Tensor product of modules

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.
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 "φ", "ψ" 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 .
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:
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 is natural isomorphism, then formula_8 can be recovered by taking formula_9 and then mapping the identity map.)
Similarly, given the natural identification formula_10, 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 formula_12. 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:
The universal property of a tensor product has the following important consequence:
Proof: For the first statement, let "L" be the subgroup of formula_2 generated by elements of the form in question, formula_14 and "q" the quotient map to "Q". We have: formula_15 as well as formula_16. 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. formula_17
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 formula_18 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 formula_18 is nonzero, one can construct an "R"-bilinear map formula_20 to an abelian group formula_21 such that formula_22. This works because if formula_23, then formula_24.
For example, to see that formula_25, is nonzero, take formula_21 to be formula_27 and formula_28. This says that the pure tensors formula_29 as long as formula_30 is nonzero in formula_31.
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 formula_32 can naturally be furnished with the "R"-scalar multiplication by extending
to the whole formula_2 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, formula_2 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 formula_2 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 formula_2 becomes a right "S"-module.
Tensor product of linear maps and a change of base ring.
Given linear maps formula_40 of right modules over a ring "R" and formula_41 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 formula_44 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 formula_47 this shows every tensor product of modules is a quotient of a tensor product of abelian groups.
See also: .
Several modules.
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: ("M"1 ⊗ "M"2) ⊗ "M"3 is naturally isomorphic to "M"1 ⊗ ("M"2 ⊗ "M"3). 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 "R"1, "R"2, "R"3, "R" be rings, not necessarily commutative.
Modules over commutative rings.
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 formula_77 and formula_78. Then "M" is the direct sum formula_79
and the same for "N". By the distributive property, one has:
i.e., formula_81 are the "R"-basis of formula_2. 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 formula_85 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 formula_2; in particular, formula_87 would not even be defined. If "M", "N" are bi-modules, then formula_2 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 formula_87.
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 formula_92 is given by formula_93 See also: tensor-hom adjunction.
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 formula_111, we have the natural isomorphism:
This says that the functor formula_113 is a left adjoint to the forgetful functor formula_114, which restricts an "S"-action to an "R"-action. Because of this, formula_115 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.
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 formula_124). Then:
Indeed, any formula_126 is of the form
If formula_128 is the order of formula_129, then we compute:
Similarly, one sees
Here are some identities useful for calculation: Let "R" be a commutative ring, "I", "J" ideals, "M", "N" "R"-modules. Then
Example: If "G" is an abelian group, formula_136; this follows from 1.
Example: formula_137; this follows from 3. In particular, for distinct prime numbers "p", "q",
Tensor products can be applied to control the order of elements of groups. Let G be an abelian group. Then the multiples of 2 in
are zero.
Example: Let formula_140 be the group of "n"-th roots of unity. It is a cyclic group and cyclic groups are classified by orders. Thus, non-canonically, formula_141 and thus, when "g" is the gcd of "n" and "m",
Example: Consider formula_143 Since formula_144 is obtained from formula_145 by imposing formula_146-linearity on the middle, we have the surjection
whose kernel is generated by elements of the form formula_148
where "r", "s", "x", "u" are integers and "s" is nonzero. Since
the kernel actually vanishes; hence, formula_150
However, consider formula_151 and formula_152. As formula_153-vector space, formula_151 has dimension 4, but formula_152 has dimension 2. 
Thus, formula_151 and formula_152 are not isomorphic. 
Example: We propose to compare formula_158 and formula_159. Like in the previous example, we have: formula_160 as abelian group and thus as formula_161-vector space (any formula_47-linear map between formula_161-vector spaces is formula_161-linear). As formula_161-vector space, formula_166 has dimension (cardinality of a basis) of continuum. Hence, formula_167 has a formula_161-basis indexed by a product of continuums; thus its formula_161-dimension is continuum. Hence, for dimension reason, there is a non-canonical isomorphism of formula_161-vector spaces:
Consider the modules formula_172 for formula_173 irreducible polynomials such that formula_174 Then,
Another useful family of examples comes from changing the scalars. Notice that
Good examples of this phenomenon to look at are when formula_177
Construction.
The construction of takes a quotient of a free abelian group with basis the symbols , used here to denote the ordered pair , 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 to the coset containing ; 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 formula_2 is the quotient group of formula_183 by the subgroup generated by formula_184, where formula_185 is the image of formula_186 under formula_187 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 . Alternately, the general construction can be given a Z("R")-module structure by defining the scalar action by 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 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 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 (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 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 through linearity by formula_194; 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 formula_195 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 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 formula_198 means tensoring "p" times over "R". By definition, an element of formula_199 is a tensor field of type ("p", "q").
As "R"-modules, formula_200 is the dual module of formula_201
To lighten the notation, put formula_202 and so formula_203. When "p", "q" ≥ 1, for each ("k", "l") with 1 ≤ "k" ≤ "p", 1 ≤ "l" ≤ "q", there is an "R"-multilinear map:
where formula_205 means formula_206 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 .
Relationship to flat modules.
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 formula_212 the tensor functor is covariant in both inputs.
It can be shown that formula_213 and formula_214 are always right exact functors, but not necessarily left exact (formula_215 where the first map is multiplication by formula_216, is exact but not after taking the tensor with formula_217). By definition, a module "T" is a flat module if formula_218 is an exact functor.
If formula_219 and formula_220 are generating sets for "M" and "N", respectively, then formula_221 will be a generating set for formula_222 Because the tensor functor formula_213 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 formula_213 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 formula_225 is exact in both positions, and the two given generating sets are bases, then formula_221 indeed forms a basis for formula_227
Additional structure.
If "S" and "T" are commutative "R"-algebras, then will be a commutative "R"-algebra as well, with the multiplication map defined by 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
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
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.
Generalization.
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 "X""i" and "y" in "Y""j",
For example, if "C" is a chain complex of flat abelian groups and if "G" is an abelian group, then the homology group of formula_230 is the homology group of "C" with coefficients in "G" (see also: universal coefficient theorem.)
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 formula_232 are viewed as locally free sheaves on "M".
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.