General remarks

We have examined the origins and properties of the tensor and exterior algebras of a module. The symmetric algebra, which concerns us now, fits into the same general pattern and, as in the other cases, we shall motivate its investigation by means of an appropriate universal problem. However, from a different standpoint, one can say that the exterior algebra can be obtained from the tensor algebra by making it anticommutative. Viewed from this position, the symmetric algebra is the result of making the tensor algebra commutative.

Because much of the theory of the symmetric algebra runs closely parallel to that of the exterior algebra, we can frequently borrow proofs and adapt them without difficulty, and when this is the case we shall restrict the amount of detail that is given.

There is, however, one area in this account of symmetric algebras which is not foreshadowed in Chapter 5. The symmetric algebra of a module can be regarded as a generalization of a polynomial ring; and for a polynomial ring the partial differentiations with respect to the indeterminates generate a commutative algebra. When this observation is analysed it leads naturally to the algebra of differential operators and this is one of the topics we shall discuss. In the next chapter this particular algebra will be examined from a more general standpoint and, as a result, we shall be able to complete the analogy between exterior and symmetric algebras.