Open main menu

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. 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 productEdit

For a ring R, a right R-module M, a left R-module N, and an abelian group G, a map φ: M × NG 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 : GG to the function φgφ, which goes from LR(M, N; G) to LR(M, N; G′).

Remarks
  1. Property (Dl) states the left and property (Dr) the right distributivity of φ over addition.
  2. Property (A) resembles some associative property of φ.
  3. Every ring R is an R-R-bimodule. So the ring multiplication (r, r′) ↦ rr in R is an R-balanced product R × RR.

DefinitionEdit

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:[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 object and balanced product with the same properties will be isomorphic to MR N and ⊗. Indeed, the mapping ⊗ is called canonical, or more explicitly: the canonical mapping (or balanced product) of the tensor product.[2]

The definition does not prove the existence of MR 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. (if a priori one is given this is natural isomorphism, then   can be recovered by taking   and then mapping the identity map.)

Similarly, given the natural identification  ,[3] one can also define MR N by the formula

 

This is known as the tensor-hom adjunction; see also § Properties.

For each x in M, y in N, one writes

xy

for the image of (x, y) under the canonical map  . It is often called a pure tensor. Strictly speaking, the correct notation would be xR y but it is conventional to drop R here. Then, immediately from the definition, there are relations:

x ⊗ (y + y′) = xy + xy (Dl)
(x + x′) ⊗ y = xy + x′ ⊗ y (Dr)
(xr) ⊗ y = x ⊗ (ry) (A)

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 MN 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:

 

induced by

 [4]

The resulting map is surjective since pure tensors xy 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.

See also: Tensor product § Tensor product of linear maps.

Several modulesEdit

(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

M1M2M3

is that each trilinear map on

M1 × M2 × M3Z

corresponds to a unique linear map

M1M2M3Z.

The binary tensor product is associative: (M1M2) ⊗ M3 is naturally isomorphic to M1 ⊗ (M2M3). The tensor product of three modules defined by the universal property of trilinear maps is isomorphic to both of these iterated tensor products.

PropertiesEdit

Modules over general ringsEdit

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:[5]
 
  • Since R is an R-R-bimodule, we have   with the ring multiplication   as its canonical balanced product.

Modules over commutative ringsEdit

Let R be a commutative ring, and M, N and P be R-modules. Then

  • (identity)  
  • (associativity)  [6] Thus   is well-defined.
  • (symmetry)   In fact, for any permutation σ of the set {1, ..., n}, there is a unique isomorphism:
 
  • (distributive property)   In fact,
 
for an index set I of arbitrary cardinality.
  • (commutes with finite product) for any finitely many  ,
 
  • (commutes with localization) for any multiplicatively closed subset S of R,
 
as  -module. Since   is an R-algebra and  , this is a special case of:
  • (commutes with base extension) If S is an R-algebra, writing  ,
 [7]
cf. § Extension of scalars.
  • (commutes with direct limit) for any direct system of R-modules Mi,
 
  • (tensoring is right exact) if
 
is an exact sequence of R-modules, then
 
is an exact sequence of R-modules, where   This is a consequence of:
  • (adjoint relation)  .
  • (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 § As linearity-preserving maps for the non-commutative case);[8] 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

 [9]

where   is given by   See also: tensor-hom adjunction.

Tensor product of an R-module with the fraction fieldEdit

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 scalarsEdit

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 the representation theory, when R, S are group algebras, the above relation becomes the Frobenius reciprocity.

ExamplesEdit

  •   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.
  •  

ExamplesEdit

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:[10]

 

Indeed, any   is of the form

 

If   is the order of  , then we compute:

 

Similarly, one sees

 

Here are some useful identities: Let R be a commutative ring, I, J ideals, M, N R-modules. Then

  1.  . If M is flat,  .
  2.  
  3.  .

Proof: Tensoring with M the exact sequence   gives

 

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  

ConstructionEdit

The construction of MN takes a quotient of a free abelian group with basis the symbols mn, 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

  1. m ∗ (n + n′) + mn + mn
  2. −(m + m′) ∗ n + mn + m′ ∗ n
  3. (m · r) ∗ nm ∗ (r · n)

where m, m′ in M, n, n′ in N, and r in R. The quotient map which takes mn =(m, n) to the coset containing mn; 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 ⋅ (mn) − m ∗ (rn). Alternately, the general construction can be given a Z(R)-module structure by defining the scalar action by r ⋅ (mn) = m ⊗ (rn) 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 mapsEdit

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 moduleEdit

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.[11] The canonical structure is the pointwise operations of addition and scalar multiplication. Thus, E is the set of all R-linear maps ER (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 EE∗∗ 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 pairingEdit

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 mapEdit

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 : FR E → HomR(E, F) such that θ(fe′) is the map ef ⋅ ⟨e′, e.[12]
  • Given left R-module E and right R-module F, there is a canonical homomorphism θ : FR E → HomR(E, F) such that θ(fe) is the map e′ ↦ f ⋅ ⟨e, e′⟩.[13]

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 : FR E → LR(F × E, R) such that θ(f′ ⊗ e′) is the map (f, e) ↦ ⟨f, f′⟩ ⋅ ⟨e′, e.[citation needed] Thus, an element of a tensor product ξFR E may be thought of giving rise to or acting as an R-bilinear map F × ER.

TraceEdit

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).

As R-modules,   is the dual module of  [14]

To lighten the notation, put   and so  .[15] When p, q ≥ 1, for each (k, l) with 1 ≤ kp, 1 ≤ lq, 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.

Relationship to flat modulesEdit

In general,

 

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.

By fixing a right R module M, a functor

 

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

 

Unlike the Hom bifunctor   the tensor functor is covariant in both inputs.

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  

Additional structureEdit

If S and T are commutative R-algebras, then SR T will be a commutative R-algebra as well, with the multiplication map defined by (m1m2) (n1n2) = (m1n1m2n2) 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

rssr

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.

GeneralizationEdit

Tensor product of complexes of modulesEdit

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,

 [16]

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.)

Tensor product of sheaves of modulesEdit

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   are viewed as locally free sheaves on M.[17]

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.

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.

See alsoEdit

NotesEdit

  1. ^ Hazewinkel, et al. (2004), p. 95, Prop. 4.5.1
  2. ^ Bourbaki, ch. II §3.1
  3. ^ 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  
  4. ^ Bourbaki, ch. II §3.2.
  5. ^ Bourbaki, ch. II §3.8
  6. ^ The first three properties (plus identities on morphisms) say that the category of R-modules, with R commutative, forms a symmetric monoidal category.
  7. ^ Proof: (using associativity in a general form)  
  8. ^ Bourbaki, ch. II §4.4
  9. ^ Bourbaki, ch.II §4.1 Proposition 1
  10. ^ Example 3.6 of http://www.math.uconn.edu/~kconrad/blurbs/linmultialg/tensorprod.pdf
  11. ^ Bourbaki, ch. II §2.3
  12. ^ Bourbaki, ch. II §4.2 eq. (11)
  13. ^ Bourbaki, ch. II §4.2 eq. (15)
  14. ^ Helgason, Lemma 2.3'
  15. ^ This is actually the definition of differential one-forms, global sections of  , in Helgason, but is equivalent to the usual definition that does not use module theory.
  16. ^ May & ch. 12 §3
  17. ^ See also Encyclopedia of Mathematics - Tensor bundle

ReferencesEdit

  • Bourbaki, Algebra
  • Helgason, Sigurdur (1978), Differential geometry, Lie groups and symmetric spaces, Academic Press, ISBN 0-12-338460-5
  • Northcott, D.G. (1984), Multilinear Algebra, Cambridge University Press, ISBN 613-0-04808-4.
  • Hazewinkel, Michiel; Gubareni, Nadezhda Mikhaĭlovna; Gubareni, Nadiya; Kirichenko, Vladimir V. (2004), Algebras, rings and modules, Springer, ISBN 978-1-4020-2690-4.
  • Peter May (1999), A concise course in algebraic topology, University of Chicago Press.