Any reader with a basic grounding in algebraic topology will recall the important rôle that the Mayer-Vietoris Sequence can play in that subject. There is an analogue of the Mayer-Vietoris Sequence in local cohomology theory, and it can play a foundational role in this subject. It is our intention in this chapter to present the basic theory of the Mayer-Vietoris Sequence in local cohomology, and to prepare for several uses of the idea during the subsequent development.

The Mayer-Vietoris Sequence involves two ideals, and so throughout this chapter, b will denote a second ideal of R (in addition to a). Let M be an R-module. The Mayer-Vietoris Sequence provides, among other things, a long exact sequence

of local cohomology modules. Its potential for use in arguments that employ induction on the number of elements in a generating set for an ideal c of <I>R can be explained as follows. Suppose that c is generated by n elements c1,…,cn, where n > 1. Set a = Rc1 + … Rcn−1 and b = Rcn, so that c = a + b. Each of a and b can be generated by fewer than n elements, but at first sight it seems that the ideal a ∩ b, which also appears in the Mayer-Vietoris Sequence, could present difficulties. However, and so by 1.1.3; hence for all i ∈ ℕ0 (see 1.2.3). Moreover, in our situation,

can be generated by n − 1 elements. Thus a, b and ab can all be generated by fewer than n elements, and an appropriate inductive hypothesis would apply to all of them.