Tensor Products

No algebra course would be complete without introducing the student to the language of category theory. Some properties of the objects of algebra are defined by their internal structure, while other properties describe how the object sits in a morphism-closed environment. Universal mapping properties are of the latter sort. Their relation to initial and terminal objects of another suitably-chosen category is emphasized. The tensor product in the category of right R-modules is defined in two ways: as a constructed object, and as a unique solution to a universal mapping problem. From the tensor product one derives functors which are adjoint to the “Hom” functors. Another feature is that tensor products can also be defined for F-algebras. The key facts that tensor products “distribute” over direct sums and that there is a uniform way to define multiple tensor products, allows one to define the tensor algebra. In the category of F-algebras generated by n elements, this algebra becomes an initial object. This graded algebra, T(V), is uniquely determined by an F-vector space V and has two important homomorphic offspring: the symmetric algebra, S(V) (modeled by polynomial rings), and the exterior algebra, E(V), (modeled by an algebra on poset chains). In the category of vector spaces, T, S and E, and their restrictions to the homogenous summands, are all functors—that is, morphisms among vector spaces induce morphisms among the algebras and their components of fixed degree. Herein lie the basic theorems concerning multilinear forms.