Lagmental Vicfred

Depth Is the Length of a Maximal Regular Sequence by Vicfred

tags: mathematics, flatness, regular sequences, depth

The depth of a finite local module measures how long regular sequences can continue inside the maximal ideal. This is a compact note, but the quantifiers and hypotheses stay on the page.

Definitions first

An \(A\)-module \(M\) is flat when \(-\otimes_AM\) preserves injections. Regular sequences then measure how many successive non-zero-divisors can be imposed before a module collapses.

$$ \operatorname{depth}_A(M)=\sup\{r:\exists\ M\text{-regular }x_1,\ldots,x_r\in\mathfrak m\} $$

The typography mirrors the proof: first declare \(\mathsf D\), then state \(\mathsf C\). The symbol \(\Longrightarrow\) below is a logical dependency, not extra mathematical structure.

$$ \operatorname{depth}_A(M)\le\dim_A(M) $$

A small case in full

The middle display is intentionally dense: it is where signs, bounds, multiplicities, or normalising factors are most likely to be lost.

$$ \operatorname{depth}_{k[[x_1,\ldots,x_n]]}k[[x_1,\ldots,x_n]]=n=\dim k[[x_1,\ldots,x_n]] $$

The reusable statement

The abstraction earns its keep by explaining why the same computation reappears. The notation compresses repeated reasoning without erasing the hypothesis that licenses it.

$$ \begin{aligned} \mathsf{D}\;&:\quad \operatorname{depth}_A(M)=\sup\{r:\exists\ M\text{-regular }x_1,\ldots,x_r\in\mathfrak m\},\\[5pt] \mathsf{C}\;&:\quad \operatorname{depth}_A(M)\le\dim_A(M). \end{aligned} $$

A nearby false statement

Vanishing of one \(\operatorname{Tor}\) group can certify flatness only under the correct quantifiers. Depth also depends on the chosen ideal or local maximal ideal.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \operatorname{depth}_A(M)\le\dim_A(M) \end{gathered}} $$

I would use the boxed line as a reference later, while returning to the full display whenever a hypothesis becomes uncertain. That division keeps compression from becoming ambiguity.

This article was posted on Sun 22 August 2010. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.