Lagmental Vicfred

A Regular Sequence Uses Successive Non-Zero-Divisors by Vicfred

tags: mathematics, flatness, regular sequences, depth

Each element in a regular sequence must remain a non-zero-divisor after quotienting by the earlier ones. I want the notation, the mechanism, and the failure mode visible at the same time.

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.

$$ x_1,\ldots,x_r\in A $$

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.

$$ x_i\ \text{non-zero-divisor on }M/(x_1,\ldots,x_{i-1})M $$

Compute before generalising

A worked instance is useful here because it exposes every index that the compressed statement hides.

$$ \left.\begin{aligned}A&=k[x,y,z],\\M&=A,\quad(x_1,x_2)=(x,y)\end{aligned}\right\}\Longrightarrow x,y\text{ is an }M\text{-regular sequence} $$

The global view

The aligned summary deliberately puts the datum and conclusion on different rows. Mathematically, this is the distinction between specifying an object and proving a property of it.

$$ \begin{aligned} \mathsf{D}\;&:\quad x_1,\ldots,x_r\in A,\\[5pt] \mathsf{C}\;&:\quad x_i\ \text{non-zero-divisor on }M/(x_1,\ldots,x_{i-1})M. \end{aligned} $$

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.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] x_i\ \text{non-zero-divisor on }M/(x_1,\ldots,x_{i-1})M \end{gathered}} $$

This is enough machinery for one note: an exact object, a worked case, a structural law, and a clearly marked boundary. Each layer can now be tested independently.

This article was posted on Sun 17 August 2014. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.