Tensor-hom adjunction

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

\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):

{\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:

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

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

\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]

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

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

has components

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

given by evaluation: For

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

\varepsilon (\phi \otimes x)=\phi (x).

The components of the unit

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

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

are defined as follows: For y in Y,

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

is a right S-module homomorphism given by

\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,

\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

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

Likewise,

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),

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

is a right S-module homomorphism defined by

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

and therefore

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

The Ext and Tor functors

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

References

^ May, J.P.; Sigurdsson, J. (2006). Parametrized Homotopy Theory. A.M.S. p. 253. ISBN 0-8218-3922-5.

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

[MaySigurdsson-1] May, J.P.; Sigurdsson, J. (2006). Parametrized Homotopy Theory. A.M.S. p. 253. ISBN 0-8218-3922-5.

[1]

Tensor-hom adjunction

General statement

Counit and unit

The Ext and Tor functors

See also

References