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 (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.

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:

φ(m, n + n′) = φ(m, n) + φ(m, n′)	(Dlφ)
φ(m + m′, n) = φ(m, n) + φ(m′, n)	(Drφ)
φ(m ⋅ r, n) = φ(m, r ⋅ n)	(Aφ)

The set of all such balanced products over R from M × N to G is denoted by L_R(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 L_R(M, N; G) into an abelian group.

For M and N fixed, the map G ↦ L_R(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 L_R(M, N; G) to L_R(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′) ↦ 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

$M \otimes_R N$

is an abelian group together with a balanced product (as defined above)

$\otimes : M \times N \to M \otimes_{R} N$

which is universal in the following sense:^[1]

For every abelian group G and every balanced product

$f: M \times N \to G\,$

there is a unique group homomorphism

$\tilde{f}: M \otimes_R N \to G$

such that

$\tilde{f} \circ \otimes = f.$

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 M ⊗_R 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 M ⊗_R N; see below for a construction.

The tensor product can also be defined as a representing object for the functor G → L_R(M,N;G); explicitly, this means there is a natural isomorphism:

$\operatorname{Hom}_{\mathbb{Z}} (M \otimes_R N, G) \simeq \operatorname{L}_R(M, N; G), \, g \mapsto g \circ \otimes.$

This is a succinct way of stating the universal mapping property given above. (A priori, if one is given this is natural isomorphism, then $\otimes$ can be recovered by taking $G = M \otimes_R N$ and then mapping the identity map.)

Similarly, given the natural identification $\operatorname{L}_R(M, N; G) = \operatorname{Hom}_R(M, \operatorname{Hom}_{\mathbb{Z}}(N, G))$,^[3] one can also define M ⊗_R N by the formula

$\operatorname{Hom}_{\mathbb{Z}} (M \otimes_R N, G) \simeq \operatorname{Hom}_R(M, \operatorname{Hom}_{\mathbb{Z}}(N, G))$.

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

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

x ⊗ y

for the image of (x, y) under the canonical map $\otimes: M \times N \to M \otimes_R N$. 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:

Proposition — Every element of $M \otimes_R N$ can be written, non-uniquely, as

$\sum_i x_i \otimes y_i, \, x_i \in M, y_i \in N.$

In other words, the image of $\otimes$ generates $M \otimes_R N$. Furthermore, if f is a function defined on elements $x \otimes y$ with values in an abelian group G, then f extends uniquely to the homomorphism defined on the whole $M \otimes_R N$ if and only if $f(x \otimes y)$ is Z-bilinear in x and y.

Proof: For the first statement, let L be the subgroup of $M \otimes_R N$ generated by elements of the form in question, $Q = (M \otimes_R N) / L$ and q the quotient map to Q. We have: $0 = q \circ \otimes$ as well as $0 = 0 \circ \otimes$. 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. $\square$

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 $M \otimes_R N$ can naturally be furnished with the R-scalar multiplication by extending

$r \cdot (x \otimes y) := rx \otimes y = x \otimes ry$,

to the whole $M \otimes_R N$ 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, $M \otimes_R N$ satisfies a universal property similar to the above: for any R-module G, there is a natural isomorphism:

$\operatorname{Hom}_R(M \otimes_R N, G) \simeq \{$ R-bilinear maps from $M \times N$ to $G \}, \, g \mapsto g \circ \otimes$.

If R is not necessarily commutative but if M has a left action by a ring S (for example, R), then $M \otimes_R N$ can be given the left S-module structure, like above, by the formula

$s \cdot (x \otimes y) := sx \otimes y$.

If N has a right action by a ring S, then, in the analogous way, $M \otimes_R N$ becomes a right S-module.

Tensor product of linear maps and a change of base ring

Given linear maps $f: M \to M'$ of right modules over a ring R and $g: N \to N'$ of left modules, there is a unique group homomorphism

$f \otimes g: M \otimes _R N \to M' \otimes_R N'$

such that $(f \otimes g)(x \otimes y) = f(x) \otimes g(y)$.

The construction has a consequence that tensoring is a functor: each right R-module M determines the functor

$M \otimes_R -: R-\mathbf{Mod} \to \mathbf{Ab}$

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 $f: R \to S$ 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:

$M \otimes_R N \to M \otimes_S N$

induced by $M \times N \overset{\otimes_S} \to M \otimes_S N$.^[4] (The resulting map is surjective since pure tensors x ⊗ y generate the whole module.) In particular, taking R to be Z, 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 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 ⊗ M₃

is that each trilinear map on

M₁ × M₂ × M₃ → Z

corresponds 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.

• For a R₁-R₂-bimodule M₁₂ and a left R₂-module M₂₀ the tensor product

$M_{12}\otimes_{R_2} M_{20}$

is a left R₁-module.

• For a right R₂-module M₀₂ and an R₂-R₃-bimodule M₂₃ the tensor product

$M_{02}\otimes_{R_2} M_{23}$

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

$(M_{01} \otimes_{R_1} M_{12}) \otimes_{R_2} M_{20} = M_{01} \otimes_{R_1} (M_{12}\otimes_{R_2} M_{20})$.^[5]

• Since R is an R-R-bimodule, we have the tensor product

$R\otimes_R R = R$

with the ring multiplication

$mn =: m\otimes_R n$

as its canonical balanced product.

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

• (identity)

$R \otimes_R M = M$

• (associativity)

$(M \otimes_R N) \otimes_R P = M \otimes_R (N \otimes_R P)$^[6]

Thus

$M \otimes_R N \otimes_R P := M \otimes_R (N \otimes_R P)$

is well-defined.

• (symmetry)

$M \otimes_R N = N \otimes_R M$

In fact, for any permutation σ of the set { 1, 2, ..., n }, there is a unique isomorphism

$M_1 \otimes_R \cdots \otimes_R M_n \to M_{\sigma(1)} \otimes_R \cdots \otimes_R M_{\sigma(n)}$

under which $x_1 \otimes \cdots \otimes x_n$ maps to $x_{\sigma(1)} \otimes \cdots \otimes x_{\sigma(n)}.$

• (distributive property)

$M \otimes_R (N \oplus P) = (M \otimes_R N) \oplus (M \otimes_R P)$

In fact,

$M \otimes_R (\bigoplus_{i \in I} N_i) = \bigoplus_{i \in I} M \otimes_R N_i$

for an index set I of arbitrary cardinality.

• (commutes with finite product) for any finitely many $N_i$,

$M \otimes_R \prod_{i = 1}^n N_i = \prod_{i = 1}^nM \otimes_R N_i$

• (commutes with localization) for any multiplicatively closed subset S of R,

$S^{-1}(M \otimes_R N) = S^{-1}M \otimes_{S^{-1}R} S^{-1}N$

as $S^{-1} R$-module. Since $S^{-1} R$ is an R-algebra and $S^{-1} - = S^{-1} R \otimes_R -$, this is a special case of:

• (commutes with base extension) If S is an R-algebra, writing $-_{S} = S \otimes_R -$,

$(M \otimes_R N)_S = M_S \otimes_S N_S;$^[7]

cf. § Extension of scalars.

• (commutes with direct limit) for any direct system of R-modules M_i,

$(\varinjlim M_i) \otimes_R N = \varinjlim (M_i \otimes_R N).$

• (tensoring is right exact) if $0 \to N' \overset{f}\to N \overset{g}\to N'' \to 0$ is an exact sequence of R-modules, then

$M \otimes_R N' \overset{1 \otimes f}\to M \otimes_R N \overset{1 \otimes g}\to M \otimes_R N'' \to 0$

is an exact sequence of R-modules, where $(1 \otimes f)(x \otimes y) = x \otimes f(y).$ This is a consequence of:

• (adjoint relation) $\operatorname{Hom}_R(M \otimes_R N, P) = \operatorname{Hom}_R(M, \operatorname{Hom}_R(N, P))$.
• (tensor-hom relation) there is a canonical R-linear map:

$\operatorname{Hom}_R(M, N) \otimes P \to \operatorname{Hom}_R(M, N \otimes P),$

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:

$\operatorname{Hom}_R(M, N) \otimes \operatorname{Hom}_R(M', N') \to \operatorname{Hom}_R(M \otimes M', N \otimes N')$

which is an isomorphism if either $(M, N)$ or $(M, M')$ is a pair of finitely generated projective modules.

To give a practical example, suppose M, N are free modules with bases $e_i, i \in I$ and $f_j, j \in J$. Then M is the direct sum $M = \bigoplus_{i \in I} R e_i$ and the same for N. By the distributive property, one has:

$M \otimes_R N = \bigoplus_{i, j} R(e_i \otimes f_j)$;

i.e., $e_i \otimes f_j, \, i \in I, j \in J$ are the R-basis of $M \otimes_R N$. 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,

$\mathbb{Q} \otimes_{\mathbb{Z}} \mathbb{Z}/p^n = 0$

(cf. "examples"). On the other hand,

$(\varprojlim \mathbb{Z}/p^n) \otimes_{\mathbb{Z}} \mathbb{Q} = \mathbb{Z}_p \otimes_{\mathbb{Z}} \mathbb{Q} = \mathbb{Z}_p[p^{-1}] = \mathbb{Q}_p$

where $\mathbb{Z}_p, \mathbb{Q}_p$ 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 $M \otimes_R N$; in particular, $N \otimes_R M$ would not even be defined. If M, N are bi-modules, then $M \otimes_R N$ 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 $N \otimes_R M$.

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

$(M \otimes_R N) \otimes_S P = M \otimes_R (N \otimes_S P)$

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

$\operatorname{Hom}_S(M \otimes_R N, P) = \operatorname{Hom}_R(M, \operatorname{Hom}_S(N, P)), \, f \mapsto f'$^[9]

where $f'$ is given by $f'(x)(y) = f(x \otimes y).$ See also: tensor-hom adjunction.

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 $\operatorname{Hom}_S (S, -) = -$, we have the natural isomorphism:

$\operatorname{Hom}_S (M \otimes_R S, P) = \operatorname{Hom}_R (M, \operatorname{Res}_R(P)).$

This says that the functor $- \otimes_R S$ is a left adjoint to the forgetful functor $\operatorname{Res}_R$, which restricts an S-action to an R-action. Because of this, $- \otimes_R S$ 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.

Example: $R^n \otimes_R S = S^n$ for any R-algebra S (i.e., a free module remains free after extending scalars.)

Example: For a commutative ring $R$ and a commutative R-algebra S, we have:

• $S \otimes_R R[x_1, \dots, x_n] = S[x_1, \dots, x_n]$; in fact, more generally,
• $S \otimes_R (R[x_1, \dots, x_n]/I) = S[x_1, \dots, x_n]/ IS[x_1, \dots, x_n], \, I$ an ideal.

Example: Using the fact $\mathbb{C} = \mathbb{R}[x]/(x^2 + 1)$, by the previous example and the Chinese remainder theorem, we have: as ring

$\mathbb{C} \otimes_{\mathbb{R}} \mathbb{C} = \mathbb{C}[x]/(x^2 + 1) = \mathbb{C}[x]/(x+\mathrm i) \times \mathbb{C}[x]/(x-\mathrm i) = \mathbb{C}^2$

(this gives an example when a tensor product is a direct product.)

Example: $\mathbb{R} \otimes_{\mathbb{Z}} \mathbb{Z}[\mathrm i] = \mathbb{C}[\mathrm i] = \mathbb{C}$ where i is the imaginary unit.

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

$\mathbb{Q} \otimes_{\mathbb{Z}} G = 0$.^[10]

Indeed, any element x of $\mathbb{Q} \otimes_{\mathbb{Z}} G$ is of the form

$x = \sum_i r_i \otimes g_i$

where $r_i \in \mathbb{Q}, g_i \in G$. If $n_i$ is the order of $g_i$, then we compute:

$x = \sum (r_i / n_i )n_i \otimes g_i = \sum r_i / n_i \otimes n_i g_i = 0.$

Similarly, one sees

$\mathbb{Q}/\mathbb{Z} \otimes_{\mathbb{Z}} \mathbb{Q}/\mathbb{Z} = 0.$

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

1. $R/I \otimes_R M = M/IM$. If M is flat, $IM = I \otimes_R M$.
2. $M/IM \otimes_{R/I} N/IN = M \otimes_R N \otimes_R R/I.$
3. $R/I \otimes_R R/J = R/(I + J)$.

Proof: Tensoring with M the exact sequence $0 \to I \to R \to R/I \to 0$ gives

$I \otimes_R M \overset{f}\to R \otimes_R M = M \to R/I \otimes_R M \to 0$

where f is given by $i \otimes x \mapsto ix$. 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

$R/I \otimes_R R/J = {R/J \over I(R/J) }= {R/J \over (I + J)/J} = R/(I+J)$. $\square$

Example: If G is an abelian group, $G \otimes_{\mathbb{Z}} \mathbb{Z}/n = G/nG$; this follows from 1.

Example: $\mathbb{Z}/n \otimes_{\mathbb{Z}} \mathbb{Z}/m = \mathbb{Z}/{\operatorname{gcd}(n, m)}$; this follows from 3.

Example: Let $\mu_n$ be the group of n-th roots of unity. It is a cyclic group and cyclic groups are classified by orders. Thus, non-canonically, $\mu_n \approx \mathbb{Z}/n$ and thus, when g is the gcd of n and m,

$\mu_n \otimes_{\mathbb{Z}} \mu_m \approx \mu_g.$

Example: Consider $\mathbb{Q} \otimes_{\mathbb{Z}} \mathbb{Q}$. Since $\mathbb{Q} \otimes_{\mathbb{Q}} \mathbb{Q}$ is obtained from $\mathbb{Q} \otimes_{\mathbb{Z}} \mathbb{Q}$ by imposing $\mathbb{Q}$-linearity on the middle, we have the surjection

$\mathbb{Q} \otimes_{\mathbb{Z}} \mathbb{Q} \to \mathbb{Q} \otimes_{\mathbb{Q}} \mathbb{Q}$

whose kernel is generated by elements of the form ${r \over s} x \otimes y - x \otimes {r \over s} y$ where r, s, x, u are integers and s is nonzero. Since

${r \over s} x \otimes y = {r \over s} x \otimes {s \over s} y = x \otimes {r \over s} y,$

the kernel actually vanishes; hence, $\mathbb{Q} \otimes_{\mathbb{Z}} \mathbb{Q} = \mathbb{Q} \otimes_{\mathbb{Q}} \mathbb{Q} = \mathbb{Q}.$

Example: We propose to compare $\mathbb{R} \otimes_{\mathbb{Z}} \mathbb{R}$ and $\mathbb{R} \otimes_{\mathbb{R}} \mathbb{R}$. Like in the previous example, we have: $\mathbb{R} \otimes_{\mathbb{Z}} \mathbb{R} = \mathbb{R} \otimes_{\mathbb{Q}} \mathbb{R}$ as abelian group and thus as Q-vector space (any Z-linear map between Q-vector spaces is Q-linear). As Q-vector space, $\mathbb{R}$ has dimension (cardinarity of a basis) of continuum. Hence, $\mathbb{R} \otimes_{\mathbb{Q}} \mathbb{R}$ has a Q-basis indexed by a product of continuums; thus its Q-dimension is continuum. Hence, for dimension reason, there is a non-canonical isomorphism of Q-vector spaces:

$\mathbb{R} \otimes_{\mathbb{Z}} \mathbb{R} \approx \mathbb{R} \otimes_{\mathbb{R}} \mathbb{R}$.

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

1. −m ∗ (n + n′) + m ∗ n + m ∗ n′
2. −(m + m′) ∗ n + m ∗ n + m′ ∗ n
3. (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,

$\otimes: M \times N \to M \otimes_R N, \, (m, n) \mapsto [m * n]$

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:

$M \times R \times N {{{} \atop \overset{\sigma \times 1}\rightarrow}\atop{\underset{1 \times \tau} \rightarrow \atop {}}} M \times N \overset{\otimes}\to M \otimes_R N,$

together with the requirements

$m \otimes (n + n') = m \otimes n + m \otimes n',$

$(m + m') \otimes n = m \otimes n + m' \otimes n.$

If S is a subring of a ring R, then $M \otimes_R N$ is the quotient group of $M \otimes_S N$ by the subgroup generated by $xr \otimes_S y - x \otimes_S ry, \, r \in R, x \in M, y \in N$, where $x \otimes_S y$ is the image of $(x, y)$ under $\otimes: M \times N \to M \otimes_{S} N.$ 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 ⋅ (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 Hom_R(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 E → R (also called linear forms), with operations

$(\phi + \psi)(u) = \phi(u) + \psi(u), \quad \phi, \psi \in E^*, u \in E$

$(r \cdot \phi) (u) = r \cdot \phi(u), \quad \phi \in E^*, u \in E, r \in R,$

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

$\langle \cdot , \cdot \rangle : E^* \times E \to R : (e', e) \mapsto \langle e', e \rangle = e'(e)$

$\langle \cdot , \cdot \rangle : F \times F^* \to R : (f, f') \mapsto \langle f, f' \rangle = f'(f) .$

The pairing is left R-linear in its left argument, and right R-linear in its right argument:

$\langle r \cdot g, h \cdot s \rangle = r \cdot \langle g, h \rangle \cdot s, \quad r, s \in R .$

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.

• Given right R-module E and right R-module F, there is a canonical homomorphism θ : F ⊗_R E^∗ → Hom_R(E, F) such that θ(f ⊗ e′) is the map e ↦ f ⋅ ⟨e′, e⟩.^[12]
• Given left R-module E and right R-module F, there is a canonical homomorphism θ : F ⊗_R E → Hom_R(E^∗, F) such that θ(f ⊗ e) 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 : F^∗ ⊗_R E^∗ → L_R(F × E, R) such that θ(f′ ⊗ e′) is the map (f, e) ↦ ⟨f, f′⟩ ⋅ ⟨e′, e⟩. Thus, an element of a tensor product ξ ∈ F^∗ ⊗_R E^∗ may be thought of giving rise to or acting as an R-bilinear map F × E → R.

Trace

Let R be a commutative ring and E an R-module. Then there is a canonical R-linear map:

$E^* \otimes_R E \to R$

induced by $\phi \otimes x \mapsto \phi(x)$; it is the unique R-linear corresponding to the duality pairing.

If E is a finitely generated projective R-module, then one can identify $E^* \otimes_R E = \operatorname{End}_R(E)$ through the canonical homomorphism mentioned above and then the above is the trace map:

$\operatorname{tr}: \operatorname{End}_R(E) \to R.$

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

$\mathfrak{T}^p_q = \Gamma(M, T M)^{\otimes p} \otimes_R \Gamma(M, T^* M)^{\otimes q}$

where Γ means the space of sections and the superscript $\otimes p$ means tensoring p times over R. By definition, an element of $\mathfrak{T}^p_q$ is a tensor field of type (p, q).

As R-modules, $\mathfrak{T}^q_p$ is the dual module of $\mathfrak{T}^p_q.$^[14]

To lighten the notation, put $E = \Gamma(M, T M)$ and so $E^* = \Gamma(M, T^* M)$.^[15] When p, q ≥ 1, for each (k, l) with 1 ≤ k ≤ p, 1 ≤ l ≤ q, there is an R-multilinear map:

$E^p \times {E^*}^q \to E^{p-1} \times {E^*}^{q-1}, \, (X_1, \dots, X_p, \omega_1, \dots, \omega_q) \mapsto \langle X_k, \omega_l \rangle (X_1, \dots, \widehat{X_l}, \dots, X_p, \omega_1, \dots, \widehat{\omega_l}, \dots, \omega_q)$

where $E^p$ means $\prod_1^p E$ and the hat means a term is omitted. By the universal property, it corresponds to a unique R-linear map:

$C^k_l: \mathfrak{T}^p_q \to \mathfrak{T}^{p-1}_{q-1}.$

It is called the contraction of tensors in the index (k, l). Unwinding what the universal property says one sees:

$C^k_l(X_1 \otimes \cdots \otimes X_p \otimes \omega_1 \otimes \cdots \otimes \omega_q) = \langle X_k, \omega_l \rangle X_1 \otimes \cdots \widehat{X_l} \cdots \otimes X_p \otimes \omega_1 \otimes \cdots \widehat{\omega_l} \cdots \otimes \omega_q.$

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, $-\otimes_R-:\mathrm{Mod}\mbox{--}R\times R\mbox{--}\mathrm{Mod}\rightarrow \mathrm{Ab}$ 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 $M\otimes_R-:R\mbox{--}\mathrm{Mod}\rightarrow \mathrm{Ab}$ arises, and symmetrically a left R module N could be fixed to create a functor $-\otimes_RN:\mathrm{Mod}\mbox{--}R\rightarrow \mathrm{Ab}$. Unlike the Hom bifunctor $\mathrm{Hom}_R(-,-)$, the tensor functor is covariant in both inputs.

It can be shown that M⊗- and -⊗N are always right exact functors, but not necessarily left exact. By definition, a module T is a flat module if T⊗- is an exact functor.

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 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 $\{m_i \otimes n_j \mid i\in I, j \in J\}$ indeed forms a basis for M ⊗_F N.

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 $\{m_i \otimes n_j \mid i\in I, j \in J\}$ indeed forms a basis for M⊗_F N.

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 − 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.

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

$(X \otimes_R Y)_n = \sum_{i + j = n} X_i \otimes_R Y_j,$

with the differential given by: for x in X_i and y in Y_j,

$d_{X \otimes Y} (x \otimes y) = d_X(x) \otimes y + (-1)^i x \otimes d_Y(y).$^[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 $C \otimes_{\mathbb{Z}} G$ 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)

$(T M)^{\otimes p} \otimes_{O} (T^* M)^{\otimes q}$

where O is the sheaf of rings of smooth functions on M and the bundles $TM, T^*M$ 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.

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.

See also

• Tor functor
• Tensor product of algebras
• Tensor product of fields

Notes

1. Hazewinkel, et al. (2004), p. 95, Prop. 4.5.1
2. Bourbaki, ch. II §3.1
3. First, if R = Z, then the claimed identification is given by $f \mapsto f'$ with $f'(x)(y) = f(x, y)$. In general, $\operatorname{Hom}_{\mathbb{Z}}(N, G)$ has the structure of a right R-module by $(g \cdot r)(y) = g(r y)$. Thus, for any Z-bilinear map f, f′ is R-linear $\Leftrightarrow f'(xr) = f'(x) \cdot r \Leftrightarrow f(xr, y) = f(x, ry).$
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) $(M \otimes_R N)_S = (S \otimes_S M) \otimes_R N = M_S \otimes_R N = M_S \otimes_S S \otimes_R N = M_S \otimes_S N_S$
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 $T^*M$, 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

References

• 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.