In
commutative algebra
Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominent ...
, a regular sequence is a sequence of elements of a
commutative ring
In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not sp ...
which are as independent as possible, in a precise sense. This is the algebraic analogue of the geometric notion of a
complete intersection
In mathematics, an algebraic variety ''V'' in projective space is a complete intersection if the ideal of ''V'' is generated by exactly ''codim V'' elements. That is, if ''V'' has dimension ''m'' and lies in projective space ''P'n'', there shou ...
.
Definitions
For a commutative ring ''R'' and an ''R''-
module
Module, modular and modularity may refer to the concept of modularity. They may also refer to:
Computing and engineering
* Modular design, the engineering discipline of designing complex devices using separately designed sub-components
* Modul ...
''M'', an element ''r'' in ''R'' is called a non-zero-divisor on ''M'' if ''r m'' = 0 implies ''m'' = 0 for ''m'' in ''M''. An ''M''-regular sequence is a sequence
:''r''
1, ..., ''r''
''d'' in ''R''
such that ''r''
''i'' is a not a zero-divisor on ''M''/(''r''
1, ..., ''r''
''i''-1)''M'' for ''i'' = 1, ..., ''d''. Some authors also require that ''M''/(''r''
1, ..., ''r''
''d'')''M'' is not zero. Intuitively, to say that
''r''
1, ..., ''r''
''d'' is an ''M''-regular sequence means that these elements "cut ''M'' down" as much as possible, when we pass successively from ''M'' to ''M''/(''r''
1)''M'', to ''M''/(''r''
1, ''r''
2)''M'', and so on.
An ''R''-regular sequence is called simply a regular sequence. That is, ''r''
1, ..., ''r''
''d'' is a regular sequence if ''r''
1 is a non-zero-divisor in ''R'', ''r''
2 is a non-zero-divisor in the ring ''R''/(''r''
1), and so on. In geometric language, if ''X'' is an
affine scheme
In commutative algebra, the prime spectrum (or simply the spectrum) of a ring ''R'' is the set of all prime ideals of ''R'', and is usually denoted by \operatorname; in algebraic geometry it is simultaneously a topological space equipped with the ...
and ''r''
1, ..., ''r''
''d'' is a regular sequence in the ring of regular functions on ''X'', then we say that the closed subscheme ⊂ ''X'' is a
complete intersection
In mathematics, an algebraic variety ''V'' in projective space is a complete intersection if the ideal of ''V'' is generated by exactly ''codim V'' elements. That is, if ''V'' has dimension ''m'' and lies in projective space ''P'n'', there shou ...
subscheme of ''X''.
Being a regular sequence may depend on the order of the elements. For example, ''x'', ''y''(1-''x''), ''z''(1-''x'') is a regular sequence in the polynomial ring C
'x'', ''y'', ''z'' while ''y''(1-''x''), ''z''(1-''x''), ''x'' is not a regular sequence. But if ''R'' is a
Noetherian In mathematics, the adjective Noetherian is used to describe Category_theory#Categories.2C_objects.2C_and_morphisms, objects that satisfy an ascending chain condition, ascending or descending chain condition on certain kinds of subobjects, meaning t ...
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 algebraic num ...
and the elements ''r''
''i'' are in the maximal ideal, or if ''R'' is a
graded ring
In mathematics, in particular abstract algebra, a graded ring is a ring such that the underlying additive group is a direct sum of abelian groups R_i such that R_i R_j \subseteq R_. The index set is usually the set of nonnegative integers or the se ...
and the ''r''
''i'' are homogeneous of positive degree, then any permutation of a regular sequence is a regular sequence.
Let ''R'' be a Noetherian ring, ''I'' an ideal in ''R'', and ''M'' a finitely generated ''R''-module. The
depth of ''I'' on ''M'', written depth
''R''(''I'', ''M'') or just depth(''I'', ''M''), is the supremum of the lengths of all ''M''-regular sequences of elements of ''I''. When ''R'' is a Noetherian local ring and ''M'' is a finitely generated ''R''-module, the depth of ''M'', written depth
''R''(''M'') or just depth(''M''), means depth
''R''(''m'', ''M''); that is, it is the supremum of the lengths of all ''M''-regular sequences in the maximal ideal ''m'' of ''R''. In particular, the depth of a Noetherian local ring ''R'' means the depth of ''R'' as a ''R''-module. That is, the depth of ''R'' is the maximum length of a regular sequence in the maximal ideal.
For a Noetherian local ring ''R'', the depth of the zero module is ∞, whereas the depth of a nonzero finitely generated ''R''-module ''M'' is at most 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 t ...
of ''M'' (also called the dimension of the support of ''M'').
[N. Bourbaki. ''Algèbre Commutative. Chapitre 10.'' Springer-Verlag (2007). Th. X.4.2.]
Examples
*Given an integral domain
any nonzero
gives a regular sequence.
*For a prime number ''p'', the local ring Z
(''p'') is the subring of the rational numbers consisting of fractions whose denominator is not a multiple of ''p''. The element ''p'' is a non-zero-divisor in Z
(''p''), and the quotient ring of Z
(''p'') by the ideal generated by ''p'' is the field Z/(''p''). Therefore ''p'' cannot be extended to a longer regular sequence in the maximal ideal (''p''), and in fact the local ring Z
(''p'') has depth 1.
*For any field ''k'', the elements ''x''
1, ..., ''x''
''n'' in the polynomial ring ''A'' = ''k''
1, ..., ''x''''n''">'x''1, ..., ''x''''n''form a regular sequence. It follows that the
localization
Localization or localisation may refer to:
Biology
* Localization of function, locating psychological functions in the brain or nervous system; see Linguistic intelligence
* Localization of sensation, ability to tell what part of the body is a ...
''R'' of ''A'' at the maximal ideal ''m'' = (''x''
1, ..., ''x''
''n'') has depth at least ''n''. In fact, ''R'' has depth equal to ''n''; that is, there is no regular sequence in the maximal ideal of length greater than ''n''.
*More generally, let ''R'' be a
regular local ring In commutative algebra, a regular local ring is a Noetherian local ring having the property that the minimal number of generators of its maximal ideal is equal to its Krull dimension. In symbols, let ''A'' be a Noetherian local ring with maximal ide ...
with maximal ideal ''m''. Then any elements ''r''
1, ..., ''r''
''d'' of ''m'' which map to a basis for ''m''/''m''
2 as an ''R''/''m''-vector space form a regular sequence.
An important case is when the depth of a local ring ''R'' is equal to its
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 t ...
: ''R'' is then said to be
Cohen-Macaulay. The three examples shown are all Cohen-Macaulay rings. Similarly, a finitely generated ''R''-module ''M'' is said to be Cohen-Macaulay if its depth equals its dimension.
Non-Examples
A simple non-example of a regular sequence is given by the sequence
of elements in