There is some big mistake i'm doing that I can find.
Take an affine scheme $X= Spec A$ smooth over $k$ of dimension 1, and consider $\Omega_X$, kahler differentials. I want to understand 'how many' global sections there are.
On one hand, by the very definition this should be $\Omega_A$, which I can explicitly compute with generators and generally has positive dimension (even infinite, for example $A=k[x]$ yields itself).
On the other hand, I know that the genus of $X$ is $H^1(X,O_X)=0$, because the scheme is affine and thus has null cohomology. But the dimension of $H^0(X,\Omega_X)$ equals the genus, so is zero!!
I am sure that I am getting wrong in the second part, but where?
Thank you, Andrea
As said in the comment : $\dim H^0(X,\Omega^1_X)$ is the genus of the curve $X$ only if $X$ is projective, but $\operatorname{Spec}k[x]$ isn't.