Tensor product of modules

In mathematics, the tensor product of modules is a construction that allows arguments about bilinear maps (roughly speaking, "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 left-module and a right-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.

Multilinear mappings

For a ring "R", a right "R"-module "M_R", a left "R"-module "_RN", and an abelian group "Z", a bilinear map or balanced product from "M" × "N" to "Z" is a function "φ": "M" × "N" → "Z" such that for all "m","m′" in "M", "n","n′" in "N", and "r" in "R":

# "φ"("m"+"m′","n") = "φ"("m","n") + "φ"("m′","n")
# "φ"("m","n"+"n′") = "φ"("m","n") + "φ"("m","n′")
# "φ"("m"·"r","n") = "φ"("m","r"·"n")

The set of all such bilinear maps from "M" × "N" to "Z" is denoted by Bilin("M","N";"Z").

Property 3 differs slightly from the definition for vector spaces. This is necessary because "Z" is only assumed to be an abelian group, so "r"·"φ"("m","n") would not make sense.

If "φ", "ψ" are bilinear maps, then "φ" + "ψ" is a bilinear map, and -"φ" is a bilinear map, when these operations are defined pointwisely. This turns the set Bilin("M","N";"Z") into an abelian group. The neutral element is the zero mapping.

For "M" and "N" fixed, the map "Z" ↦ Bilin("M","N";"Z") 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" : "Z" → "Z′" to the function ["φ" ↦ "g" ∘ "φ"] , which goes from Bilin("M","N";"Z") to Bilin("M","N";"Z′").

Definition

Let "M","N" and "R" be as in the previous section. The tensor product over "R"

:$M\; otimes\_R\; N$

is an abelian group together with a bilinear map (in the sense defined above)

:$otimes\; :\; M\; imes\; N\; o\; M\; otimes\_\{R\}\; N$

which is universal in the following sense:

For every abelian group "Z" and every bilinear
$f:\; M\; imes\; N\; o\; Z,$ there is a unique group homomorphism :$ilde\{f\}:\; M\; otimes\_R\; N\; o\; Z$such that:$ilde\{f\}\; circ\; otimes\; =\; f.$

As with all universal properties, the above property defines the tensor product up to (a unique) isomorphism. It does not prove its existence; see below for a construction.

The tensor product can also be defined as a representing object for the functor "Z" → Bilin_"R"("M","N";"Z"). This is equivalent to the universal mapping property given above.

Examples

Consider the rational numbers Q and the integers modulo "n" Z_"n". As with any abelian group, both can be considered as modules over the integers, Z.Let "B": Q × Z_"n" → "M" be a Z-bilinear operator. Then "B"("q", "k") = "B"("q"/"n", "nk") = "B"("q"/"n", 0) = 0, so every bilinear operator is identically zero. Therefore, if we define "P" to be the trivial module, and "T" to be the zero bilinear function, then we see that the properties for the tensor product are satisfied. Therefore, the tensor product of Q and Z_"n" is {0}.

Construction

To construct the tensor product, we can proceed as for vector spaces; the construction carries over without any changes. Roughly speaking, the construction of "M" ⊗ "N" takes a quotient of a free module with basis the symbols "m" ⊗ "n" for "m" in "M" and "n" in "N" by the submodule generated by all elements of the form:
# −("m"+"m′") ⊗ "n" + "m" ⊗ "n" + "m′" ⊗ "n"
# −"m" ⊗ ("n"+"n′") + "m" ⊗ "n" + "m" ⊗ "n′"
# ("m"·"r") ⊗ "n" − "m" ⊗ ("r"·"n")where "m","m′" in "M", "n","n′" in "N", and "r" in "R". In the quotient module, the function which takes ("m","n") to "m" ⊗ "n" is bilinear, and the submodule has been chosen minimally so that this map is bilinear.

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 "Z" which is both linear and bilinear is the zero map.

everal modules

It is possible to generalize the definition to a tensor product of any number of spaces. For example, the universal property of

:"M"₁ ⊗ "M"₂ ⊗ "M"₃

is that every trilinear map on

:"M"₁ × "M"₂ × "M"₃ → "Z"

corresponds to a unique linear map

:"M"₁ ⊗ "M"₂ ⊗ "M"₃ → "Z".

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.

Additional structure

The tensor product, as defined, is an abelian group, not an "R"-module. In general, it is impossible to put an "R"-module structure on the tensor product. However, if "M" is an ("S","R")-bimodule, then "M"⊗"N" is a left "S"-module, and similarly, if "N" is an ("R","T")-bimodule, then "M"⊗"N" is a right "T"-module. If "M" and "N" each have bimodule structures as above, then "M"⊗"N" is an ("S","T")-bimodule. In particular, if "R" is a commutative ring, then "M"⊗"N" will always be an "R"-module.

If {"m"_"i"}_"i"∈"I" and {"n"_"j"}_"j"∈"J" are generating sets for "M" and "N", respectively, then {"m"_"i"⊗"n"_"j"}_{"i"∈"I","j"∈"J"} will be a generating set for "M"⊗"N". Because the tensor product is right exact, not in general exact, this may not be a minimal generating set, even if the original generating sets are minimal. However, if the tensor products are taken over a field, then we are in the case of vector spaces as above, and if the two given generating sets are bases, we will get a basis for "M" ⊗ "N".

If "S" and "T" are commutative "R"-algebras, then "S" ⊗ "T" will be a commutative "R"-algebra as well, with the multiplication map defined by ("m"₁ ⊗ "m"₂)("n"₁ ⊗ "n"₂) = ("m"₁"n"₁ ⊗ "m"₂"n"₂) and extended by linearity. In this setting, the tensor product become a fibered coproduct in the category of "R"-algebras. Note that any commutative ring is a "Z"-algebra, so we may always take "M" ⊗_Z "N".

It is also possible to generalize the definition to tensor products of modules over the same ring. If the ring is non-commutative, distinguish right modules and left modules. We will write "_RM" for a left module, and "M_R" for a right module. If a module "M" has both a left module structure over a ring "R" and a right module structure over a ring "S", and in addition for every "m" in "M", "r" in "R" and "s" in "S" we have "r(ms) = (rm)s", then we will say "M" is a bimodule, and will denote it by "_RM_S". Note that every left-module is a bimodule with Z acting by "mn" = "m + m + ... + m" as the right ring, and vice versa.

When defining the tensor product, the ring in question should be specified: most modules can be considered as modules over several different rings or over the same ring with a different actions of the ring on the module elements.

The most general form of the tensor product definition is as follows: let "M_R" and "_RN" be a right and a left module, respectively. Their tensor product over "R" is an abelian group "P" together with an "R"-bilinear operator "T": "M" × "N" → "P" such that for every "R"-bilinear operator "B": "M" × "N" → "O" there is a unique group homomorphism "L": "P" → "O" such that "L" o "T" = "B". "P" need not be a module over "R". However, if "_S1M_R" is an "S₁-R"-bimodule, then there is a unique left "S₁"-module structure on "P" which is compatible with "T". Similarly, if "_RM_S2" is an "R-S₂"-bimodule, then there is a unique right "S₂"-module structure on "P" which is compatible with "T". If "M" and "N" are both bimodules, then "P" is also a bimodule, again in a unique way. ("P", "T") are unique up to a unique isomorphism, and are called the "tensor product" of "M" and "N".

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" − "sr"

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.

ee also

*Tor functor
*Tensor product of algebras
*Tensor product of fields

References

* Multilinear Algebra, Northcott D.G, 1984 Cambridge University Press