 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, CohenMacaulay rings and modules, for which the equality holds.
Contents
Definition
Let R be a commutative Noetherian ring, I an ideal of R and M a finite Rmodule with the property that IM is properly contained in M. Then the Idepth of M, also commonly called the grade of M, is defined as
By definition, the depth of a ring R is its depth as a module over itself.
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 and M is a finitely generated Rmodule. Then all maximal regular sequences x_{1},..., x_{n} for M, where each x_{i} belongs to , have the same length n equal to the 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 the 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 and M is a finitely generated Rmodule. If the projective dimension of M is finite, then the Auslander–Buchsbaum formula states
 pd_{R}(M) + depth(M) = depth(R).
Depth zero rings
A commutative Noetherian local ring R has depth zero if and only if its maximal ideal is an associated prime, or, equivalently, when there is a nonzero element x of R such that (that is, x annihilates ). This means, essentially, that the closed point is an embedded component.
For example, the ring k[x,y] / (x^{2},xy) (where k is a field), which represents a line (x = 0) 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.
References
 Eisenbud, David (1995), Commutative algebra with a view toward algebraic geometry, Graduate Texts in Mathematics, 150, Berlin, New York: SpringerVerlag, ISBN 9780387942698, MR1322960
 Winfried Bruns; Jürgen Herzog, Cohen–Macaulay rings. Cambridge Studies in Advanced Mathematics, 39. Cambridge University Press, Cambridge, 1993. xii+403 pp. ISBN 0521410681
Categories: Module theory
 Commutative algebra
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