Exterior Powers

Let R be a commutative ring. Unless indicated otherwise, all modules are R-modules and all tensor products are taken over R, so we abbreviate ⊗R to ⊗. A bilinear function out of M1 × M2 turns into a linear function out of the tensor product M1 ⊗ M2. In a similar way, a multilinear function out of M1 × · · · ×Mk turns into a linear function out of the k-fold tensor product M1 ⊗ · · · ⊗Mk. We will concern ourselves with the case when the component modules are all the same: M1 = · · · = Mk = M . The tensor power M⊗k universally linearizes all the multilinear functions on Mk. A function on Mk can have its variables permuted to give new functions on Mk. When looking at permutations of the variables, two important types of functions on Mk occur: symmetric and alternating. These will be defined in Section 2. We will introduce in Section 3 the module which universally linearizes the alternating multilinear functions on Mk: the exterior power Λk(M). It is a certain quotient module of M⊗k. The special case of exterior powers of finite free modules will be examined in Section 4. Exterior powers will be extended from modules to linear maps in Section 5. Applications of exterior powers to determinants are in Section 6 and to linear independence are in Section 7. Section 8 will introduce a product Λk(M)× Λ`(N)→ Λk+`(M) Finally, in Section 9 we will combine all the exterior powers of a fixed module into a noncommutative ring called the exterior algebra of the module. The exterior power construction is important in geometry, where it provides the language for discussing differential forms on manifolds. (A differential form on a manifold is related to exterior powers of the dual space of the tangent space of a manifold at each of its points.) Exterior powers also arise in representation theory, as one of several ways of creating new representations of a group from a given representation of the group. In linear algebra, exterior powers provide an algebraic mechanism for detecting linear relations among vectors and for studying the “geometry” of the subspaces of vector space.