For a ring R, a right R-module M, a left R-module N, and an abelian group G, a map φ: M × N → G is said to be R-balanced, R-middle-linear or an R-balanced product if for all m, m′ in M, n, n′ in N, and r in R the following hold:
The set of all such balanced products over R from M × N to G is denoted by LR(M, N; G).
If φ, ψ are balanced products, then each of the operations φ + ψ and −φ defined pointwise is a balanced product. This turns the set LR(M, N; G) into an abelian group.
For M and N fixed, the map G ↦ LR(M, N; G) is a functor from the category of abelian groups to the category of sets. The morphism part is given by mapping a group homomorphism g : G → G′ to the function φ ↦ g ∘ φ, which goes from LR(M, N; G) to LR(M, N; G′).
Property (Dl) states the left and property (Dr) the right distributivity of φ over addition.
For every abelian group G and every balanced product
there is a unique group homomorphism
As with all universal properties, the above property defines the tensor product uniquely up to a unique isomorphism: any other object and balanced product with the same properties will be isomorphic to M ⊗RN and ⊗. Indeed, the mapping ⊗ is called canonical, or more explicitly: the canonical mapping (or balanced product) of the tensor product.
The definition does not prove the existence of M ⊗RN; see below for a construction.
for the image of (x, y) under the canonical map . It is often called a pure tensor. Strictly speaking, the correct notation would be x ⊗Ry but it is conventional to drop R here. Then, immediately from the definition, there are relations:
x ⊗ (y + y′) = x ⊗ y + x ⊗ y′
(x + x′) ⊗ y = x ⊗ y + x′ ⊗ y
(x ⋅ r) ⊗ y = x ⊗ (r ⋅ y)
The universal property of a tensor product has the following important consequence:
Proposition — Every element of can be written, non-uniquely, as
In other words, the image of generates . Furthermore, if f is a function defined on elements with values in an abelian group G, then f extends uniquely to the homomorphism defined on the whole if and only if is -bilinear in x and y.
Proof: For the first statement, let L be the subgroup of generated by elements of the form in question, and q the quotient map to Q. We have: as well as . Hence, by the uniqueness part of the universal property, q = 0. The second statement is because to define a module homomorphism, it is enough to define it on the generating set of the module.
The proposition says that one can work with explicit elements of the tensor products instead of invoking the universal property directly each time. This is very convenient in practice. For example, if R is commutative, then can naturally be furnished with the R-scalar multiplication by extending
to the whole by the previous proposition (strictly speaking, what is needed is a bimodule structure not commutativity; see a paragraph below). Equipped with this R-module structure, satisfies a universal property similar to the above: for any R-module G, there is a natural isomorphism:
If R is not necessarily commutative but if M has a left action by a ring S (for example, R), then can be given the left S-module structure, like above, by the formula
If N has a right action by a ring S, then, in the analogous way, becomes a right S-module.
Tensor product of linear maps and a change of base ringEdit
Given linear maps of right modules over a ring R and of left modules, there is a unique group homomorphism
The construction has a consequence that tensoring is a functor: each right R-module M determines the functor
from the category of left modules to the category of abelian groups that sends N to M ⊗ N and a module homomorphism f to the group homomorphism 1 ⊗ f.
If is a ring homomorphism and if M is a right S-module and N a left S-module, then there is the canonical surjective homomorphism:
The resulting map is surjective since pure tensors x ⊗ y generate the whole module. In particular, taking R to be this shows every tensor product of modules is a quotient of a tensor product of abelian groups.
(This section need to be updated. For now, see § Properties for the more general discussion.)
It is possible to extend the definition to a tensor product of any number of modules over the same commutative ring. For example, the universal property of
M1 ⊗ M2 ⊗ M3
is that each trilinear map on
M1 × M2 × M3 → Z
corresponds to a unique linear map
M1 ⊗ M2 ⊗ M3 → Z.
The binary tensor product is associative: (M1 ⊗ M2) ⊗ M3 is naturally isomorphic to M1 ⊗ (M2 ⊗ M3). The tensor product of three modules defined by the universal property of trilinear maps is isomorphic to both of these iterated tensor products.
If R is not commutative, the order of tensor products could matter in the following way: we "use up" the right action of M and the left action of N to form the tensor product ; in particular, would not even be defined. If M, N are bi-modules, then has the left action coming from the left action of M and the right action coming from the right action of N; those actions need not be the same as the left and right actions of .
The associativity holds more generally for non-commutative rings: if M is a right R-module, N a (R, S)-module and P a left S-module, then
as abelian group.
The general form of adjoint relation of tensor products says: if R is not necessarily commutative, M is a right R-module, N is a (R, S)-module, P is a right S-module, then as abelian group
where f is given by . Since the image of f is IM, we get the first part of 1. If M is flat, f is injective and so is an isomorphism onto its image. 2. follows from 1. and 3. is because
Example: If G is an abelian group, ; this follows from 1.
Example:; this follows from 3.
Example: Let be the group of n-th roots of unity. It is a cyclic group and cyclic groups are classified by orders. Thus, non-canonically, and thus, when g is the gcd of n and m,
Example: Consider Since is obtained from by imposing -linearity on the middle, we have the surjection
whose kernel is generated by elements of the form
where r, s, x, u are integers and s is nonzero. Since
the kernel actually vanishes; hence,
Example: We propose to compare and . Like in the previous example, we have: as abelian group and thus as -vector space (any -linear map between -vector spaces is -linear). As -vector space, has dimension (cardinality of a basis) of continuum. Hence, has a -basis indexed by a product of continuums; thus its -dimension is continuum. Hence, for dimension reason, there is a non-canonical isomorphism of -vector spaces:
Consider the modules for irreducible polynomials such that Then,
Another useful family of examples comes from changing the scalars. Notice that
Good examples of this phenomenon to look at are when
The construction of M ⊗ N takes a quotient of a free abelian group with basis the symbols m ∗ n, used here to denote the ordered pair(m, n), for m in M and n in N by the subgroup generated by all elements of the form
−m ∗ (n + n′) + m ∗ n + m ∗ n′
−(m + m′) ∗ n + m ∗ n + m′ ∗ n
(m · r) ∗ n − m ∗ (r · n)
where m, m′ in M, n, n′ in N, and r in R. The quotient map which takes m ∗ n =(m, n) to the coset containing m ∗ n; that is,
is balanced, and the subgroup has been chosen minimally so that this map is balanced. The universal property of ⊗ follows from the universal properties of a free abelian group and a quotient.
More category-theoretically, let σ be the given right action of R on M; i.e., σ(m, r) = m · r and τ the left action of R of N. Then the tensor product of M and N over R can be defined as the coequalizer:
together with the requirements
If S is a subring of a ring R, then is the quotient group of by the subgroup generated by , where is the image of under In particular, any tensor product of R-modules can be constructed, if so desired, as a quotient of a tensor product of abelian groups by imposing the R-balanced product property.
In the construction of the tensor product over a commutative ring R, the R-module structure can be built in from the start by forming the quotient of a free R-module by the submodule generated by the elements given above for the general construction, augmented by the elements r ⋅ (m ∗ n) − m ∗ (r ⋅ n). Alternately, the general construction can be given a Z(R)-module structure by defining the scalar action by r ⋅ (m ⊗ n) = m ⊗ (r ⋅ n) when this is well-defined, which is precisely when r ∈ Z(R), the centre of R.
The direct product of M and N is rarely isomorphic to the tensor product of M and N. When R is not commutative, then the tensor product requires that M and N be modules on opposite sides, while the direct product requires they be modules on the same side. In all cases the only function from M × N to G that is both linear and bilinear is the zero map.
The dual module of a right R-module E, is defined as HomR(E, R) with the canonical left R-module structure, and is denoted E∗. The canonical structure is the pointwise operations of addition and scalar multiplication. Thus, E∗ is the set of all R-linear maps E → R (also called linear forms), with operations
The dual of a left R-module is defined analogously, with the same notation.
There is always a canonical homomorphism E → E∗∗ from E to its second dual. It is an isomorphism if E is a free module of finite rank. In general, E is called a reflexive module if the canonical homomorphism is an isomorphism.
In the general case, each element of the tensor product of modules gives rise to a left R-linear map, to a right R-linear map, and to an R-bilinear form. Unlike the commutative case, in the general case the tensor product is not an R-module, and thus does not support scalar multiplication.
Given right R-module E and right R-module F, there is a canonical homomorphism θ : F ⊗RE∗ → HomR(E, F) such that θ(f ⊗ e′) is the map e ↦ f ⋅ ⟨e′, e⟩.
Given left R-module E and right R-module F, there is a canonical homomorphism θ : F ⊗RE → HomR(E∗, F) such that θ(f ⊗ e) is the map e′ ↦ f ⋅ ⟨e, e′⟩.
Both cases hold for general modules, and become isomorphisms if the modules E and F are restricted to being finitely generated projective modules (in particular free modules of finite ranks). Thus, an element of a tensor product of modules over a ring R maps canonically onto an R-linear map, though as with vector spaces, constraints apply to the modules for this to be equivalent to the full space of such linear maps.
Given right R-module E and left R-module F, there is a canonical homomorphism θ : F∗ ⊗RE∗ → LR(F × E, R) such that θ(f′ ⊗ e′) is the map (f, e) ↦ ⟨f, f′⟩ ⋅ ⟨e′, e⟩. Thus, an element of a tensor product ξ ∈ F∗ ⊗RE∗ may be thought of giving rise to or acting as an R-bilinear map F × E → R.
Let R be a commutative ring and E an R-module. Then there is a canonical R-linear map:
induced through linearity by ; it is the unique R-linear map corresponding to the natural pairing.
If E is a finitely generated projective R-module, then one can identify through the canonical homomorphism mentioned above and then the above is the trace map:
When R is a field, this is the usual trace of a linear transformation.
Example from differential geometry: tensor fieldEdit
The most prominent example of a tensor product of modules in differential geometry is the tensor product of the spaces of vector fields and differential forms. More precisely, if R is the (commutative) ring of smooth functions on a smooth manifold M, then one puts
where Γ means the space of sections and the superscript means tensoring p times over R. By definition, an element of is a tensor field of type (p, q).
To lighten the notation, put and so . When p, q ≥ 1, for each (k, l) with 1 ≤ k ≤ p, 1 ≤ l ≤ q, there is an R-multilinear map:
where means and the hat means a term is omitted. By the universal property, it corresponds to a unique R-linear map:
It is called the contraction of tensors in the index (k, l). Unwinding what the universal property says one sees:
Remark: The preceding discussion is standard in textbooks on differential geometry (e.g., Helgason). In a way, the sheaf-theoretic construction (i.e., the language of sheaf of modules) is more natural and increasingly more common; for that, see the section § Tensor product of sheaves of modules.
It can be shown that and are always right exact functors, but not necessarily left exact ( where the first map is multiplication by , is exact but not after taking the tensor with ). By definition, a module T is a flat module if is an exact functor.
If and are generating sets for M and N, respectively, then will be a generating set for Because the tensor functor sometimes fails to be left exact, this may not be a minimal generating set, even if the original generating sets are minimal. If M is a flat module, the functor is exact by the very definition of a flat module. If the tensor products are taken over a field F, we are in the case of vector spaces as above. Since all F modules are flat, the bifunctor is exact in both positions, and the two given generating sets are bases, then indeed forms a basis for
If S and T are commutative R-algebras, then S ⊗RT will be a commutative R-algebra as well, with the multiplication map defined by (m1 ⊗ m2) (n1 ⊗ n2) = (m1n1 ⊗ m2n2) and extended by linearity. In this setting, the tensor product become a fibered coproduct in the category of R-algebras.
If M and N are both R-modules over a commutative ring, then their tensor product is again an R-module. If R is a ring, RM is a left R-module, and the commutator
rs − sr
of any two elements r and s of R is in the annihilator of M, then we can make M into a right R module by setting
mr = rm.
The action of R on M factors through an action of a quotient commutative ring. In this case the tensor product of M with itself over R is again an R-module. This is a very common technique in commutative algebra.
For example, if C is a chain complex of flat abelian groups and if G is an abelian group, then the homology group of is the homology group of C with coefficients in G (see also: universal coefficient theorem.)