In mathematics, the tensor product of modules is a construction that allows arguments about bilinear maps (e.g. multiplication) to be carried out in terms of linear maps. The module construction is analogous to the construction of the tensor product of vector spaces, but can be carried out for a pair of modules over a commutative ring resulting in a third module, and also for a pair of a right-module and a left-module over any ring, with result an abelian group. Tensor products are important in areas of abstract algebra, homological algebra, algebraic topology, algebraic geometry, operator algebras and noncommutative geometry. The universal property of the tensor product of vector spaces extends to more general situations in abstract algebra. The tensor product of an algebra and a module can be used for extension of scalars. For a commutative ring, the tensor product of modules can be iterated to form the tensor algebra of a module, allowing one to define multiplication in the module in a universal way.
Balanced product
For a ring R, a right R-module M, a left R-module N, and an abelian group G, a map 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 to G is denoted by .
If , are balanced products, then each of the operations and −φ defined pointwise is a balanced product. This turns the set into an abelian group.
For M and N fixed, the map is a functor from the category of abelian groups to itself. The morphism part is given by mapping a group homomorphism to the function , which goes from to .
Remarks Properties (Dl) and (Dr) express biadditivity of φ, which may be regarded as distributivity of φ over addition. Property (A) resembles some associative property of φ. Every ring R is an R-bimodule. So the ring multiplication in R is an R-balanced product .
Definition For a ring R, a right R-module M, a left R-module N, the tensor product over R
is an abelian group together with a balanced product (as defined above)
which is universal in the following sense:Hazewinkel, et al. (2004), p. 95, Prop. 4.5.1
For every abelian group G and every balanced product there is a unique group homomorphism such that
As with all universal properties, the above property defines the tensor product uniquely up to a unique isomorphism: any other abelian group and balanced product with the same properties will be isomorphic to 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 ; see below for a construction.
The tensor product can also be defined as a representing object for the functor ; explicitly, this means there is a natural isomorphism:
This is a succinct way of stating the universal mapping property given above. (If a priori one is given this natural isomorphism, then can be recovered by taking and then mapping the identity map.)
Similarly, given the natural identification ,First, if , then the claimed identification is given by with . In general, has the structure of a right R-module by . Thus, for any -bilinear map f, f′ is R-linear . one can also define by the formula
This is known as the tensor-hom adjunction; see also .
For each x in M, y in N, one writes
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 ⊗R y but it is conventional to drop R here. Then, immediately from the definition, there are relations: x ⊗ (y + y′) = x ⊗ y + x ⊗ y′ (Dl⊗) (x + x′) ⊗ y = x ⊗ y + x′ ⊗ y (Dr⊗) (x ⋅ r) ⊗ y = x ⊗ (r ⋅ y) (A⊗)
The universal property of a tensor product has the following important consequence:
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.
Application of the universal property of tensor products
Determining whether a tensor product of modules is zero
In practice, it is sometimes more difficult to show that a tensor product of R-modules is nonzero than it is to show that it is 0. The universal property gives a convenient way for checking this.
To check that a tensor product is nonzero, one can construct an R-bilinear map to an abelian group such that . This works because if , then .
For example, to see that , is nonzero, take to be and . This says that the pure tensors as long as is nonzero in .
For equivalent modules
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 and the left and right actions by R on modules are considered to be equivalent, 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
Analogously, if N has a right action by a ring S, then becomes a right S-module.
Tensor product of linear maps and a change of base ring 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 and a module homomorphism f to the group homomorphism .
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:
induced by
The resulting map is surjective since pure tensors 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.
Several modules (This section need to be updated. For now, see 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
is that each trilinear map on
corresponds to a unique linear map
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.
Properties
Modules over general rings Let R1, R2, R3, R be rings, not necessarily commutative. For an R1-R2-bimodule M12 and a left R2-module M20, is a left R1-module. For a right R2-module M02 and an R2-R3-bimodule M23, is a right R3-module. (associativity) For a right R1-module M01, an R1-R2-bimodule M12, and a left R2-module M20 we have: Since R is an R-R-bimodule, we have with the ring multiplication as its canonical balanced product.
Modules over commutative rings Let R be a commutative ring, and M, N and P be R-modules. Then (in the below, "=" denotes canonical isomorphisms; this attitude is permissible since a tensor product is defined only up to unique isomorphisms) Identity Associativity Symmetry In fact, for any permutation σ of the set {1, ..., n}, there is a unique isomorphism: The first three properties (plus identities on morphisms) say that the category of R-modules, with R commutative, forms a symmetric monoidal category. Distribution over direct sums In fact, for an index set I of arbitrary cardinality. Since finite products coincide with finite direct sums, this implies: Distribution over finite products For any finitely many , Base extension If S is an R-algebra, writing , Proof: (using associativity in a general form) cf. . A corollary is: Distribution over localization For any multiplicatively closed subset S of R, as an -module, since is an R-algebra and . Commutativity with direct limits For any direct system of R-modules Mi, Adjunction A corollary is: Right-exactness If is an exact sequence of R-modules, then is an exact sequence of R-modules, where Tensor-hom relation There is a canonical R-linear map: which is an isomorphism if either M or P is a finitely generated projective module (see for the non-commutative case); more generally, there is a canonical R-linear map: which is an isomorphism if either or is a pair of finitely generated projective modules.
To give a practical example, suppose M, N are free modules with bases and . Then M is the direct sum and the same for N. By the distributive property, one has:
i.e., are the R-basis of . Even if M is not free, a free presentation of M can be used to compute tensor products.
The tensor product, in general, does not commute with inverse limit: on the one hand,
(cf. "examples"). On the other hand,
where are the ring of p-adic integers and the field of p-adic numbers. See also "profinite integer" for an example in the similar spirit.
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 is given by .
Tensor product of an R-module with the fraction field Let R be an integral domain with fraction field K. For any R-module M, as R-modules, where is the torsion submodule of M. If M is a torsion R-module then and if M is not a torsion module then . If N is a submodule of M such that is a torsion module then as R-modules by . In , if and only if or . In particular, where . where is the localization of the module at the prime ideal (i.e., the localization with respect to the nonzero elements).
Extension of scalars
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 , we have the natural isomorphism:
This says that the functor is a left adjoint to the forgetful functor , which restricts an S-action to an R-action. Because of this, is often called the extension of scalars from R to S. In representation theory, when R, S are group algebras, the above relation becomes Frobenius reciprocity.
Examples , for any R-algebra S (i.e., a free module remains free after extending scalars.) For a commutative ring and a commutative R-algebra S, we have: in fact, more generally, where is an ideal. Using , the previous example and the Chinese remainder theorem, we have as rings This gives an example when a tensor product is a direct product. .
Examples The structure of a tensor product of quite ordinary modules may be unpredictable.
Let G be an abelian group in which every element has finite order (that is G is a torsion abelian group; for example G can be a finite abelian group or ). Then:Example 3.6 of http://www.math.uconn.edu/~kconrad/blurbs/linmultialg/tensorprod.pdf