After dimension, depth is the most fundamental numerical invariant of a Noetherian local ring R or a finite R-module M. While depth is defined in terms of regular sequences, it can be measured by the (non-)vanishing of certain Ext modules. This connection opens commutative algebra to the application of homological methods. Depth is connected with projective dimension and several notions of linear algebra over Noetherian rings.

Equally important is the description of depth (and its global relative grade) in terms of the Koszul complex which, in a sense, holds an intermediate position between arithmetic and homological algebra.

This introductory chapter also contains a section on graded rings and modules. These allow a decomposition of their elements into homogeneous components and therefore have a more accessible structure than rings and modules in general.

Regular sequences

Let M be a module over a ring R. We say that x ∈ R is an M-regular element if xz = 0 for z ∈ M implies z = 0, in other words, if x is not a zero-divisor on M. Regular sequences are composed of successively regular elements:

Definition 1.1.1 A sequence x = x1,…,xn of elements of R is called an M-regular sequence or simply an M-sequence if the following conditions are satisfied: (i) xi is an M/(x1, …, xi−1)M-regular element for i = 1,…, n, and (ii) M/xM ≠ 0.

In this situation we shall sometimes say that M is an x-regular module. A regular sequence is an R-sequence.

A weak M-sequence is only required to satisfy condition (i).

Very often R will be a local ring with maximal ideal m, and M ≠ 0 a finite R-module.