In
commutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
and
homological
Homology may refer to:
Sciences
Biology
*Homology (biology), any characteristic of biological organisms that is derived from a common ancestor
*Sequence homology, biological homology between DNA, RNA, or protein sequences
*Homologous chromo ...
algebra, depth is an important invariant of
rings and
modules
Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a s ...
. Although depth can be defined more generally, the most common case considered is the case of modules over a commutative
Noetherian In mathematics, the adjective Noetherian is used to describe objects that satisfy an ascending or descending chain condition on certain kinds of subobjects, meaning that certain ascending or descending sequences of subobjects must have finite lengt ...
local ring In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebrai ...
. 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
In commutative algebra, the Krull dimension of a commutative ring ''R'', named after Wolfgang Krull, is the supremum of the lengths of all chains of prime ideals. The Krull dimension need not be finite even for a Noetherian ring. More generally th ...
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
:
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 David or Dai Rees may refer to:
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* Dave Rees (born 1969), American drummer for SNFU and Wheat Chiefs
* David Rees (cartoonist) (born 1972), American cartoonist and televis ...
, the depth can also be characterized using the notion of a
regular sequence.
Theorem (Rees)
Suppose that
is a commutative Noetherian
local ring In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebrai ...
with the maximal
ideal
Ideal may refer to:
Philosophy
* Ideal (ethics), values that one actively pursues as goals
* Platonic ideal, a philosophical idea of trueness of form, associated with Plato
Mathematics
* Ideal (ring theory), special subsets of a ring considered ...
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
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 In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebrai ...
with the maximal
ideal
Ideal may refer to:
Philosophy
* Ideal (ethics), values that one actively pursues as goals
* Platonic ideal, a philosophical idea of trueness of form, associated with Plato
Mathematics
* Ideal (ring theory), special subsets of a ring considered ...
and
is a finitely generated
-module. If the projective dimension of
is finite, then the
Auslander–Buchsbaum formula states
:
Depth zero rings
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
*
* 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
Module theory
Commutative algebra