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.
a finitely generated
is properly contained in
, also commonly called the grade of
, is defined as By definition, the depth of a local ring
with a maximal ideal
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.
is a commutative Noetherian local ring with the maximal ideal
is a finitely generated
Then all maximal regular sequences
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.
is a commutative Noetherian local ring with the maximal ideal
is a finitely generated
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
This means, essentially, that the closed point is an embedded component.
is a field), which represents a line (
) 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.