All rings considered in this book will have identity elements.

Throughout the book, R will always denote a non-trivial commutative Noetherian ring, and a will denote an ideal of R. We shall only assume that R has additional properties (such as being local) when these are explicitly stated; however, the phrase ‘(R,m) is a local ring’ will mean that R is a commutative Noetherian quasi-local ring with unique maximal ideal m.

For an ideal c of R, we denote Supp(R/c) = {p ∈ Spec(R) : p ⊇ c} by Var(c), and refer to this as the variety of c.

By a multiplicatively closed subset of R, we shall mean a subset of R which is closed under multiplication and contains 1. It should be noted (and this comment is relevant for the final chapter) that, if S is a non-empty subset of R which is closed under multiplication, then, even if S does not contain 1, we can form the commutative ring S−1R and, for an R-module M, the S−1Rmodule S−1M. In fact, S−1R ≅ (S ∪ {1})−1R, and, in S−1R, the element sr/s, for r ∈ R and s ∈ S, is independent of the choice of such s; similar comments apply to S−1M.

The symbol ℤ will always denote the ring of integers; in addition, ℕ (respectively ℕ0) will always denote the set of positive (respectively non-negative) integers. The field of rational (respectively real, complex) numbers will be denoted by ℚ (respectively ℝ, ℂ).