Tensor Products of Modules

The tensor product is one of algebra's most powerful and ubiquitous constructions, providing the definitive way to linearize multilinear operations. While initially abstract, its utility spans from simplifying computations in linear algebra to forming the bedrock of modern differential geometry and quantum mechanics. Mastering the tensor product unlocks a deeper understanding of how algebraic structures interact, particularly through the process of extension of scalars, which allows you to change the ring over which your modules are defined.

1. The Universal Property: Defining via What It Does

At its heart, the tensor product solves a universal problem. Suppose you have a right $R$-module $M$ and a left $R$-module $N$ over a ring $R$. A map $f: M \times N \to L$ (where $L$ is an abelian group) is called $R$-bilinear if it is additive in each argument and satisfies $f(mr, n) = f(m, rn)$ for all $r \in R$.

The tensor product $M \otimes_R N$ is an abelian group equipped with a specific bilinear map $\otimes: M \times N \to M \otimes_R N$ defined by $(m, n) \mapsto m \otimes n$. Its defining characteristic is the universal property for bilinear maps: For any abelian group $L$ and any $R$-bilinear map $f: M \times N \to L$, there exists a unique group homomorphism $\tilde{f}: M \otimes_R N \to L$ such that $f = \tilde{f} \circ \otimes$. In other words, every bilinear map out of $M \times N$ factors uniquely through the tensor product.

$$\begin{array}{ccc} M \times N & \xrightarrow{\otimes} & M \otimes_R N \\ & \underset{f}{\searrow} & \downarrow{\exists ! \, \tilde{f}} \\ & & L \end{array}$$

This property is the definition. Any construction (e.g., via a quotient of the free abelian group on $M \times N$) is merely a proof of existence. The elements $m \otimes n$ are called simple tensors, but not every element of $M \otimes_R N$ is a simple tensor; general elements are finite sums $\sum_i m_i \otimes n_i$.

2. Basic Properties and "Natural" Isomorphisms

From the universal property, fundamental isomorphisms follow naturally. These are the tools you use to manipulate tensor products in proofs and calculations.

• Right Exactness: The tensor product functor $- \otimes_R N$ is right exact. For any exact sequence of right $R$-modules $M' \xrightarrow{\alpha} M \xrightarrow{\beta} M'' \to 0$, the resulting sequence $M' \otimes_R N \xrightarrow{\alpha \otimes 1} M \otimes_R N \xrightarrow{\beta \otimes 1} M'' \otimes_R N \to 0$ is exact. It is not generally left exact.
• Distributivity over Direct Sums: The tensor product distributes over arbitrary direct sums: $(\bigoplus_{i \in I} M_i) \otimes_R N \cong \bigoplus_{i \in I} (M_i \otimes_R N)$. This is extremely useful for breaking down complex tensors.
• Associativity: Given modules $M_R$, $_R N_S$, and $_S L$, we have a natural isomorphism $(M \otimes_R N) \otimes_S L \cong M \otimes_R (N \otimes_S L)$.
• Commutativity/Base Change: For a commutative ring $R$, $M \otimes_R N \cong N \otimes_R M$. This allows us to treat tensors more flexibly.

3. Tensor Products of Free Modules

The behavior is most intuitive when modules are free. If $M$ is free with basis $\{e_i\}_{i \in I}$ and $N$ is free with basis $\{f_j\}_{j \in J}$ over a commutative ring $R$, then $M \otimes_R N$ is a free $R$-module with basis $\{e_i \otimes f_j\}_{(i,j) \in I \times J}$.

For example, if $V$ and $W$ are finite-dimensional vector spaces over a field $\mathbb{F}$ with bases $\{v_1,..., v_m\}$ and $\{w_1,..., w_n\}$, then $V \otimes_{\mathbb{F}} W$ is a vector space of dimension $mn$ with basis $\{v_i \otimes w_j\}$. Any element $T \in V \otimes W$ can be uniquely represented by an $m \times n$ matrix of coefficients from $\mathbb{F}$, linking the tensor product directly to the space of bilinear maps $V \times W \to \mathbb{F}$.

4. Extension of Scalars: Changing the Ring

One of the most important applications is extension of scalars. Given a ring homomorphism $\phi: R \to S$, we can turn any $R$-module $M$ into an $S$-module. How? By forming $S \otimes_R M$, where $S$ is viewed as a right $R$-module via $\phi$ ($s \cdot r = s\phi(r)$). The resulting abelian group becomes a left $S$-module via the action $s' \cdot (s \otimes m) = (s's) \otimes m$.

Think of this as "enlarging" or "adapting" the module $M$ to be understood over the new ring $S$. A classic example is complexification: extending a real vector space $V_{\mathbb{R}}$ to a complex vector space $\mathbb{C} \otimes_{\mathbb{R}} V$. The new space has the same real dimension but is now equipped with a complex structure, allowing you to talk about eigenvalues and eigenvectors of real-linear maps.

5. Key Applications

The tensor product's ability to linearize multilinearity makes it indispensable.

• Multilinear Algebra: The tensor product constructs spaces of multilinear maps. The space of $k$-multilinear forms on a vector space $V$ is isomorphic to the dual space $(V^{\otimes k})^*$. Higher tensor powers $V^{\otimes k} = V \otimes ... \otimes V$ form the arena for studying tensors of rank $k$.
• Differential Geometry: Tensor products are used to define the tensor bundles on a manifold. The tangent bundle $TM$ and cotangent bundle $T^*M$ are combined via tensor products to form bundles of $(p,q)$-tensors: $T^{(p,q)}M = TM^{\otimes p} \otimes T^*M^{\otimes q}$. A Riemannian metric, for instance, is a specific section of $T^*M \otimes T^*M$.
• Representation Theory: For a group $G$, if $V$ and $W$ are representations (modules over the group algebra), their tensor product $V \otimes W$ is a new representation under the action $g \cdot (v \otimes w) = (g\cdot v) \otimes (g\cdot w)$. This is how composite quantum systems are modeled in physics—the state space is the tensor product of the individual particles' state spaces.

Common Pitfalls

1. Assuming Pure Tensors are General: A major mistake is assuming every element in $M \otimes N$ is of the form $m \otimes n$. In general, elements are sums of such simple tensors, and a representation as a sum is not unique. For instance, in $\mathbb{Q} \otimes_{\mathbb{Z}} \mathbb{Z}/2\mathbb{Z}$, we have $1 \otimes 1 = \frac{1}{2} \otimes 0 = 0$.
2. Ignoring the Ring of Scalars: The notation $M \otimes N$ is incomplete; you must always know the base ring $R$ in $M \otimes_R N$. Changing $R$ changes the group. $\mathbb{C} \otimes_{\mathbb{R}} \mathbb{C}$ is a 4-dimensional real space, while $\mathbb{C} \otimes_{\mathbb{C}} \mathbb{C}$ is a 1-dimensional complex space.
3. Misapplying Commutativity: The "commutative" isomorphism $M \otimes_R N \cong N \otimes_R M$ typically requires $R$ to be commutative, or requires careful treatment of side module structures in the non-commutative case. Don't swap factors carelessly.
4. Forgetting Right Exactness: Because tensor product is not left exact, it does not generally preserve injections. For example, the injection $2\mathbb{Z} \hookrightarrow \mathbb{Z}$ becomes the zero map $2\mathbb{Z} \otimes_{\mathbb{Z}} \mathbb{Z}/2\mathbb{Z} \to \mathbb{Z} \otimes_{\mathbb{Z}} \mathbb{Z}/2\mathbb{Z}$ since $2a \otimes \bar{1} = a \otimes \bar{2} = a \otimes \bar{0} = 0$.

Summary

• The tensor product $M \otimes_R N$ is defined by its universal property: it converts $R$-bilinear maps $M \times N \to L$ into linear maps $M \otimes_R N \to L$.
• It satisfies key natural isomorphisms for distributivity over direct sums and associativity, and the functor $-\otimes_R N$ is right exact.
• For free modules, the tensor product is straightforward: the product of bases forms a basis for the tensor product.
• Extension of scalars via $S \otimes_R M$ is a fundamental technique to adapt a module over a new ring, with complexification of real vector spaces as a prime example.
• Its applications are vast, forming the foundation for multilinear forms, the tensor bundles of differential geometry, and the combined representations in representation theory.