Concept
In commutative and homological algebra, depth is an important invariant of rings and modules. Although depth can be defined more generally, the most common case considered is the case of modules over a commutative Noetherian local ring. In this case, the depth of a module is related with its projective dimension by the Auslander–Buchsbaum formula. A more elementary property of depth is the inequality where denotes the Krull dimension of the module . Depth is used to define classes of rings and modules with good properties, for example, Cohen-Macaulay rings and modules, for which equality holds. Let be a commutative ring, an ideal of and a finitely generated -module with the property that is properly contained in . (That is, some elements of are not in .) Then the -depth of , also commonly called the grade of , is defined as By definition, the depth of a local ring with a maximal ideal is its -depth as a module over itself. If is a Cohen-Macaulay local ring, then depth of is equal to the dimension of . By a theorem of David Rees, the depth can also be characterized using the notion of a regular sequence. Suppose that is a commutative Noetherian local ring with the maximal ideal and is a finitely generated -module. Then all maximal regular sequences for , where each belongs to , have the same length equal to the -depth of . The projective dimension and the depth of a module over a commutative Noetherian local ring are complementary to each other. This is the content of the Auslander–Buchsbaum formula, which is not only of fundamental theoretical importance, but also provides an effective way to compute the depth of a module. Suppose that is a commutative Noetherian local ring with the maximal ideal and is a finitely generated -module. If the projective dimension of is finite, then the Auslander–Buchsbaum formula states A commutative Noetherian local ring has depth zero if and only if its maximal ideal is an associated prime, or, equivalently, when there is a nonzero element of such that (that is, annihilates ).