Regular sequence

In commutative algebra, a commutative ring R is said to have depth d iff there exists an R-sequence of length d on R and no longer R-sequence. More generally, an R-module M is likewise said to have depth d iff there exists an R-regular sequence on M of length d and no longer such sequence. The concept is inherently module-theoretic and so there is no harm in approaching it from this point of view.

A specific ordered sequence of (non-zero non-unit) elements r₁, r₂, ..., r_d from R is said to be an R-regular sequence on M iff for each i, r_i is not a zerodivisor on the quotient R-module M/(r₁, r₂, ..., r_i-1)M, 1 &le i &le d. In particular this means that r₁ is required to be a nonzerodivisor on M.

The sequence r₁, r₂, ..., r_d may be a regular sequence on M and yet not be a regular sequence under a permutation. However it is a theorem that if R is local any R-sequence is regular if and only if every permutation of it is regular.

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, ..., 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 module equals its Krull dimension: the module is then said to be Cohen-Macaulay.