Depth (ring theory)

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 dim "M" denotes the Krull dimension of the module "M". Depth is used to define classes of rings and modules with good properties, for example, Cohen-Macaulay rings and modules, for which equality holds.
Definition.
Let "R" be a commutative ring, "I" an ideal of "R" and "M" a finite "R"-module with the property that "IM" is properly contained in "M". Then the "I"-depth of "M", also commonly called the grade of "M", is defined as 
By definition, the depth of a local ring "R" with a maximal ideal formula_3 is its formula_3-depth as a module over itself. If "R" is a Cohen-Macaulay local ring, then depth of "R" is equal to the dimension of "R".
By a theorem of David Rees, the depth can also be characterized using the notion of a regular sequence.
Theorem (Rees).
Suppose that "R" is a commutative Noetherian local ring with the maximal ideal formula_3 and "M" is a finitely generated "R"-module. Then all maximal regular sequences "x""1"..., "x""n" for "M", where each "x""i" belongs to formula_3, have the same length "n" equal to the formula_3-depth of "M".
Depth and projective dimension.
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 "R" is a commutative Noetherian local ring with the maximal ideal formula_3 and "M" is a finitely generated "R"-module. If the projective dimension of "M" is finite, then the Auslander–Buchsbaum formula states
Depth zero rings.
A commutative Noetherian local ring "R" has depth zero if and only if its maximal ideal formula_3 is an associated prime, or, equivalently, when there is a nonzero element "x" of "R" such that formula_11 (that is, "x" annihilates formula_3). This means, essentially, that the closed point is an embedded component.
For example, the ring formula_13 (where "k" is a field), which represents a line (formula_14) with an embedded double point at the origin, has depth zero at the origin, but dimension one: this gives an example of a ring which is not Cohen–Macaulay.