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 (module homomorphisms). 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 and algebraic geometry. The universal property of the tensor product of vector spaces extends to more general situations in abstract algebra. It allows the study of bilinear or multilinear operations via linear operations. 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.
Contents
- Balanced product
- Definition
- Tensor product of linear maps and a change of base ring
- Several modules
- Properties
- Tensor product of an R module with the fraction field
- Extension of scalars
- Examples
- Construction
- As linear maps
- Dual module
- Duality pairing
- An element as a bilinear map
- Trace
- Example from differential geometry tensor field
- Relationship to flat modules
- Additional structure
- Tensor product of complexes of modules
- Tensor product of sheaves of modules
- References
Balanced product
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 the operations φ + ψ and −φ defined pointwise are each 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.
- Property (A) resembles some associative property of φ.
- Every ring R is an R-R-bimodule. So the ring multiplication (r, r′) ↦ r ⋅ r′ in R is an R-balanced product R × R → R.
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:
For every abelian group G and every balanced productAs 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 ⊗R N 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 ⊗R N; see below for a construction.
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:
This is a succinct way of stating the universal mapping property given above. (A priori, if one is given this is natural isomorphism, then
Similarly, given the natural identification
This is known as the tensor-hom adjunction; see also § Properties.
For each x in M, y in N, one writes
x ⊗ yfor the image of (x, y) under the canonical map
The universal property of a tensor product has the following important consequence:
Proof: For the first statement, let L be the subgroup of
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
to the whole
If R is not necessarily commutative but if M has a left action by a ring S (for example, R), then
If N has a right action by a ring S, then, in the analogous way,
Tensor product of linear maps and a change of base ring
Given linear maps
such that
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
induced by
See also: Tensor product § Tensor product of linear maps.
Several modules
(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 ⊗ M3is that each trilinear map on
M1 × M2 × M3 → Zcorresponds 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.
Properties
Let R1, R2, R3, R be rings, not necessarily commutative.
Let R be a commutative ring, and M, N and P be R-modules. Then
To give a practical example, suppose M, N are free modules with bases
i.e.,
The tensor product, in general, does not commute with inverse limit: on the one hand,
(cf. "examples"). On the other hand,
where
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
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
Tensor product of an R-module with the fraction field
Let R be an integral domain with fraction field K.
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
This says that the functor
Example:
Example: For a commutative ring
Example: Using the fact
(this gives an example when a tensor product is a direct product.)
Example:
See also: Weil restriction.
Examples
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 Q/Z). Then
Indeed, any element x of
where
Similarly, one sees
Here are some useful identities: Let R be a commutative ring, I, J ideals, M, N R-modules. Then
-
R / I ⊗ R M = M / I M . If M is flat,I M = I ⊗ R M . -
M / I M ⊗ R / I N / I N = M ⊗ R N ⊗ R R / I . -
R / I ⊗ R R / J = R / ( I + J ) .
Proof: Tensoring with M the exact sequence
where f is given by
Example: If G is an abelian group,
Example:
Example: Let
Example: Consider
whose kernel is generated by elements of the form
the kernel actually vanishes; hence,
Example: We propose to compare
Construction
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
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.
As linear maps
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.
Dual module
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.
Duality pairing
We denote the natural pairing of its dual E∗ and a right R-module E, or of a left R-module F and its dual F∗ as
The pairing is left R-linear in its left argument, and right R-linear in its right argument:
An element as a (bi)linear map
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.
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.
Trace
Let R be a commutative ring and E an R-module. Then there is a canonical R-linear map:
induced by
If E is a finitely generated projective R-module, then one can identify
When R is a field, this is the usual trace of a linear transformation.
Example from differential geometry: tensor field
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
As R-modules,
To lighten the notation, put
where
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.
Relationship to flat modules
In general,
By fixing a right R module M, a functor
It can be shown that M⊗- and -⊗N are always right exact functors, but not necessarily left exact (
If {mi}i∈I and {nj}j∈J are generating sets for M and N, respectively, then {mi⊗nj}i∈I,j∈J will be a generating set for M⊗N. Because the tensor functor M⊗R- 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 M⊗R- 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 -⊗R- is exact in both positions, and the two given generating sets are bases, then
When the tensor products are taken over a field F so that -⊗- is exact in both positions, and the generating sets are bases of M and N, it is true that
See also: pure submodule.
Additional structure
If S and T are commutative R-algebras, then S ⊗R T 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 − srof 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.
Tensor product of complexes of modules
If X, Y are complexes of R-modules (R a commutative ring), then their tensor product is the complex given by
with the differential given by: for x in Xi and y in Yj,
For example, if C is a chain complex of flat abelian groups and if G is an abelian group, then the homology group of
Tensor product of sheaves of modules
In this setup, for example, one can define a tensor field on a smooth manifold M as a (global or local) section of the tensor product (called tensor bundle)
where O is the sheaf of rings of smooth functions on M and the bundles
The exterior bundle on M is the subbundle of the tensor bundle consisting of all antisymmetric covariant tensors. Sections of the exterior bundle are differential forms on M.
See also: Tensor product bundle.
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.