General remarks

A coalgebra is a concept which is dual (in a sense that belongs to Category Theory) to that of an associative algebra, and consequently to almost every result that we have concerning algebras there is a corresponding result for coalgebras. Now it sometimes happens that an algebra and a coalgebra are built on the same underlying set. When this occurs, and provided the algebra and the coalgebra interact suitably, the result is called a Hopf algebra. Our concern with these matters stems from the fact that, when M is an R-module, both E(M) and S(M) are Hopf algebras of particular interest.

Whenever we have a coalgebra the linear forms on it can be considered as the elements of an algebra. The algebra which arises in this way from E(M) is known as the Grassmann algebra of M; for S(M) the resulting algebra has very close connections with the algebra of differential operators which was described in Chapter 6.

Throughout this chapter we shall follow our usual practice of using R to denote a commutative ring with an identity element; and when the tensor symbol ⊗ is used it is understood that the underlying ring is always R unless there is an explicit statement indicating the contrary.

Finally, it happens to be convenient, before introducing the notion of a coalgebra, to reformulate the definition of an R-algebra. This reformulation is carried out in Section (7.1).