In
mathematics, the tensor-hom adjunction is that the
tensor product
and
hom-functor
form an
adjoint pair:
![{\displaystyle \operatorname {Hom} (Y\otimes X,Z)\cong \operatorname {Hom} (Y,\operatorname {Hom} (X,Z)).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b12081802f137e17aa9de103a99a7e214b28bfd0)
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}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1a9699ccecb062a7440cd50bee6e672c74a0e111)
Fix an
-bimodule
and define functors
and
as follows:
![{\displaystyle F(Y)=Y\otimes _{R}X\quad {\text{for }}Y\in {\mathcal {D}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3c9b607a9f57b81fcc9dbf379ba10bda451a85ed)
![{\displaystyle G(Z)=\operatorname {Hom} _{S}(X,Z)\quad {\text{for }}Z\in {\mathcal {C}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ed827509e05bbd30cbb9724e1241506bdbc55ad1)
Then
is left
adjoint to
. 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)).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/83a3b61f24c85fa28e85c16cd6ac9e0fab13fce5)
This is actually an isomorphism of
abelian groups. More precisely, if
is an
-bimodule and
is a
-bimodule, then this is an isomorphism of
-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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a3caeec84ee55e05731ec9857d9f599c20369eb7)
has
components
![{\displaystyle \varepsilon _{Z}:\operatorname {Hom} _{S}(X,Z)\otimes _{R}X\to Z}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e187589949ea42c4d28fcefa240dc2f2b5d5cce)
given by evaluation: For
![{\displaystyle \phi \in \operatorname {Hom} _{S}(X,Z)\quad {\text{and}}\quad x\in X,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/de8127375f69cd5746e1489bded3017ae2a26856)
![{\displaystyle \varepsilon (\phi \otimes x)=\phi (x).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b527ff7cfdc600f0ab18f244c01056ad0a55f547)
The
components of the unit
![{\displaystyle \eta :1_{\mathcal {D}}\to GF}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cc912fb67aea3aaa16396c90b133c622827b3625)
![{\displaystyle \eta _{Y}:Y\to \operatorname {Hom} _{S}(X,Y\otimes _{R}X)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a66af43e4ae0338ca6a585851524d13f95c45f1f)
are defined as follows: For
in
,
![{\displaystyle \eta _{Y}(y)\in \operatorname {Hom} _{S}(X,Y\otimes _{R}X)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/81dd81ad300919c16076f9797482bf835c16a0b3)
is a right
-module homomorphism given by
![{\displaystyle \eta _{Y}(y)(t)=y\otimes t\quad {\text{for }}t\in X.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bb509ace0729d19a5138f6ac99d6b84abd9cba5b)
The
counit and unit equations can now be explicitly verified. For
in
,
![{\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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9073a6c883717e0eccf2e9da5471e69e36b0fcb0)
is given on
simple tensors of
by
![{\displaystyle \varepsilon _{FY}\circ F(\eta _{Y})(y\otimes x)=\eta _{Y}(y)(x)=y\otimes x.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/acaca44af9a6c0583f80a32ec598fe42b41f3629)
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).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cf0b0600ed7128748d0f0198e37cb2634edaccd9)
For
in
,
![{\displaystyle G(\varepsilon _{Z})\circ \eta _{GZ}(\phi )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d814f1e5b9d37f9d2c4965284ce3661e1cbb5d87)
is a right
-module homomorphism defined by
![{\displaystyle G(\varepsilon _{Z})\circ \eta _{GZ}(\phi )(x)=\varepsilon _{Z}(\phi \otimes x)=\phi (x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0f25221b0e23e0c276aec1c44a498514a314af6f)
and therefore
![{\displaystyle G(\varepsilon _{Z})\circ \eta _{GZ}(\phi )=\phi .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3f7bc9c4e28248424c151a5ac350c1aa78981907)
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.
See also
References
-
^
May, J.P.; Sigurdsson, J. (2006). Parametrized Homotopy Theory. A.M.S. p. 253.
ISBN
0-8218-3922-5.