I would like to gain some intuition regarding the modules of Kähler differentials $\Omega^j_{A/k}$ of an affine algebra $A$ over a (say - algebraically closed) field $k$.
Let us recall the definition: let $A^e = A\mathrel{\otimes_k} A$, let $f:A^e\to A$ be the map defined by $f(a\otimes b) = ab$, and let $I = \ker f$. Then $\Omega^1_{A/k} = I/I^2$. And, $\Omega^j_{A/k} = \bigwedge^j \Omega^1_{A/k}$.
An important theorem regarding Kähler differentials says: If $k \to A$ is smooth of relative dimension $n$, then $\Omega^n_{A/k}$ is a projective module of finite rank.
My question:
I was wondering if anyone could provide some examples of:
How does the module of Kähler differentials look for some singular varieties? For example, what is $\Omega^1_{A/k}$ for $A = k[x,y]/(y^2-x^3)$?
Can anyone provide an example of a non-singular affine variety with coordinate ring $A$, such that $k \to A$ is smooth of relative dimension $n$, and $\Omega^n_{A/k}$ is projective but not free?
I would be happy for any concrete example that will help my intuition on the subject.
Thanks!