General remarks

The exterior algebra is one of the most interesting and useful of the algebras that can be derived from a module. As we shall see it is an anticommutative algebra having intimate connections with the theory of determinants. Our aim in this chapter will be to establish all its main properties; and in so doing we shall follow a pattern which can be used again, with only small modifications, to study a related algebra, namely the symmetric algebra of a module.

As usual R and S are reserved to denote commutative rings with an identity element, and we again allow ourselves the freedom to omit the suffix from the tensor symbol when there is no uncertainty concerning the ground ring. Algebras are understood to be associative and to have an identity element; and homomorphisms of rings and algebras are required to preserve identity elements. Finally T(M) denotes the tensor algebra of an R-module M.

The exterior algebra

Let M be an R-module. We propose to define its exterior algebra, and, as in the case of the tensor algebra, we shall do this by means of a universal problem. However, because we are now in a position to make use of the properties of tensor algebras, the details on this occasion will be much simpler.

Suppose that φ: M → A is an R-linear mapping of M into an R-algebra A, and suppose that (φ(m))2=0 for all m ∈ M. If now h: A → B is a homomorphism of R-algebras, then h ∘ φ is an R-linear mapping of M into B with the property that, for every m ∈ M, the square of (h ∘ φ)(m) is zero.