-hom adjunction" reflects their relationship: tensor is the left adjoint, while hom is the right adjoint.

General statement

Say R and S are rings, and consider the right module categories :
Fix an -bimodule X and define functors F: D → C and G: C → D as follows:
Then F is left adjoint to G. This means there is a natural isomorphism
This is actually an isomorphism of abelian groups. More precisely, if Y is an bimodule and Z is a bimodule, then this is an isomorphism of bimodules. This is one of the motivating examples of the structure in a closed bicategory.

Counit and unit

Like all adjunctions, the tensor-hom adjunction can be described by its counit and unit natural transformations. Using the notation from the previous section, the counit
has components
given by evaluation: For
The components of the unit
are defined as follows: For y in Y,
is a right S-module homomorphism given by
The counit and unit equations can now be explicitly verified. For
Y in C,
is given on simple tensors of Y⊗X by
Likewise,
For φ in HomS,
is a right S-module homomorphism defined by
and therefore

The Ext and Tor functors

The Hom functor commutes with arbitrary limits, while the tensor product functor commutes with arbitrary colimits that exist in their domain category. However, in general, fails to commute with colimits, and fails to commute with limits; this failure occurs even among finite limits or colimits. This failure to preserve short exact sequences motivates the definition of the Ext functor and the Tor functor.