The module of differentials is universal for derivations out of an algebra. The formulas are more useful when each symbol has a job rather than merely decorating the theorem.
The mathematical object
An \(A\)-module \(M\) produces a quasi-coherent sheaf \(\widetilde M\) on \(\operatorname{Spec}A\). Localisation gives its sections on basic opens and its stalks at primes.
There are two layers here: the object \(\mathsf D\) and the law \(\mathsf C\). Writing them separately makes the direction of \(\Longrightarrow\) visible and keeps an accidental converse from slipping in.
One explicit computation
This is the algebraic core of the note. Once this line is correct, the surrounding interpretation has something solid to refer to.
Why the identity matters
The compact alignment is a local map of the argument: assumptions on the first row, consequence on the second. Any generalisation must preserve that dependency.
Where it can fail
Sheafification repairs local compatibility, but it does not make every sheaf quasi-coherent. Affine module methods apply only to the quasi-coherent class.
The result is compact enough to reuse without pretending that the caveat has disappeared. The worked line remains the quickest consistency check.