I have the following (it can be very silly) question.
Suppose I have a commutative algebra $A$ over a field $k$ of $char(k)=0$ which defines a $n$-dimensional smooth variety $X=Spec(A)$.
What are the conditions on $A$, when $\Omega^n(A)\simeq A$ (not canonically) as $A$-modules?
Here by $\Omega^n(A)$ I mean $\Lambda^n(\Omega^1A)$, and $\Omega^1A$ denotes the Kahler differentials. In the category of $C^\infty$-manifolds isomorphism between top forms and ring of functions on a manifold means that the manifold is orientable. So I guess I am asking under which conditions on $A$, $Spec(A)$ is "orientable".