In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the tensor-hom adjunction is that the
tensor product
In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otimes W ...
and
hom-functor
In mathematics, specifically in category theory, hom-sets (i.e. sets of morphisms between objects) give rise to important functors to the category of sets. These functors are called hom-functors and have numerous applications in category theory and ...
form an
adjoint pair
In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are kn ...
:
:
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
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
, and consider the right
module
Module, modular and modularity may refer to the concept of modularity. They may also refer to:
Computing and engineering
* Modular design, the engineering discipline of designing complex devices using separately designed sub-components
* Mo ...
categories (an analogous statement holds for left modules):
:
Fix an (''R'',''S'')-bimodule ''X'' and define functors ''F'': ''D'' → ''C'' and ''G'': ''C'' → ''D'' as follows:
:
:
Then ''F'' is left
adjoint
In mathematics, the term ''adjoint'' applies in several situations. Several of these share a similar formalism: if ''A'' is adjoint to ''B'', then there is typically some formula of the type
:(''Ax'', ''y'') = (''x'', ''By'').
Specifically, adjoin ...
to ''G''. This means there is a
natural isomorphism
In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natur ...
:
This is actually an isomorphism of
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commut ...
s. 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
In mathematics, a bicategory (or a weak 2-category) is a concept in category theory used to extend the notion of category to handle the cases where the composition of morphisms is not (strictly) associative, but only associative ''up to'' an isomor ...
.
[
]
Counit and unit
Like all adjunctions, the tensor-hom adjunction can be described by its counit and unit
natural transformation
In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natur ...
s. Using the notation from the previous section, the counit
:
has
component
Circuit Component may refer to:
•Are devices that perform functions when they are connected in a circuit.
In engineering, science, and technology Generic systems
* System components, an entity with discrete structure, such as an assem ...
s
:
given by evaluation: For
:
:
The
component
Circuit Component may refer to:
•Are devices that perform functions when they are connected in a circuit.
In engineering, science, and technology Generic systems
* System components, an entity with discrete structure, such as an assem ...
s of the unit
:
:
are defined as follows: For
in
,
:
is a right
-module homomorphism given by
:
The
counit and unit equations can now be explicitly verified. For
in
,
:
is given on
simple tensor
In mathematics, the modern component-free approach to the theory of a tensor views a tensor as an abstract object, expressing some definite type of multilinear concept. Their properties can be derived from their definitions, as linear maps or m ...
s of
by
:
Likewise,
:
For
in
'',''
:
is a right
-module homomorphism defined by
:
and therefore
:
The Ext and Tor functors
The
Hom functor
In mathematics, specifically in category theory, hom-sets (i.e. sets of morphisms between objects) give rise to important functors to the category of sets. These functors are called hom-functors and have numerous applications in category theory and ...
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 sequence
An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next.
Definition
In the context ...
s motivates the definition of the
Ext functor
In mathematics, the Ext functors are the derived functors of the Hom functor. Along with the Tor functor, Ext is one of the core concepts of homological algebra, in which ideas from algebraic topology are used to define invariants of algebraic stru ...
and the
Tor functor
In mathematics, the Tor functors are the derived functors of the tensor product of modules over a ring. Along with the Ext functor, Tor is one of the central concepts of homological algebra, in which ideas from algebraic topology are used to con ...
.
See also
*
Currying
In mathematics and computer science, currying is the technique of translating the evaluation of a function that takes multiple arguments into evaluating a sequence of functions, each with a single argument. For example, currying a function f that ...
*
Ext functor
In mathematics, the Ext functors are the derived functors of the Hom functor. Along with the Tor functor, Ext is one of the core concepts of homological algebra, in which ideas from algebraic topology are used to define invariants of algebraic stru ...
*
Tor functor
In mathematics, the Tor functors are the derived functors of the tensor product of modules over a ring. Along with the Ext functor, Tor is one of the central concepts of homological algebra, in which ideas from algebraic topology are used to con ...
*
Change of rings
In algebra, given a ring homomorphism f: R \to S, there are three ways to change the coefficient ring of a module; namely, for a left ''R''-module ''M'' and a left ''S''-module ''N'',
*f_! M = S\otimes_R M, the induced module.
*f_* M = \operatorn ...
References
*
{{Category theory
Adjoint functors
Commutative algebra