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The quantum relative entropy is additive on tensor products:This follows by expanding $\log ({\sigma }_A\otimes {\sigma }_B)=\log {\sigma }_A\otimes I+I\otimes \log {\sigma }_B$ and its counterpart for $\rho$.

For an affine variety with coordinate ring $A$ and an $A$-module $M$, localization defines sections on principal open subsets by the displayed formula. Restrictions are the natural localization maps. Equivalently, a section is a family of stalk values which locally come from one fraction $m/f^r$. These data form a quasi-coherent sheaf, with global sections $M$ and stalk $M_{{\mathfrak{m}}_P}$ at $P$. Sheaf tensor products correspond to tensor products over $A$; pullback of a sheaf of modules corresponds to extension of scalars, and affine direct image to restriction of scalars.

The Appell–Humbert theorem classifies holomorphic line bundles on a complex torus by a Riemann form together with a compatible semicharacter of its lattice.

A stabilizer group excludes $-I$, so each element squares to $I$ and is Hermitian; an imaginary-phase Pauli element would square to $-I$. A nonidentity stabilizer contains at least one nonidentity local Pauli matrix and has zero trace, since the trace of a tensor product is the product of local traces. Its eigenvalues are therefore $\pm 1$ with equal multiplicities. This trace fact also gives $\dim {\mathcal{X}}_S=2^n/|S|$ from the stabilizer-projector formula.

For a ring homomorphism $A\to B$, extension of scalars takes an invertible module to an invertible $B$-module. A direct-summand presentation in a finite free module remains a direct-summand presentation after tensoring. At a prime $\mathfrak{q}\subset B$ contracting to $\mathfrak{p}$, localization gives $B_{\mathfrak{q}}{\otimes }_{A_{\mathfrak{p}}}{L}_{\mathfrak{p}}\cong B_{\mathfrak{q}}$. Compatibility with tensor products, identities and composition makes the Picard group of a ring a covariant functor.

A bilinear square finite element uses polynomials of degree at most one separately in each coordinate and takes values at the four corners as degrees of freedom. On the unit square its nodal shape functions are $(1-\xi )(1-\eta )$, $\xi (1-\eta )$, $\xi \eta$, and $(1-\xi )\eta$, in anticlockwise corner order. Each is a tensor product of one-dimensional linear shape functions. A function in this space need not be affine, since $\xi \eta$ has total degree two. On a physical square of side $h$, its Laplacian stiffness matrix has diagonal entries $2/3$, adjacent-corner entries $-1/6$ and opposite-corner entries $-1/3$. Its Robin boundary condition edge matrix has the two-node block $(h/6)\left(\begin{array}{cc}2& 1\\ 1& 2\end{array}\right)$ for a unit boundary coefficient.

A quantum system with two specified subsystems has a tensor product Hilbert space. A joint density operator determines subsystem $A$'s measurement statistics through its partial trace ${\rho }_A={\mathrm{Tr}}_B{\rho }_{AB}$. A joint pure state need not give a pure reduced state: the Bell state $(|00⟩+|11⟩)/\sqrt{2}$ gives ${\rho }_A=I/2$.

The Brauer group of a field $k$ consists of Morita equivalence classes of central simple $k$-algebras. Multiplication is induced by tensor product, the identity is $[k]$, and the inverse of $[A]$ is the class of the opposite algebra $A^{\mathrm{op}}$.

Category theory studies mathematical structures through the morphisms between them. A category specifies which objects of a category and arrows are allowed; a functor translates these objects and arrows into another category; a natural transformation compares two such translations coherently. For example, the category of groups records both groups and group homomorphisms, so it remembers which maps respect multiplication.

The usefulness appears in constructions we already know. A Cartesian product combines two sets, while a direct product of groups combines two groups. Both satisfy the same universal property of a categorical product: any pair of maps into the factors comes from exactly one map into the product. This gives a reusable way to construct maps and establish isomorphisms, without depending on a particular choice of coordinates or element labels. In a partially ordered set regarded as a category, the corresponding categorical product is a greatest lower bound. The shared definition exposes a connection between pairing data and finding the best common bound.

Start with the computed examples under category, functor, natural transformation, and product in a category. Their arrow equations are practical tests: do both routes in a commutative diagram produce the same result? Later, the same language organizes tensor products, adjunctions, and the fundamental group of a topological space. Emily Riehl's textbook develops these basic notions before using them to compare constructions across different subjects. ^[ref]

In their 1945 paper, Samuel Eilenberg and Saunders Mac Lane set out the notions of category, functor, and natural transformation. The introduction uses the relation between a vector space and its dual space to ask which comparisons are natural, setting up a precise language for compatibility with maps. ^[ref]

The product cells of finite CW complexes give the tensor product of their cellular chain complexes. The boundary formula is the displayed signed product rule. The sign comes from moving the outward normal of the second factor past the $p$ oriented tangent directions of the first factor. This is the cellular counterpart of the Eilenberg–Zilber theorem.

In a field of characteristic two, $1+1=0$. The usual splitting of a tensor product into its symmetric square and exterior square by $(I\pm \tau )/2$ is unavailable.

For finite-dimensional group representations, the character of their tensor product is the pointwise product of their characters. In a tensor-product basis, the diagonal entries of $A\otimes B$ are $A_{ii}{B}_{jj}$, so their sum is $(\mathrm{tr}A)(\mathrm{tr}B)$. This gives a direct route from a character table to tensor-product decompositions.

Clebsch-Gordan coefficients are the change-of-basis coefficients between an uncoupled tensor-product basis and a basis adapted to the irreducible decomposition of the tensor product.

The tensor product of spin-$j_1$ and spin-$j_2$ irreducible representations decomposes into one copy of each spinThe change-of-basis entries between uncoupled and coupled states are Clebsch--Gordan coefficients.

Let $A,B$ be unital associative algebras over a field, each a simple ring, and $J\ne 0$ a two-sided ideal of their tensor product. Choose $0\ne x={\sum }_{i=1}^n{a}_i\otimes b_i\in J$ of minimal two-factor tensor rank. Both factor families are linearly independent. Since $A{a}_{1}A=A$, choose ${\sum }_j{u}_j{a}_1{v}_j=1$ and replace $x$ by ${\sum }_j(u_j\otimes 1)x(v_j\otimes 1)$. The result has first coefficient $1$, is nonzero by the linear independence of the $b_i$, and retains minimal two-factor tensor rank; its first factor family is therefore still linearly independent. Now apply the same construction in $B$ to make the first second-factor coefficient $1$. The result is again nonzero of minimal two-factor tensor rank, and both its coefficient families are linearly independent.

If $Z(A)=k$, commutation with $a\otimes 1$ gives a tensor in $J$ expressible with at most $n-1$ pure tensors, hence zero by minimality. Independence forces every $c_i$ to be central, hence a scalar multiple of $1$. Independence of $1,c_2,\dots ,c_n$ then forces $n=1$. Thus $1\otimes 1\in J$, proving simplicity of the tensor product without a splitting-field argument.

For classes on $X$ and $Y$, apply the induced maps on cohomology of the coordinate projections and take their cup product on $X\times Y$. The result has degree $|a|+|b|$. On a tensor product of graded rings, the sign rule $(a\otimes b)(c\otimes d)=(-1{)}^{|b||c|}ac\otimes bd$ makes this a ring homomorphism.

Complexification replaces each fiber of a real vector bundle by its tensor product with $\mathbb{C}$ over $\mathbb{R}$. The complexification of a real vector bundle of rank one is a complex line bundle. For the tautological bundle over Real projective space, this complex line bundle squares to the trivial vector bundle.

The complexification of a real vector space $V$ allows complex scalar coefficients while preserving its real linear relations. It can be constructed as the tensor product $V{\otimes }_{\mathbb{R}}\mathbb{C}$, or as formal sums $u+iv$ with $u,v\in V$. A real basis of $V$ becomes a complex basis of $V_{\mathbb{C}}$, so its complex dimension equals the original real dimension. Every real linear map extends uniquely by complex linearity. In particular a real representation extends by viewing its real matrices as complex matrices. Complexifying the realification of a complex representation produces the original complex representation together with its complex conjugate; the coordinate change $(u,v)\mapsto (u+iv,u-iv)$ exhibits the two blocks.

Commuting Hamiltonian terms on separate factors are simultaneously diagonalized by tensor products of their eigenvectors and have additive eigenvalues.

If $f$ is homogeneous of degree one, multiplication by powers of the invertible element $f$ identifies every homogeneous component of $M_f$ with $(M_f{)}_0$. A homogeneous element $m$ of degree $a$ is written $f^a(f^{-a}m)$. Thus the displayed map is an isomorphism of graded modules. In particular, the degree-zero part of a tensor product of localized graded modules is the tensor product of their degree-zero parts over $(S_f{)}_0$. This proves tensor compatibility of graded sheafification on degree-one-generated Proj.