Depth (ring theory) explained

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

depth(M)\leq\dim(M),

where

\dimM

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

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

depth_I(M)=min\{i:\operatorname{Ext}^i(R/I,M)\ne0\}.

By definition, the depth of a local ring

with a maximal ideal

ak{m}

is its

ak{m}

-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)

Suppose that

is a commutative Noetherian local ring with the maximal ideal

ak{m}

and

is a finitely generated

-module. Then all maximal regular sequences

x_1,\ldots,x_n

for

, where each

x_i

belongs to

ak{m}

, have the same length

equal to the

ak{m}

-depth of

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

is a commutative Noetherian local ring with the maximal ideal

ak{m}

and

is a finitely generated

-module. If the projective dimension of

is finite, then the Auslander–Buchsbaum formula states

pd_R(M)+depth(M)=depth(R).

Depth zero rings

A commutative Noetherian local ring

has depth zero if and only if its maximal ideal

ak{m}

is an associated prime, or, equivalently, when there is a nonzero element

such that

xak{m}=0

(that is,

annihilates

ak{m}

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

For example, the ring

k[x,y]/(x^2,xy)

(where

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

- Winfried Bruns; Jürgen Herzog, Cohen–Macaulay rings. Cambridge Studies in Advanced Mathematics, 39. Cambridge University Press, Cambridge, 1993. xii+403 pp.