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Tensor product of modules

In module theory, the tensor product of two modules M and N over a commutative ring R with identity is an R-module M \otimes_R N, together with an R-bilinear map \otimes: M \times N \to M \otimes_R N, that satisfies a universal property: for any R-module P and any R-bilinear map f: M \times N \to P, there exists a unique R-linear map \tilde{f}: M \otimes_R N \to P such that f = \tilde{f} \circ \otimes.^[1] This construction generalizes the tensor product of vector spaces to the broader setting of modules, providing a canonical way to combine them while preserving bilinearity in each argument.^[2] The tensor product can be explicitly constructed as the quotient of the free R-module on the set M \times N by the submodule generated by the relations enforcing bilinearity: (m + m') \otimes n = m \otimes n + m' \otimes n, m \otimes (n + n') = m \otimes n + m \otimes n', and (rm) \otimes n = m \otimes (rn) = r(m \otimes n) for all m, m' \in M, n, n' \in N, and r \in R.^[2] Elements of M \otimes_R N are finite R-linear combinations of pure tensors m \otimes n, though not all elements need to be pure tensors in general.^[3] Key properties include symmetry (M \otimes_R N \cong N \otimes_R M), associativity ((M \otimes_R N) \otimes_R P \cong M \otimes_R (N \otimes_R P)), and right exactness, meaning that if $0 \to A \to B \to C \to 0 is a short exact sequence of R-modules, then M \otimes_R A \to M \otimes_R B \to M \otimes_R C \to 0 is exact.^[1] It also distributes over direct sums: M \otimes_R (N \oplus P) \cong (M \otimes_R N) \oplus (M \otimes_R P).^[2] Originally developed for vector spaces in the context of multilinear algebra and physics—such as in Cauchy's work on stress tensors (1822) and later formalized by Ricci and Levi-Civita (1901)—the tensor product for modules was systematized in the mid-20th century, notably by the Bourbaki group in 1948.^[2] In modern algebra, it plays a central role in base change for modules (e.g., S \otimes_R M extends scalars from R to an R-algebra S), homological algebra (e.g., Tor functors measure deviations from exactness), and algebraic geometry (e.g., tensor products of sheaves on schemes).^[3] For free modules with bases, the tensor product is free with the product basis, facilitating computations in examples like \mathbb{Z}/m\mathbb{Z} \otimes_\mathbb{Z} \mathbb{Z}/n\mathbb{Z} \cong \mathbb{Z}/\gcd(m,n)\mathbb{Z}.^[2]

Fundamentals

Definition

Let R be a ring and let M be a right R-module and N a left R-module. The tensor product M \otimes_R N is an abelian group (regarded as a \mathbb{Z}-module) together with a group homomorphism \otimes: M \times N \to M \otimes_R N, written (m, n) \mapsto m \otimes n, that is bilinear in the sense of being additive in each variable separately and satisfying the balanced scalar multiplication condition (m r) \otimes n = m \otimes (r n) for all m \in M, n \in N, and r \in R.^[2] This bilinear map has the universal property: for any abelian group P and any bilinear map f: M \times N \to P, there exists a unique group homomorphism g: M \otimes_R N \to P such that g(m \otimes n) = f(m, n) for all m \in M, n \in N.^[2] Bilinearity means that the map \otimes satisfies (m_1 + m_2) \otimes n = m_1 \otimes n + m_2 \otimes n and m \otimes (n_1 + n_2) = m \otimes n_1 + m \otimes n_2 for all m, m_1, m_2 \in [M](/page/Module) and n, n_1, n_2 \in [N](/page/Module), along with the balanced condition above, which ensures compatibility with the ring action from opposite sides.^[2] When R is commutative, both [M](/page/Module) and [N](/page/Module) can be regarded as two-sided modules, and the tensor product acquires an R-module structure via r (m \otimes n) = (r m) \otimes n = m \otimes (r n).^[2] The notation M \otimes_R N specifies the base ring R, distinguishing it from tensor products over other rings, while simple tensors are denoted m \otimes n; in general, elements of M \otimes_R N are finite sums of such simple tensors, subject to the bilinearity relations.^[2] The concept originated as the tensor product of vector spaces in multilinear algebra, and was generalized to modules over commutative rings by Bourbaki in the 1940s.^[2]

Universal Property

The tensor product M \otimes_R N of a right R-module M and a left R-module N is characterized up to unique isomorphism by its universal property with respect to bilinear maps. Specifically, there is an R-balanced bilinear map \iota: M \times N \to M \otimes_R N given by \iota(m, n) = m \otimes n, such that for any abelian group P and any R-balanced bilinear map f: M \times N \to P, there exists a unique group homomorphism \tilde{f}: M \otimes_R N \to P satisfying f = \tilde{f} \circ \iota. When R is commutative, M \otimes_R N becomes an R-module, and the universal property strengthens to R-modules P with unique R-linear maps \tilde{f}. This property identifies M \otimes_R N as the representing object for the functor that sends R-modules to the set of R-balanced bilinear maps from M \times N.^[2] A key corollary of this universal property concerns the vanishing of the tensor product. The module M \otimes_R N is zero if and only if every R-balanced bilinear map M \times N \to P is the zero map for every abelian group P. Indeed, if M \otimes_R N = 0, then the universal map \iota is zero, so any bilinear map factoring through it must also be zero; conversely, if all such bilinear maps vanish, then in particular the induced homomorphism from the universal property applied to \iota itself (taking P = M \otimes_R N) is the identity, which must then be zero, forcing M \otimes_R N = 0. Equivalently, in terms of homological algebra, M \otimes_R N = 0 if and only if, for a projective resolution \cdots \to P_1 \to P_0 \to M \to 0 of M, the induced map P_1 \otimes_R N \to P_0 \otimes_R N is surjective, as the tensor product is the cokernel of this map.^[2] The universal property also implies that the tensor product functor preserves isomorphisms. If \phi: M \to M' is an isomorphism of right R-modules and \psi: N \to N' is an isomorphism of left R-modules, then there is a unique isomorphism \phi \otimes \psi: M \otimes_R N \to M' \otimes_R N' such that (\phi \otimes \psi)(m \otimes n) = \phi(m) \otimes \psi(n), obtained by applying the universal property to the bilinear map (m, n) \mapsto \phi(m) \otimes \psi(n). This follows directly from the uniqueness clause, as the composite \psi \circ \iota' \circ (\phi \times \mathrm{id}_N) (or symmetrically) induces the desired linear map, which is an isomorphism since \phi and \psi are.^[2]^[4] The construction extends naturally to finite tensor products of multiple modules via iteration, leveraging the universal property and associativity. For modules over a commutative ring R, the k-fold tensor product M_1 \otimes_R \cdots \otimes_R M_k of R-modules M_1, \dots, M_k is defined inductively as (M_1 \otimes_R \cdots \otimes_R M_{k-1}) \otimes_R M_k, equipped with the composite multilinear map M_1 \times \cdots \times M_k \to M_1 \otimes_R \cdots \otimes_R M_k. This satisfies the universal property for R-multilinear maps: for any R-module P and multilinear map f: M_1 \times \cdots \times M_k \to P, there is a unique R-linear map \tilde{f}: M_1 \otimes_R \cdots \otimes_R M_k \to P such that f = \tilde{f} \circ \iota. Moreover, the iterated tensor product is independent of parenthesization up to canonical isomorphism, again by the universal property applied to associativity isomorphisms.^[2]

Construction

Bilinear Maps Approach

One standard construction of the tensor product M \otimes_R N of an R-module M and an R-module N proceeds by forming the free R-module on the set M \times N and quotienting by appropriate relations to enforce bilinearity (or balanced bilinearity in the non-commutative case). Let F denote the free R-module generated by the symbols e_{(m,n)} for all m \in M and n \in N, so elements of F are finite formal R-linear combinations \sum r_i e_{(m_i, n_i)} with r_i \in R. Define the submodule K \subseteq F generated by the elements enforcing additivity and homogeneity:

e_{(m + m', n)} - e_{(m, n)} - e_{(m', n)}, \quad e_{(m, n + n')} - e_{(m, n)} - e_{(m, n')},

e_{(r m, n)} - r e_{(m, n)}, \quad e_{(m, r n)} - r e_{(m, n)}

for all m, m' \in M, n, n' \in N, and r \in R. The tensor product is then the quotient module T = F / K, with the canonical surjection \pi: F \to T.^[1]^[5] This construction induces a natural R-balanced map \beta: M \times N \to T given by \beta(m, n) = \pi(e_{(m,n)}), often denoted m \otimes n. Since K is generated precisely by the relations that make \beta additive in each variable and homogeneous (with the balancing condition (m r) \otimes n = m \otimes (r n) for non-commutative R), the map \beta is R-bilinear when R is commutative and R-balanced otherwise. Elements of T can thus be represented as finite sums \sum r_i (m_i \otimes n_i) subject to these relations, and the notation M \otimes_R N is standard for this module.^[1]^[6] To verify that T satisfies the universal property, consider any R-module P and any R-balanced map f: M \times N \to P. By the universal property of the free module F, there exists a unique R-linear map \tilde{f}: F \to P such that \tilde{f}(e_{(m,n)}) = f(m,n). Since f respects the bilinearity relations, \tilde{f} vanishes on the generators of K, hence factors uniquely through the quotient as an R-linear map \overline{f}: T \to P with f = \overline{f} \circ \beta. This confirms that T \cong M \otimes_R N realizes the universal bilinear (or balanced) mapping object.^[5]^[1] This bilinear maps approach offers the advantage of working over arbitrary (possibly non-commutative) rings R, where the balancing relation explicitly captures the interaction between left and right module structures, distinguishing it from constructions reliant on commutativity. It also directly underscores the "balanced" nature of the tensor product, emphasizing its role as a universal object for maps that commute with scalar multiplication from both sides.^[6]^[7]

Free Resolutions Approach

The free resolutions approach to computing the tensor product of modules M and N over a ring R utilizes projective resolutions within homological algebra. To determine M \otimes_R N, first construct a projective resolution of one module, say N:

\cdots \to P_1 \xrightarrow{d_1} P_0 \to N \to 0,

where each P_i is a projective R-module and the sequence is exact except at N. This resolution exists for any R-module N.^[8] Tensor the resolution with M, forming the chain complex

\cdots \to M \otimes_R P_1 \xrightarrow{M \otimes_R d_1} M \otimes_R P_0 \to 0.

The tensor product is then the 0-th homology group of this complex:

M \otimes_R N \cong H_0(M \otimes_R P_\bullet) = \ker(M \otimes_R d_0) / \operatorname{im}(M \otimes_R d_1).

This isomorphism holds independently of the choice of projective resolution for N, up to chain homotopy equivalence. The higher homology groups H_i(M \otimes_R P_\bullet) for i \geq 1 vanish if the tensor functor -\otimes_R M is exact, which occurs precisely when M is flat.^[9] In cases where the resolved module is projective, such as free modules or vector spaces over a field (where all modules are free), the resolution is trivial: $0 \to P_0 \to N \to 0 with P_0 \cong N. Tensoring then yields M \otimes_R N directly as the kernel of the zero map, simplifying computations to a direct product of bases.^[10] This approach introduces the Tor derived functors of the tensor product, defined as \Tor_i^R(M, N) = H_i(M \otimes_R P_\bullet). In particular, \Tor_0^R(M, N) \cong M \otimes_R N, while \Tor_1^R(M, N) captures the failure of exactness, arising as the kernel of M \otimes_R P_0 \to M \otimes_R N modulo the image from higher terms; it obstructs the preservation of short exact sequences under tensoring.^[8]

Properties

Over General Rings

The tensor product of modules over an arbitrary ring R exhibits functoriality in both arguments. Specifically, for R-modules M, M' and N, N', where for non-commutative R, M (resp. N) is a right (resp. left) R-module, and R-linear maps f: M \to M', g: N \to N', there exists a unique R-linear map f \otimes g: M \otimes_R N \to M' \otimes_R N' defined on elementary tensors by f \otimes g (m \otimes n) = f(m) \otimes g(n), which extends bilinearly and respects the relations in the tensor product.^[11] This construction ensures that the tensor product operation is compatible with module homomorphisms, preserving the algebraic structure under composition.^[11] A key property is the right exactness of the tensor functor. For any R-module N, the functor -\otimes_R N: {}_R\Mod \to {}_R\Mod is right exact, meaning that if $0 \to A \to B \to C \to 0 is a short exact sequence of R-modules, then the induced sequence A \otimes_R N \to B \otimes_R N \to C \otimes_R N \to 0 is exact.^[11] In general, this functor is not left exact, as the map $0 \to A \otimes_R N need not be injective unless additional conditions, such as flatness of N, hold. The dual functor M \otimes_R -, fixing M, is likewise right exact.^[11] The tensor product distributes over direct sums in each variable. For R-modules M and a family of R-modules (N_i)_{i \in I}, there is a canonical isomorphism M \otimes_R \bigoplus_{i \in I} N_i \cong \bigoplus_{i \in I} (M \otimes_R N_i), induced by the bilinear maps sending m \otimes (n_i)_i \mapsto (m \otimes n_i)_i. The symmetric statement holds with roles reversed. For direct products, distributivity holds when the index set is finite, as finite direct sums and products coincide in the category of modules; for infinite products, it requires conditions such as M being finitely presented.^[12] Associativity provides a natural isomorphism between iterated tensor products: for R-modules M, N, P, (M \otimes_R N) \otimes_R P \cong M \otimes_R (N \otimes_R P), given by the bilinear map (m \otimes n) \otimes p \mapsto m \otimes (n \otimes p), which is an isomorphism respecting the universal properties. For change of rings, given a ring homomorphism \phi: R \to S, the tensor product S \otimes_R M equips the R-module M with a natural S-module structure via the action s' \cdot (s \otimes m) = (s' s \otimes m) for s, s' \in S, m \in M. This construction, known as extension of scalars, makes -\otimes_R S: {}_R\Mod \to {}_S\Mod a covariant functor.^[13]

Over Commutative Rings

When the base ring R is commutative, the tensor product of modules acquires additional symmetries and compatibilities that simplify many constructions and computations. Specifically, for R-modules M and N, there exists a canonical isomorphism M \otimes_R N \cong N \otimes_R M of R-modules, induced by the R-bilinear flip map (m, n) \mapsto (n, m) via the universal property of the tensor product.^[2] This symmetry renders the functor \otimes_R commutative in the sense that the order of factors does not affect the isomorphism class, a property absent in the non-commutative case.^[2] A key compatibility arises with localization: if S \subset [R](/page/R) is a multiplicative subset, then for any R-module [M](/page/Module), there is a canonical isomorphism of S^{-1}R-modules M \otimes_R S^{-1}R \cong S^{-1}M, where the map sends m \otimes (s^{-1} r) to (s^{-1} r) m = (r m)/s.^[14] This isomorphism explicitly inverts elements of S in the tensor product, allowing localization to commute with tensoring and facilitating the study of modules at prime ideals.^[14] In the special case where R is an integral domain with fraction field K = \operatorname{Frac}(R), the tensor product M \otimes_R K embeds the torsion-free part of M into a vector space over K; the natural map M \to M \otimes_R K given by m \mapsto m \otimes 1 has kernel precisely the torsion submodule \{ m \in M \mid \exists 0 \neq r \in R, rm = 0 \}, and the image is isomorphic to M / \operatorname{Tor}(M) as an R-submodule of the K-vector space M \otimes_R K.^[2]^[15] Base change with respect to ideals also simplifies under commutativity. For an ideal I \subset R and R-module M, the canonical isomorphism (R/I) \otimes_R M \cong M / IM of R/I-modules sends \overline{r} \otimes m \mapsto \overline{r m}, where \overline{r} = r + I.^[2] This identifies the tensor product with the quotient by the submodule generated by I, providing a direct way to descend modules modulo ideals.^[2] If M and N are R-algebras (i.e., associative unital ring extensions of R), then M \otimes_R N inherits a natural R-algebra structure via the multiplication (m_1 \otimes n_1)(m_2 \otimes n_2) = (m_1 m_2) \otimes (n_1 n_2), extended R-linearly to sums; the unit is $1_M \otimes 1_N.^[16] This makes the tensor product the coproduct in the category of commutative R-algebras, preserving multiplicative structure while leveraging the underlying module tensor product.^[16]

Examples

Algebraic Examples

A fundamental example of the tensor product arises in the category of vector spaces over a field k. For finite-dimensional vector spaces V and W over k, with bases \{v_i\}_{i=1}^m and \{w_j\}_{j=1}^n respectively, the tensor product V \otimes_k W is isomorphic to the vector space k^{m n} with basis \{v_i \otimes w_j\}_{i,j}, and thus \dim_k (V \otimes_k W) = (\dim_k V) \cdot (\dim_k W).^[17] This isomorphism follows from the universal property, where the bilinear map sending (v_i, w_j) to v_i \otimes w_j generates the space, and relations ensure linearity in each factor. Over a commutative ring R, the tensor product of free modules preserves freeness and rank in a multiplicative manner. Specifically, if M \cong R^m and N \cong R^n as R-modules, then M \otimes_R N \cong R^{m n}, where the isomorphism arises by "flattening" the m \times n matrix of elementary tensors e_i \otimes f_j into a basis for the free module of rank m n.^[18] This extends the vector space case, as free modules over R behave analogously to vector spaces when R is a field. For cyclic modules over the integers, the tensor product captures interactions between torsion elements. Consider \mathbb{Z}/p\mathbb{Z} \otimes_\mathbb{Z} \mathbb{Z}/q\mathbb{Z}, where p and q are positive integers; this is isomorphic to \mathbb{Z}/d\mathbb{Z} with d = \gcd(p, q), and hence zero if p and q are coprime.^[19] The isomorphism is induced by the bilinear map (a \mod p, b \mod q) \mapsto ab \mod d, which factors through the relations p(a \otimes b) = a \otimes (p b) = 0 and similarly for q. A classic illustration of how tensor products interact with torsion occurs with rational coefficients. For any positive integer n, the module \mathbb{Z}/n\mathbb{Z} \otimes_\mathbb{Z} \mathbb{Q} = 0, since every elementary tensor k \otimes (r/s) satisfies n(k \otimes r/s) = (n k) \otimes (r/s) = k \otimes (n r/s) = 0, annihilating the generators.^[20] Although \mathbb{Q} is flat over \mathbb{Z}, this computation shows that tensoring with \mathbb{Q} kills torsion subgroups. Pathological behavior emerges in non-integral domains with nilpotent elements. Let k be a field and R = k/(x^2), so x^2 = 0 in R; the ideal (x) = x R is the socle of R. Then the modules M = N = R/(x) \cong k (as k-vector spaces, with trivial x-action) satisfy M \otimes_R N \cong R/(x) \cong k, via the general isomorphism R/I \otimes_R R/J \cong R/(I + J) for ideals I, J, here with I = J = (x) so I + J = (x).^[21] This contrasts with the direct product M \times N \cong k^2, highlighting how the ring action collapses the tensor product.

Geometric Examples

In differential geometry, the tensor product construction extends naturally to bundles over manifolds, providing a framework for tensor fields that encode multilinear operations on tangent and cotangent spaces. On a smooth manifold M, the tensor bundle TM \otimes T^*M consists of fiberwise tensor products of the tangent bundle TM and cotangent bundle T^*M; its smooth sections form the space of (1,1)-tensor fields, which at each point p \in M assign a linear map T_p M \to T_p M.^[22] These fields arise as derivations or endomorphisms on local frames, facilitating the description of infinitesimal transformations on the manifold's geometry.^[22] Covariant tensor fields, sections of the bundle \bigotimes^k T^*M for k \geq 1, generalize scalar fields (k=0) and 1-forms (k=1) by acting as multilinear functionals on k tangent vectors at each point. When restricted to alternating multilinear maps, these yield sections of the exterior tensor power \wedge^k T^*M, which is canonically isomorphic to the space of differential k-forms \Omega^k(M); this identification underpins integration and Stokes' theorem on manifolds.^[22] A prominent physical application appears in general relativity, where the stress-energy tensor T on the spacetime manifold is a smooth symmetric section of T^*M \otimes T^*M (or more precisely, of the symmetric square \mathrm{Sym}^2(T^*M)), quantifying local energy density, momentum flux, and stresses that source gravitational curvature via Einstein's field equations.^[23] In local coordinates (x^i) on M, tensor fields express components relative to basis elements: a (1,1)-tensor field takes the form \sigma = \sigma^i_j \, \frac{\partial}{\partial x^i} \otimes dx^j, where \sigma^i_j are smooth functions, while the Levi-Civita connection on a Riemannian manifold involves Christoffel symbols \Gamma^k_{ij} defining a (1,2)-tensor field \Gamma = \Gamma^k_{ij} \, dx^i \otimes dx^j \otimes \frac{\partial}{\partial x^k}, used in covariant differentiation to preserve metric compatibility.^[22] At the level of multilinear algebra over finite-dimensional spaces such as \mathbb{R}^n, the tensor product V \otimes W for vector spaces V, W generalizes the outer product of vectors u \in V, v \in W, yielding a pure tensor u \otimes v, which under suitable identifications (such as with the dual space) corresponds to a rank-1 linear operator, foundational for describing matrices and higher-rank tensors in a coordinate-free manner.^[24]

Applications

Extension of Scalars

Given a ring homomorphism \phi: R \to S, the tensor product S \otimes_R M of an R-module M with S (viewed as a right R-module via \phi) inherits a natural S-module structure defined by s' \cdot (s \otimes m) = (s' s) \otimes m for s', s \in S and m \in M.^[25] This construction, known as the extension of scalars from R to S, equips M with an action compatible with the original R-module structure, as the map R \to S induces the necessary bilinearity.^[25] If S is flat as an R-module, the extension of scalars functor -\otimes_R S: \mathrm{Mod}_R \to \mathrm{Mod}_S preserves exact sequences, making it a flat functor; moreover, if the homomorphism R \to S is faithfully flat, this functor reflects exactness as well.^[25] When M is a free R-module of finite rank n, the extended module S \otimes_R M is isomorphic to the free S-module S^n, reflecting the basis extension across the ring change.^[25] The right exactness of the tensor product ensures compatibility with short exact sequences in this base change process.^[25] A concrete example arises when extending scalars from the real numbers \mathbb{R} to the complex numbers \mathbb{C} via the inclusion \mathbb{R} \hookrightarrow \mathbb{C}, yielding \mathbb{C} \otimes_\mathbb{R} \mathbb{R} \cong \mathbb{C} as \mathbb{C}-algebras.^[26] This isomorphism allows real polynomials to factor into complex linear terms, illustrating how base extension reveals roots invisible over the original base ring. In algebraic geometry, extension of scalars facilitates descent: given a faithfully flat morphism f: \mathrm{Spec}(S) \to \mathrm{Spec}(R), an S-module N descends to an R-module if it arises as M \otimes_R S for some M, with the descent datum encoded by isomorphisms compatible with the cocycle condition on the cover induced by f.^[27] This framework reconstructs global objects from local data over the base change. For group representations, extension of scalars provides change of base: if G is a finite group with a representation over a field k, tensoring with a k-algebra A yields an A-module structure on the extended space, preserving the G-action and enabling study of representations over larger coefficient rings.^[28] This operation is central to analyzing semisimplicity after base extension in modular representation theory.^[29]

Relation to Flat Modules

A module M over a ring R is defined to be flat if the functor -\otimes_R M is exact, meaning that for any exact sequence of R-modules, the sequence obtained after tensoring with M remains exact.^[30] This property ensures that tensor products with M preserve both injectivity and surjectivity of homomorphisms. Equivalently, M is flat if and only if \Tor_1^R(N, M) = 0 for every R-module N, where \Tor is derived from projective resolutions.^[31] Free modules are flat, as tensoring with a free module corresponds to a direct sum of copies of the identity functor, which is exact.^[30] Projective modules are also flat, since they are direct summands of free modules and direct summands preserve exactness in this context.^[31] However, the converse does not hold: there exist flat modules that are not projective. For instance, the ring \mathbb{Z} is flat over itself as a trivial free module, but a non-projective example is \mathbb{Q} as a \mathbb{Z}-module, which is flat but not projective since it lacks a basis over \mathbb{Z}.^[30]^[31] To illustrate flatness, consider the short exact sequence $0 \to \mathbb{Z} \xrightarrow{\times 2} \mathbb{Z} \to \mathbb{Z}/2\mathbb{Z} \to 0. Tensoring with \mathbb{Q} yields $0 \to \mathbb{Q} \xrightarrow{\times 2} \mathbb{Q} \to 0 \to 0, where multiplication by 2 is an isomorphism (hence injective and surjective onto its image), preserving exactness and confirming \mathbb{Q} is flat over \mathbb{Z}.^[30] In contrast, tensoring the same sequence with \mathbb{Z}/2\mathbb{Z} gives $0 \to \mathbb{Z}/2\mathbb{Z} \xrightarrow{0} \mathbb{Z}/2\mathbb{Z} \xrightarrow{\id} \mathbb{Z}/2\mathbb{Z} \to 0, where the first map is zero. Thus, at the first \mathbb{Z}/2\mathbb{Z}, the image of the incoming map is 0 but the kernel of the outgoing map is \mathbb{Z}/2\mathbb{Z}, showing the sequence is not exact and \mathbb{Z}/2\mathbb{Z} is not flat over \mathbb{Z}.^[32] Localization preserves flatness: for any multiplicative set S in R, the localized ring S^{-1}R is flat as an R-module.^[30] This follows from the fact that localization is an exact functor, and tensoring with S^{-1}R inverts elements of S, maintaining exact sequences.^[31] Flatness has significant applications in commutative algebra. If M is flat over R, then for any ideal I \subseteq R, the map I \otimes_R M \to R \otimes_R M \cong M is injective, reflecting that flat modules do not introduce torsion in submodules like ideals.^[30] This property is crucial in the study of Cohen-Macaulay modules, where flat resolutions help verify conditions like depth equaling dimension in local rings, aiding in the classification of regular and Gorenstein rings.^[31]

Advanced Topics

Representation as Linear Maps

One key representation of the tensor product arises from its universal property, which establishes a natural isomorphism of R-modules

\Hom_R(M \otimes_R N, P) \cong \Bil_R(M \times N, P)

for any R-module P, where \Bil_R(M \times N, P) denotes the abelian group of R-bilinear maps M \times N \to P.^[1] This bijection sends an R-linear map f: M \otimes_R N \to P to the bilinear map (m,n) \mapsto f(m \otimes n), with the inverse constructing f from a bilinear map via the universal bilinear map M \times N \to M \otimes_R N.^[1] A related perspective is provided by the tensor-hom adjunction, which yields a natural isomorphism

\Hom_R(M, \Hom_R(N, P)) \cong \Hom_R(M \otimes_R N, P)

for R-modules M, N, and P, natural in all three variables.^[33] This adjunction positions the functor - \otimes_R N as left adjoint to \Hom_R(N, -). The isomorphism is constructed by associating to a homomorphism g: M \to \Hom_R(N, P) the composite M \otimes_R N \to P via m \otimes n \mapsto g(m)(n), with the inverse sending h: M \otimes_R N \to P to the map m \mapsto (n \mapsto h(m \otimes n)).^[33] When M is a finitely generated free R-module of rank m (so M \cong R^m) and N is any R-module, the tensor product admits an explicit representation as a Hom space via the dual module M^* = \Hom_R(M, R) \cong R^m. In this case, there is a natural R-module isomorphism

M \otimes_R N \cong \Hom_R(M^*, N).

This follows from the adjunction by setting the first variable to M^*, yielding \Hom_R(M^*, \Hom_R(R, N)) \cong \Hom_R(M^* \otimes_R R, N) \cong \Hom_R(M^*, N), but since \Hom_R(R, N) \cong N and M^* \otimes_R R \cong M^*, the structure simplifies to the desired form using the freeness of M.^[21] For M finitely generated free, the endomorphism ring \End_R(M) also admits a tensor product representation: there is an R-module isomorphism

\End_R(M) \cong M \otimes_R M^*.

Explicitly, with respect to bases \{e_i\} for M and dual basis \{\varepsilon^j\} for M^*, the elementary tensors e_i \otimes \varepsilon^j form a basis for \End_R(M), corresponding to the rank-one endomorphisms sending e_k \mapsto \delta^j_k e_i. The trace on \End_R(M) is then induced via this isomorphism, pairing with the canonical trace on matrix rings when M \cong R^m.^[21] A concrete illustration occurs in the action of endomorphisms on M: under the isomorphism \End_R(M) \cong M \otimes_R M^*, matrix multiplication corresponds to the R-linear map \mu: M \otimes_R M^* \to \End_R(M) \to \Hom_R(M, M) composed with evaluation, but more directly, the action map (m \otimes \phi) \cdot n = \phi(n) m defines an R-module homomorphism M \otimes_R M^* \otimes_R M \to M that encodes left multiplication by "matrices" in this tensor representation.^[21]

Duality and Trace

The dual module of an R-module M is defined as M^* = \Hom_R(M, R), the R-module of R-linear homomorphisms from M to R.^[34] For a finitely generated projective module M, the natural evaluation map M \to (M^*)^* given by m \mapsto \mathrm{ev}_m, where \mathrm{ev}_m(\phi) = \phi(m) for \phi \in M^*, is an isomorphism, so M \cong M^{**}.^[34] This reflexivity property holds in particular for finitely generated projective modules, enabling a rich duality structure in module theory.^[34] The duality induces a natural pairing \langle \cdot, \cdot \rangle: M \times M^* \to R defined by \langle m, \phi \rangle = \phi(m), which is R-bilinear.^[34] This pairing extends uniquely to an R-linear map M \otimes_R M^* \to R, often called the contraction or evaluation map, sending the pure tensor m \otimes \phi to \phi(m).^[21] For a finitely generated free module M with dual basis \{e_i\} and \{\phi_i\} (satisfying \phi_j(e_i) = \delta_{ij}), the image of the identity endomorphism under the inverse of the natural isomorphism M \otimes_R M^* \cong \End_R(M) is \sum_i e_i \otimes \phi_i, and the contraction on this element yields 1, reflecting the dimension of M.^[21] Elements of M \otimes_R M^* correspond to bilinear maps via the universal property of the tensor product. Specifically, a pure tensor m \otimes \phi \in M \otimes_R M^* induces the evaluation map on M^* \times M \to R given by (\psi, n) \mapsto \psi(n) \cdot \phi(m), though more directly, under the isomorphism M \otimes_R M^* \cong \End_R(M), it corresponds to the rank-one endomorphism n \mapsto \phi(n) m.^[21] This identification highlights how tensor products with duals encode linear transformations without choosing bases. For a finitely generated free module M, the trace provides a canonical R-linear functional \tr: \End_R(M) \to R. Using dual bases \{e_i\} and \{\phi_i\}, for f \in \End_R(M), \tr(f) = \sum_i \phi_i(f(e_i)), which equals the contraction applied to the image of f under the isomorphism \End_R(M) \cong M \otimes_R M^*.^[21] This trace is independent of the choice of dual bases and extends the classical trace from vector spaces to modules. In the context of the pairing, it can be expressed as \tr(f) = \sum_i \langle e_i \otimes \phi_i, f \rangle, where the sum \sum_i e_i \otimes \phi_i represents the identity.^[21] In representation theory, traces arising from tensor products play a central role in character theory. For representations \rho: G \to \End_k(V) and \sigma: G \to \End_k(W) of a group G over a field k, the character of the tensor product representation on V \otimes_k W is the product of the individual characters: \chi_{V \otimes_k W}(g) = \chi_V(g) \chi_W(g) = \tr(\rho(g)) \tr(\sigma(g)).^[35] This multiplicativity extends to tensor powers, where the character of V^{\otimes n} is \chi_V(g)^n = \tr(\rho(g))^n, facilitating the decomposition of tensor powers into irreducibles and computations in symmetric groups or Lie algebras.^[35] Such traces underpin orthogonality relations and dimension formulas in semisimple representations.^[35]

Generalizations

To Chain Complexes

The tensor product of two chain complexes C_\bullet and D_\bullet over a ring R is defined as the chain complex (C \otimes D)_n = \bigoplus_{p+q=n} C_p \otimes_R D_q, where the differential is given by d_{C \otimes D} = d_C \otimes \mathrm{id}_{D_q} + (-1)^p \mathrm{id}_{C_p} \otimes d_D on the summand C_p \otimes D_q.^[36]^[37] This construction equips the category of chain complexes with a monoidal structure, preserving the algebraic properties of the underlying module tensor product.^[36] The homology of the total complex H_n(C \otimes D) is related to the Tor functor through a spectral sequence, specifically the first-quadrant spectral sequence with E_2^{p,q} = H_p (H_q(C) \otimes^\mathbb{L}_R D) converging to H_{p+q}(C \otimes^\mathbb{L}_R D), under suitable boundedness conditions on the complexes.^[38] This spectral sequence arises from filtering the tensor product complex by the bidegrees of the summands and provides a tool for computing the homology in terms of derived functor information.^[38] A key example is the computation of Tor groups: if P_\bullet \to M is a projective resolution of a module M, then \Tor_i^R(M, N) \cong H_i(P_\bullet \otimes_R N) for any module N, where the homology is taken in the tensor product complex.^[8] This illustrates how tensoring a resolution yields the derived tensor product as the homology of the resulting complex.^[8] Regarding properties, if one complex, say C_\bullet, is acyclic (as in a resolution) and the other module N is flat over R, then the tensor product complex C_\bullet \otimes_R N remains exact except possibly in degree zero, reflecting the right-exactness preserved under flatness.^[39] More generally, tensoring with a flat complex preserves exactness in the sense that the functor -\otimes E_\bullet is exact when E_\bullet is degreewise flat.^[39] In applications within homological algebra, the Künneth formula describes the homology of the tensor product when the coefficients are over a field k: H_n(C \otimes_k D) \cong \bigoplus_{p+q=n} H_p(C) \otimes_k H_q(D), since higher Tor terms vanish over a field.^[40] This isomorphism extends to a short exact sequence involving Tor for integer coefficients, but over fields, it simplifies to a direct tensor product of homologies; a dual version holds for Ext groups in cohomology.^[40]^[37]

To Sheaves of Modules

The tensor product of two sheaves of modules \mathcal{F} and \mathcal{G} on a ringed space (X, \mathcal{O}_X) is defined as the sheafification of the presheaf U \mapsto \mathcal{F}(U) \otimes_{\mathcal{O}_X(U)} \mathcal{G}(U), denoted \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{G}.^[13] This construction is characterized by the universal property that morphisms from \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{G} to another sheaf \mathcal{H} correspond to \mathcal{O}_X-bilinear morphisms from \mathcal{F} \times \mathcal{G} to \mathcal{H}.^[13] On stalks, the tensor product satisfies (\mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{G})_x \cong \mathcal{F}_x \otimes_{\mathcal{O}_{X,x}} \mathcal{G}_x for every point x \in X.^[41] In the derived setting, the derived tensor product \mathcal{F} \otimes^\mathbf{L}_{\mathcal{O}_X} \mathcal{G} is defined in the derived category D(\mathcal{O}_X) of sheaves of \mathcal{O}_X-modules, using a flat resolution of one factor to ensure the tensor product is computed correctly.^[42] If \mathcal{F} or \mathcal{G} admits a resolution by flat sheaves, the derived tensor reduces to the ordinary tensor product.^[42] The cohomology sheaves of \mathcal{F} \otimes^\mathbf{L}_{\mathcal{O}_X} \mathcal{G} are the sheaf Tor groups \Tor_i^{\mathcal{O}_X}(\mathcal{F}, \mathcal{G}), which vanish for i > 0 if one of the sheaves is flat, meaning locally the corresponding module is flat.^[42] These sheaf Tor sheaves quantify obstructions to exactness in tensor products and play a key role in the derived category, such as in computing intersection multiplicities or extensions.^[43] Key properties include coherence preservation: if \mathcal{F} and \mathcal{G} are coherent \mathcal{O}_X-modules, then \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{G} is coherent.^[44] Similarly, if one is of finite presentation and the other coherent, the tensor is coherent.^[44] Flatness for sheaves of modules is defined locally, where a sheaf is flat if its stalks are flat modules over the stalk rings; this ensures the tensor functor is exact.^[45] On an affine scheme X = \Spec R, the tensor product of quasi-coherent sheaves \widetilde{M} and \widetilde{N} associated to R-modules M and N satisfies \widetilde{M} \otimes_{\mathcal{O}_X} \widetilde{N} \cong \widetilde{M \otimes_R N}, so the global sections recover the module tensor product: \Gamma(X, \widetilde{M} \otimes_{\mathcal{O}_X} \widetilde{N}) \cong M \otimes_R N.^[46] In algebraic geometry, the tensor product applies to line bundles, which are invertible sheaves. For effective Cartier divisors D and E on a scheme X, the associated line bundles satisfy \mathcal{O}_X(D) \otimes_{\mathcal{O}_X} \mathcal{O}_X(E) \cong \mathcal{O}_X(D + E).^[47] This additivity underlies the group structure of the Picard group. Tensor products also resolve structure sheaves in contexts like Koszul complexes, where higher Tor sheaves detect singularities or non-flatness in geometric constructions.^[42]

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