Blueprint for the Adjunction Formula: Depth and Regular Sequences

1.9 Depth and Regular Sequences

This mostly follows Bruns-Herzog 1.2. We are building up to Bruns-Herzog 1.2.12.

Actually maybe we don’t need depth??

Definition 1.9.1

The depth of a module \(M\) is . . .

Theorem 1.9.2 BH 1.2.12

Any regular sequence is part of a system of parameters

Proof ▼

This is from BH chapter 1, and uses associated primes. It also uses the definition of dimension of a module,

Let \(x \in \mathfrak {m}\) an \(M\)-regular element. As \(x \in R\) is \(M\)-regular, it is not in any associated prime (Lemma 1.8.5).

Then, as the associated primes are minimal among the primes that contain the annihilator (which is the support), we have that the previous statement implies that

\[ \dim M / xM \leq \dim M - 1 \]

by corollary 1.4.12

By dimension theory (see BH A.4), this inequality is an equality.

Given a regular sequence, iterate this process. Once we get to the end of this process, we can extend this sequence (by prime avoidance?) to a sequence of \(\dim M\) elements, by definition (of system of parameters) we have a system of parameters for M.