Regular sequences and depth

For an R-module M, a sequence a₁,…,a_r is M-regular when multiplication by a_i is injective on M/(a₁,…,a_{i−1})M for each i, and the final quotient is nonzero. This last condition rules out counting a unit as a meaningful cut: a unit acts injectively but makes the quotient zero. The order is part of the definition. Global rings can have regular sequences whose permutations fail regularity, so one should not reorder equations without a theorem supporting that step.

We now specialize to a Noetherian local ring (R,m) and a finite nonzero module M, with all chosen elements in m. In this setting regular sequences can be permuted, and a regular sequence can be extended until it is maximal. The nonzero quotient requirement follows along such sequences from the local finite-module setting and Nakayama’s lemma, discussed next. For finding a first regular element, avoid the union of Ass(M), which is the zero-divisor set. Avoiding only minimal support primes can fail because an embedded associated prime may still contain the chosen element.

For the stated Noetherian local R and finite nonzero M, depth M is the supremum of lengths of M-regular sequences in m. It is finite, and every maximal such sequence has that same length. Here maximal means no further admissible element can be appended, not simply that a particular list seems long. A theorem using associated primes and regular-element reduction shows that different maximal choices agree. We state this theorem rather than substituting one successful sequence as proof that no longer sequence exists.

The fundamental inequality is depth M≤dim Supp M. If a∈m is M-regular, then depth(M/aM)=depth M−1. A module is Cohen–Macaulay when its depth equals its dimension. This equality does not require the module to be a domain or reduced: a quotient can have nilpotents and still have the maximum possible regular-sequence length. The zero module is conventionally assigned infinite depth in many texts; our computations use nonzero M so that this convention cannot distort a finite-dimensional example.

Depth is zero exactly when every element of m is a zero divisor on M. In the finite Noetherian local setting, this is equivalent to m being an associated prime: there is a nonzero element killed by the whole maximal ideal. If the support has positive dimension, this can be an embedded obstruction. The module may still have a large-looking geometric component, but no first regular cut through the chosen local point. This explains why support dimension alone cannot determine depth.

One should also separate regular sequences from regular local rings. A regular local ring has embedding dimension dim_k m/m² equal to its Krull dimension, which is a stronger structural property than merely displaying some nonzerodivisors. Cohen–Macaulay modules and rings need not be regular. When claiming a depth value, exhibit a regular sequence for a lower bound and use dimension or a zero-divisor obstruction for the upper bound. The worked examples follow that two-sided method rather than assuming the first plausible candidate list is automatically maximal.