Depth (ring theory)

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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.

Definition

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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.

Theorem (Rees)

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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  .

Depth and projective dimension

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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

 

Depth zero rings

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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  ). This means, essentially, that the closed point is an embedded component.

For example, the ring   (where   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.

References

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  • Eisenbud, David (1995), Commutative algebra with a view toward algebraic geometry, Graduate Texts in Mathematics, vol. 150, Berlin, New York: Springer-Verlag, ISBN 978-0-387-94269-8, MR 1322960
  • Winfried Bruns; Jürgen Herzog, Cohen–Macaulay rings. Cambridge Studies in Advanced Mathematics, 39. Cambridge University Press, Cambridge, 1993. xii+403 pp. ISBN 0-521-41068-1