Regular sequence (algebra)
|
|
In commutative algebra, if R is a commutative ring and M an R-module, a R-regular sequence on M is a d-tuple of (non-zero non-unit) elements
- r1, r2, ..., rd from R
such that for each i, ri is not a zerodivisor on the quotient R-module
- M/(r1, r2, ..., ri-1)M, 1 ≤ i ≤ d.
In particular this means that r1 is required to be a non-zerodivisor on M. Such a sequence is also called an R-sequence or a regular sequence, for short.
The sequence r1, r2, ..., rd may be a regular sequence on M and yet not be a regular sequence under a permutation. It is, however, a theorem that for R a local ring, an R-sequence is regular only if every permutation of it is regular.
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 R-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 dimension 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: 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.