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.

Let R be a commutative Noetherian 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 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.

Suppose that R is a commutative Noetherian local ring with the maximal ideal ${\mathfrak {m}}$ ${\mathfrak {m}}$ and M is a finitely generated R-module. Then all maximal regular sequences x₁,..., x_n for M, where each x_i belongs to ${\mathfrak {m}}$ ${\mathfrak {m}}$ , have the same length n equal to the ${\mathfrak {m}}$ ${\mathfrak {m}}$ -depth of M.

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