Hello,

this one has always been mysterious to me. The Kähler differentials $\Omega_{A/k}$ are definined, by the universal property $$Der_k(A,M)=A-Mod(\Omega_{A/k},M)$$ so for $M=A$ we get that $\Omega_{A/k}$ is the cotangent space of $spec(A)$. (or a relative version of it if k is no field).

There are two constructions of Kähler Differentials I know. The first one is $$\Omega_{A/k}=\langle df : \text{relations satisfied by any derivation} \rangle$$ I think I sort of understand this one, it says that the differential of a function just contains enough information to extract the derivation of the function out of it. And this is what a section into cotangent space should be. Something that contains just enough information to pair it with a vector-field into a function. The other construction is $$\Omega_{A/k}=I/I^2$$ Where $I$ is the Ideal of functions vanishing on the diagonal in $spec(A)\times_{spec(k)} spec(A)$.

More geometrically it says sections into cotangent space=functions vanishing on the diagonal mod higher order. But still I don't think I understand this equality on an intuitive level. Can someone explain the heuristic behind this equality? Or maybe explain $\Omega_{A/k}=I/I^2$ from another intuitive viewpoint?