Tensor product of modules

The universal property of the tensor product of vector spaces extends to more general situations in abstract algebra.

For M and N fixed, the map G ↦ LR(M, N; G) is a functor from the category of abelian groups to itself.

The tensor product can also be defined as a representing object for the functor G → LR(M,N;G); explicitly, this means there is a natural isomorphism:

Then, immediately from the definition, there are relations: The universal property of a tensor product has the following important consequence: Proposition — Every element of

The proposition says that one can work with explicit elements of the tensor products instead of invoking the universal property directly each time.

For example, if R is commutative and the left and right actions by R on modules are considered to be equivalent, then

by the previous proposition (strictly speaking, what is needed is a bimodule structure not commutativity; see a paragraph below).

satisfies a universal property similar to the above: for any R-module G, there is a natural isomorphism:

The resulting map is surjective since pure tensors x ⊗ y generate the whole module.

It is possible to extend the definition to a tensor product of any number of modules over the same commutative ring.

Then To give a practical example, suppose M, N are free modules with bases

The tensor product, in general, does not commute with inverse limit: on the one hand,

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

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[9]

Let R be an integral domain with fraction field K. The adjoint relation in the general form has an important special case: for any R-algebra S, M a right R-module, P a right S-module, using ⁠

is often called the extension of scalars from R to S. In the representation theory, when R, S are group algebras, the above relation becomes the Frobenius reciprocity.

Tensor products can be applied to control the order of elements of groups.

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 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,

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, provided the tensor product of abelian groups is already defined, the tensor product of M and N over R can be defined as the coequalizer:

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).

In the general case, not all the properties of a tensor product of vector spaces extend to modules.

Yet, some useful properties of the tensor product, considered as module homomorphisms, remain.

[11] 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

In general, E is called a reflexive module if the canonical homomorphism is an isomorphism.

Remark: The preceding discussion is standard in textbooks on differential geometry (e.g., Helgason).

is a bifunctor which accepts a right and a left R module pair as input, and assigns them to the tensor product in the category of abelian groups.

arises, and symmetrically a left R module N could be fixed to create a functor

In this setting, the tensor product become a fibered coproduct in the category of commutative R-algebras.

Sections of the exterior bundle are differential forms on M. One important case when one forms a tensor product over a sheaf of non-commutative rings appears in theory of D-modules; that is, tensor products over the sheaf of differential operators.