Tensor-hom adjunction

Say R and S are (possibly noncommutative) rings, and consider the right module categories (an analogous statement holds for left modules):

${\displaystyle {\mathcal {C}}=\mathrm {Mod} _{S}\quad {\text{and}}\quad {\mathcal {D}}=\mathrm {Mod} _{R}.}$

Fix an (R,S)-bimodule X and define functors F: D → C and G: C → D as follows:

${\displaystyle F(Y)=Y\otimes _{R}X\quad {\text{for }}Y\in {\mathcal {D}}}$

${\displaystyle G(Z)=\operatorname {Hom} _{S}(X,Z)\quad {\text{for }}Z\in {\mathcal {C}}}$

Then F is left adjoint to G. This means there is a natural isomorphism

${\displaystyle \operatorname {Hom} _{S}(Y\otimes _{R}X,Z)\cong \operatorname {Hom} _{R}(Y,\operatorname {Hom} _{S}(X,Z)).}$

This is actually an isomorphism of abelian groups. More precisely, if Y is an (A, R) bimodule and Z is a (B, S) bimodule, then this is an isomorphism of (B, A) bimodules. This is one of the motivating examples of the structure in a closed bicategory.[1]

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

${\displaystyle \varepsilon :FG\to 1_{\mathcal {C}}}$

has components

${\displaystyle \varepsilon _{Z}:\operatorname {Hom} _{S}(X,Z)\otimes _{R}X\to Z}$

given by evaluation: For

${\displaystyle \phi \in \operatorname {Hom} _{R}(X,Z)\quad {\text{and}}\quad x\in X,}$

${\displaystyle \varepsilon (\phi \otimes x)=\phi (x).}$

The components of the unit

${\displaystyle \eta :1_{\mathcal {D}}\to GF}$

${\displaystyle \eta _{Y}:Y\to \operatorname {Hom} _{S}(X,Y\otimes _{R}X)}$

are defined as follows: For y in Y,

${\displaystyle \eta _{Y}(y)\in \operatorname {Hom} _{S}(X,Y\otimes _{R}X)}$

is a right S-module homomorphism given by

${\displaystyle \eta _{Y}(y)(t)=y\otimes t\quad {\text{for }}t\in X.}$

The counit and unit equations can now be explicitly verified. For Y in C,

${\displaystyle \varepsilon _{FY}\circ F(\eta _{Y}):Y\otimes _{R}X\to \operatorname {Hom} _{S}(X,Y\otimes _{R}X)\otimes _{R}X\to Y\otimes _{R}X}$

is given on simple tensors of Y⊗X by

${\displaystyle \varepsilon _{FY}\circ F(\eta _{Y})(y\otimes x)=\eta _{Y}(y)(x)=y\otimes x.}$

Likewise,

${\displaystyle G(\varepsilon _{Z})\circ \eta _{GZ}:\operatorname {Hom} _{S}(X,Z)\to \operatorname {Hom} _{S}(X,\operatorname {Hom} _{S}(X,Z)\otimes _{R}X)\to \operatorname {Hom} _{S}(X,Z).}$

For φ in Hom_S(X, Z),

${\displaystyle G(\varepsilon _{Z})\circ \eta _{GZ}(\phi )}$

is a right S-module homomorphism defined by

${\displaystyle G(\varepsilon _{Z})\circ \eta _{GZ}(\phi )(x)=\varepsilon _{Z}(\phi \otimes x)=\phi (x)}$

and therefore

${\displaystyle G(\varepsilon _{Z})\circ \eta _{GZ}(\phi )=\phi .}$

The Hom functor ${\displaystyle \hom(X,-)}$ commutes with arbitrary limits, while the tensor product ${\displaystyle -\otimes X}$ functor commutes with arbitrary colimits that exist in their domain category. However, in general, ${\displaystyle \hom(X,-)}$ fails to commute with colimits, and ${\displaystyle -\otimes X}$ 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.

• Currying
• Ext functor
• Tor functor

• Bourbaki, Nicolas (1989), Elements of mathematics, Algebra I, Springer-Verlag, ISBN 3-540-64243-9