In mathematics, the tensor-hom adjunction is that the tensor product ${\displaystyle -\otimes X}$ and hom-functor ${\displaystyle \operatorname {Hom} (X,-)}$ form an adjoint pair:

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

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 (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 ${\displaystyle (R,S)}$-bimodule ${\displaystyle X}$ and define functors ${\displaystyle F\colon {\mathcal {D))\rightarrow {\mathcal {C))}$ and ${\displaystyle G\colon {\mathcal {C))\rightarrow {\mathcal {D))}$ 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 ${\displaystyle F}$ is left adjoint to ${\displaystyle G}$. 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 ${\displaystyle Y}$ is an ${\displaystyle (A,R)}$-bimodule and ${\displaystyle Z}$ is a ${\displaystyle (B,S)}$-bimodule, then this is an isomorphism of ${\displaystyle (B,A)}$-bimodules. This is one of the motivating examples of the structure in a closed bicategory.^[1]

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

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

has components

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

given by evaluation: For

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

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

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 ${\displaystyle y}$ in ${\displaystyle Y}$,

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

is a right ${\displaystyle S}$-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 ${\displaystyle Y}$ in ${\displaystyle {\mathcal {D))}$,

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

is given on simple tensors of ${\displaystyle Y\otimes X}$ 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 ${\displaystyle \phi }$ in ${\displaystyle \operatorname {Hom} _{S}(X,Z)}$,

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

is a right ${\displaystyle S}$-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 Ext and Tor functors

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.