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

with values in an abelian group G, then f extends uniquely to the homomorphism defined on the whole

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.

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.