What's the relationship between Kahler differentials and ordinary differential forms?



Kahler differentials and Ordinary Differentials
Let $M$ be a differentiable manifold, $A=C^\infty (M)$ its ring of global differentiable functions and $\Omega^1 (M)$ the A-module of global differential forms of class $C^\infty$. 
The A-module of Kähler differentials $\Omega\_k(A)$ is the free A-module over the symbols $db$ ($b \in A$) divided out by the relations


$d(a+b)=da+db,\quad d(ab)= adb+bda,\quad d\lambda=0 \quad(a,b\in A, \quad \lambda \in k)$


There is an obvious surjective map $\quad \Omega\_k(A) \to \Omega^1 (M)$ because the relations displayed above are valid in the classical interpretation of the calculus (Leibniz rule).
However, I do not believe at all that it is injective.


For example, if $\: M=\mathbb R$, I see absolutely no reason why $\mathrm{d}\sin(x)=\cos(x)\mathrm{d}x$ should be true
in $\Omega\_k(A) $ (beware the sirens of calculus). Things would be worse if we considered $C^\infty$
functions which, unlike the sine, are not analytic.
The same sort of reasoning applies to holomorphic manifolds and also to local rings of differentiable or holomorphic functions on manifolds.


To sum up: the differentials used in differentiable or holomorphic manifold theory are a quotient of the corresponding Kähler differentials but are not isomorphic to them.
(And I think David's claim that they are isomorphic is mistaken.)


