I'm trying to understand something relating to differentials on an elliptic curve. Let's say my elliptic curve $C = \text{Proj}(k[X,Y,Z]/Y^2Z-X^3-aXZ^2-bZ^3)$ over some characteristic $0$ field $k$.
I know that (and have the idea of why) $H^1_{dR}(C) = H^0(C,\Omega^1_{C/k}) \oplus H^1(C, \mathcal{O}_C)$. So, since $H^1_{dR}$ in particular is isomorphic to $H^{1}_{dR}(U)$ where $U = D(Z) = \text{Spec}(k[x,y]/y^2-x^3-ax-b)$, then we can deduce that $H^1_{dR}$ has a $k$-basis of $\omega, x\omega$, where $y\omega=dx$ on $U$.
Since $H^0(C,\Omega^1_{C/k})$ is $0$-th sheaf cohomology of $\Omega^1$, this is $\Omega^1_C(C)$, the space of global differentials on $C$. I know that this should be 1 dimensional, since the dimensions should be the genus of the curve. Since $\omega$ is regular at $\infty=[0:1:0]$ but $x\omega$ is not (having a pole of order $2$), is this enough to imply that the basis of $\Omega^1_C(C)$ is given by the differential $\omega$?
What I'm struggling to explicitly write out are all the details involved. If I consider the sheaf $\Omega^1_U$ and take global sections, we get differentials which can be written uniquely as
$$ \alpha=(P(x)+Q(x)y)\omega $$
for some $P(x),Q(x) \in k[x]$. I think that I should the be doing is looking at $V = D(Y) = $ $ \text{Spec} ( k[w,z] /z- w^3 - awz^2 - bz^3) $, where $w=x/y$ and $z=1/y$. So then on $D(YZ)$ we have that $\omega = dx/y = dw - \frac{w}{z}dz$ and
$$\begin{split} \alpha = & (P(\frac{w}{z})+\frac{1}{z}Q(\frac{w}{z}))(dw-\frac{w}{z}dz) \\ = & (P(\frac{w}{z})+\frac{1}{z}Q(\frac{w}{z}))dw - (\frac{w}{z}P(\frac{w}{z})+\frac{w}{z^2}Q(\frac{w}{z}))dz \end{split}$$
If we let $R=k[w,z] /(z- w^3 - awz^2 - bz^3)$, then
$$\Omega^1|_V = (Rdw\oplus Rdv)/(dz -3w^2dw - az^2dw - 2awzdz - 3bz^2dz) $$
If $Q \neq 0$ and $\deg P>0$, then the expressions $P(\frac{w}{z})+\frac{1}{z}Q(\frac{w}{z})$ and $\frac{w}{z}P(\frac{w}{z})+\frac{w}{z^2}Q(\frac{w}{z})$ are not polynomial in $w,z$ and $\alpha$ will not lie in $\Omega^1|_V$. So then we have $Q=0$ and $P(x) \in k$ and so we get that $\Omega^1_C(C)= \lbrace a \omega : a \in k \rbrace $.
My question is, does this look vaguely correct or have I just gone a bit up the left with it? Would a similar argument show, say, that the differentials $dx/y, xdx/y,...,x^{g-1}dx/y$ on a hyperelliptic curve are a basis of the global differentials there? I know that on an affine chart such as $y^2=f(x)$ we get twice as many, but the other half all have poles at infinty.
I find a lot of this sheaf cohomology and such quite conceptually challenging, and trying to assimilate it all is taking some work.
Thanks