Lagmental Vicfred

Cohen--Macaulay Rings Have No Depth Defect by Vicfred

tags: mathematics, flatness, regular sequences, depth

A local ring is Cohen--Macaulay when depth equals Krull dimension. The example is deliberately concrete; it is a test of the statement, not a substitute for it.

Start locally

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.

$$ (A,\mathfrak m)\ \text{Noetherian local} $$

The first display fixes the mathematical data. I label it \(\mathsf{data}\) mentally, while the next is the \(\mathsf{claim}\); the bridge between them is the displayed \(\Longrightarrow\), not an automatic implication.

$$ A\ \text{Cohen--Macaulay}\Longleftrightarrow\operatorname{depth}A=\dim A $$

Compute before generalising

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

$$ A=k[[x,y,z]]/(xy-z^2),\qquad\dim A=2,\qquad\operatorname{depth}A=2 $$

The global view

What survives the example is not its particular numbers but the relation encoded by the two rows below. That relation is the part worth transporting to a new setting.

$$ \begin{aligned} \mathsf{D}\;&:\quad (A,\mathfrak m)\ \text{Noetherian local},\\[5pt] \mathsf{C}\;&:\quad A\ \text{Cohen--Macaulay}\Longleftrightarrow\operatorname{depth}A=\dim A. \end{aligned} $$

Edge conditions

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.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] A\ \text{Cohen--Macaulay}\Longleftrightarrow\operatorname{depth}A=\dim A \end{gathered}} $$

With the dependency made explicit, the same pattern can be recognised safely in nearby problems. A changed hypothesis should now be easy to spot.

This article was posted on Mon 21 April 2025. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.