Regular sequence (algebra)

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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, ..., r_d from R

such that for each i, r_i is not a zerodivisor on the quotient R-module

M/(r₁, r₂, ..., r_i-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, ..., r_d 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.

[edit] 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, ..., X_d are indeterminates, then X₁, X₂, ..., X_d form a regular sequence of length d on the polynomial ring k[X₁, X₂, ..., X_d] 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.