Given a field $k$, it is well known that the global differentials $\Omega_{\mathbb{P}_k^1/k}$ of the projective line $\mathbb{P}_{k}^1$ are $\mathcal{O}(-2)$. This is usually proved by observing that the differential to one coordinate has a 2-pole on the other standard affine open. This uses that we know the line bundles over the projective line well.
So now I want to do this without refering to the twist, just by looking at the affines and glueing.
Starting with $Proj(k[x_0,x_1])=D(x_0)\cup D(x_1)$. Now $D(x_0)=Spec (k[x_0,x_1]_{x_0})_0$, i.e. the spectrum of homogenous elements of the form $P/x_0^{deg P}$, where $P\in k[x_0,x_1]$ We write this as $k[x_1/x_0]$ and get differentials $k[x_1/x_0]d(x_1/x_0)$. For notation we write $S=x_1/x_0$, so this should give us $$\Omega_{\mathbb{P}_k^1/k}(D(x_0))=K[S]dS$$ An analog calculation yiedls $$\Omega_{\mathbb{P}_k^1/k}(D(x_1))=K[T]dT$$
Now on the overlap $D(x_0)\cap D(x_1)=D(x_0x_1)$ we have $ST=1$ and $dS=-dT/T^2$ Since the differentials have to agree on the overlap we get as a condition $$P(S)dS=Q(T)dT=-\frac{P(1/T)}{T^2} dT$$ where $P$ and $Q$ are polynomials. I.e. we have on the standard open affine the data of differentials $P(S)dS$ (or $Q(T)dT$) and they have to agree on the overlap. But this seems to me like $Q$ and $P$ both have to be zero, which I know is false.
Where did I go wrong?
So you are trying to find global sections of the canonical bundle on P1. Shouldn't this be zero? I.e. Cech cohomology.