Lagmental Vicfred

Flat Modules Preserve Short Exact Sequences by Vicfred

tags: mathematics, flatness, regular sequences, depth

Flatness is the condition that tensoring introduces no new kernel. A small computation will anchor the general statement before the abstraction takes over.

The mathematical object

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.

$$ M\ \text{flat over }A $$

I keep the defining relation \(\mathsf D\) above the derived relation \(\mathsf C\). This exposes whether cancellation used \(x\ne0\) and whether the conclusion is canonical.

$$ 0\to N'\to N\Longrightarrow0\to N'\otimes_AM\to N\otimes_AM $$

One explicit computation

Here is a concrete symbolic test. Reading it from left to right reveals which transformation is reversible and which is only an implication.

$$ \operatorname{Tor}_1^A(A/I,M)=0\quad\text{for every finitely generated ideal }I\Longleftrightarrow M\text{ is flat} $$

Why the identity matters

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

$$ \begin{aligned} \mathsf{D}\;&:\quad M\ \text{flat over }A,\\[5pt] \mathsf{C}\;&:\quad 0\to N'\to N\Longrightarrow0\to N'\otimes_AM\to N\otimes_AM. \end{aligned} $$

Where it can fail

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.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] 0\to N'\to N\Longrightarrow0\to N'\otimes_AM\to N\otimes_AM \end{gathered}} $$

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 Sat 23 January 2010. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.