A faithfully flat module is flat and reflects whether another module is zero. 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.
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.
A small case in full
Here is a concrete symbolic test. Reading it from left to right reveals which transformation is reversible and which is only an implication.
The reusable statement
The invariant statement is the one that does not depend on a convenient choice of coordinates, representatives, basis, or enumeration.
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.
The final box is a summary, not a new assumption; the proof still lives in the definitions and the intervening calculation. The source keeps each scope delimiter visible for later inspection.