Regular sequence (algebra)
Encyclopedia
In commutative algebra
, if R is a commutative ring
and M an R-module
, a nonzero element r in R is called M-regular if r is not a zerodivisor on M, and M/rM is nonzero. An R-regular sequence on M is a d-tuple
such that for each i ≤ d, ri is Mi-1-regular, where Mi-1 is the quotient R-module
Such a sequence is also called an M-sequence.
It may be that r1, ..., rd is an M-sequence, and yet some permutation of the sequence is not. It is, however, a theorem that if R is a local ring
or if R is a graded ring and the ri are all homogeneous, then a sequence is an R-sequence only if every permutation of it is an R-sequence.
The depth
of R is defined as the maximum length of a regular R-sequence on R. More generally, the depth of an R-module M is the maximum length of an M-regular sequence on M. The concept is inherently module-theoretic and so there is no harm in approaching it from this point of view.
The depth of a module is always at least 0 and no greater than the Krull dimension
of the module.
An important case is when the depth of a ring equals its Krull dimension
: the ring is then said to be a Cohen-Macaulay ring
. The three examples shown are all Cohen-Macaulay rings. Similarly in the case of modules, the module M is said to be Cohen-Macaulay if its depth equals its dimension.
Commutative algebra
Commutative algebra is the branch of abstract algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra...
, if R is a commutative ring
Commutative ring
In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....
and M an R-module
Module (mathematics)
In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...
, a nonzero element r in R is called M-regular if r is not a zerodivisor on M, and M/rM is nonzero. An R-regular sequence on M is a d-tuple
- r1, ..., rd in R
such that for each i ≤ d, ri is Mi-1-regular, where Mi-1 is the quotient R-module
- M/(r1, ..., ri-1)M.
Such a sequence is also called an M-sequence.
It may be that r1, ..., rd is an M-sequence, and yet some permutation of the sequence is not. It is, however, a theorem that if R is a local ring
Local ring
In abstract algebra, more particularly in 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 number fields examined at a particular place, or...
or if R is a graded ring and the ri are all homogeneous, then a sequence is an R-sequence only if every permutation of it is an R-sequence.
The depth
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...
of R is defined as the maximum length of a regular R-sequence on R. More generally, the depth of an R-module M is the maximum length of an M-regular sequence on M. The concept is inherently module-theoretic and so there is no harm in approaching it from this point of view.
The depth of a module is always at least 0 and no greater than the Krull dimension
Krull dimension
In commutative algebra, the Krull dimension of a ring R, named after Wolfgang Krull , is the supremum of the number of strict inclusions in a chain of prime ideals. The Krull dimension need not be finite even for a Noetherian ring....
of the module.
Examples
- If k is a field, it possesses no non-zero non-unit elements so its depth as a k-module is 0.
- If k is a field and X is an indeterminate, then X is a nonzerodivisor on the formal power series ring R = k
X , but R/XR is a field and has no further nonzerodivisors. Therefore R has depth 1. - If k is a field and X1, X2, ..., Xd are indeterminates, then X1, X2, ..., Xd form a regular sequence of length d on the polynomial ring k[X1, X2, ..., Xd] and there are no longer R-sequences, so R has depth d, as does the formal power series ring in d indeterminates over any field.
An important case is when the depth of a ring equals its Krull dimension
Krull dimension
In commutative algebra, the Krull dimension of a ring R, named after Wolfgang Krull , is the supremum of the number of strict inclusions in a chain of prime ideals. The Krull dimension need not be finite even for a Noetherian ring....
: the ring is then said to be a Cohen-Macaulay ring
Cohen-Macaulay ring
In mathematics, a Cohen–Macaulay ring is a particular type of commutative ring, possessing some of the algebraic-geometric properties of a nonsingular variety, such as local equidimensionality....
. The three examples shown are all Cohen-Macaulay rings. Similarly in the case of modules, the module M is said to be Cohen-Macaulay if its depth equals its dimension.