Regular sequence

For regular Cauchy sequence, see Cauchy sequence#In constructive mathematics.

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

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.

An R-regular sequence is usually called simply a regular 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.

Examples

  1. If k is a field, it possesses no non-zero non-unit elements so its depth as a k-module is 0.
  2. 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.
  3. 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: 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.

References