Kähler differentials of affine varieties 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!

Let's do your example (1).
There is a short exact sequence $$0\to A\to A\mathrm dx\oplus A\mathrm dy\to\Omega^1\to 0$$ in which the first map maps $1$ to $2x\,\mathrm dx+3y\,\mathrm dy$. It follows that $\Omega^1$ is isomorphic to the module $$\frac{A\oplus A}{(2x,3y)}.$$
If the characteristic is $2$ or $3$, this looks like $A\oplus A/(y)$ or $A\oplus A/(y)$. If the characteristic is not one of those two, then it is clear, from this description, that if you localize at a prime not containing $x$ or $y$, then you get a free module. Now localize at one of the other ideals, and see what you get!
