Tensor-hom adjunction

In mathematics, the tensor-hom adjunction is that the tensor product and Hom functors - \otimes X and \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 "tensor-hom adjunction" is because 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 HomS(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.

Ext and Tor

The idea that the Hom functor and the tensor product functor don't lift to an exact sequence motivates the definition of the Ext functor and the Tor functor.

See also

References

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