Tensor product of algebras

Let R be a commutative ring and let A and B be R-algebras. Since A and B may both be regarded as , their tensor product :A \otimes_R B is also an R-module. The tensor product can be given the structure of a ring by defining the product on elements of the form by :(a_1\otimes b_1)(a_2\otimes b_2) = a_1 a_2\otimes b_1b_2 and then extending by linearity to all of . This ring is an R-algebra, associative and unital with the identity element given by , where 1A and 1B are the identity elements of A and B. If A and B are commutative, then the tensor product is commutative as well. The tensor product turns the category of R-algebras into a symmetric monoidal category. ==Further properties==

There are natural homomorphisms from A and B to given by :a\mapsto a\otimes 1_B :b\mapsto 1_A\otimes b These maps make the tensor product the coproduct in the category of commutative R-algebras. The tensor product is not the coproduct in the category of all R-algebras; there the coproduct is given by a more general free product of algebras. Nevertheless, the tensor product of non-commutative algebras can be described by a universal property similar to that of the coproduct: :\text{Hom}(A\otimes B,X) \cong \lbrace (f,g)\in \text{Hom}(A,X)\times \text{Hom}(B,X) \mid \forall a \in A, b \in B: [f(a), g(b)] = 0\rbrace, where [-, -] denotes the commutator. The natural isomorphism is given by identifying a morphism \phi:A\otimes B\to X on the left hand side with the pair of morphisms (f,g) on the right hand side where f(a):=\phi(a\otimes 1) and similarly g(b):=\phi(1\otimes b). ==Applications==

The tensor product of commutative algebras is of frequent use in algebraic geometry. For affine schemes X, Y, Z with morphisms from X and Z to Y, so X = Spec(A), Y = Spec(R), and Z = Spec(B) for some commutative rings A, R, B, the fiber product scheme is the affine scheme corresponding to the tensor product of algebras: :X\times_Y Z = \operatorname{Spec}(A\otimes_R B). More generally, the fiber product of schemes is defined by gluing together affine fiber products of this form. ==Examples==

• The tensor product can be used as a means of taking intersections of two subschemes in a scheme: consider the \mathbb{C}[x,y]-algebras \mathbb{C}[x,y]/f, \mathbb{C}[x,y]/g, then their tensor product is \mathbb{C}[x,y]/(f) \otimes_{\mathbb{C}[x,y]} \mathbb{C}[x,y]/(g) \cong \mathbb{C}[x,y]/(f,g), which describes the intersection of the algebraic curves f = 0 and g = 0 in the affine plane over C. • More generally, if A is a commutative ring and I,J\subseteq A are ideals, then \frac{A}{I}\otimes_A\frac{A}{J}\cong \frac{A}{I+J}, with a unique isomorphism sending (a+I) \otimes (b+J) to ab + I+J. • Tensor products can be used as a means of changing coefficients. For example, \mathbb{Z}[x,y]/(x^3 + 5x^2 + x - 1)\otimes_\mathbb{Z} \mathbb{Z}/5 \cong \mathbb{Z}/5[x,y]/(x^3 + x - 1) and \mathbb{Z}[x,y]/(f) \otimes_\mathbb{Z} \mathbb{C} \cong \mathbb{C}[x,y]/(f). • Tensor products also can be used for taking products of affine schemes over a field. For example, \mathbb{C}[x_1,x_2]/(f(x)) \otimes_\mathbb{C} \mathbb{C}[y_1,y_2]/(g(y)) is isomorphic to the algebra \mathbb{C}[x_1,x_2,y_1,y_2]/(f(x),g(y)) which corresponds to an affine surface in \mathbb{A}^4_\mathbb{C} if f and g are not zero. • Given R-algebras A and B whose underlying rings are graded-commutative rings, the tensor product A\otimes_RB becomes a graded commutative ring by defining (a\otimes b)(a'\otimes b')=(-1)^aa'\otimes bb' for homogeneous a, a', b, and b'. • The tensor product of two matrix algebras is M_m(A)\otimes_AM_n(A)\cong M_{mn}(A), the isomorphism given by extending the Kronecker product of two matrices via the universal property. ==See also==