= Solution
The <complex tautological line bundle> on $\mathbb P^1$ has fibre at a line $\ell\subseteq\mathbb C^2$ equal to $\ell$ itself:
$$
\mathcal O_{\mathbb P^1}(-1)=\{(\ell,v):v\in\ell\}\subseteq\mathbb P^1\times\mathbb C^2.
$$
Define $\mathcal O(1)=\mathcal O(-1)^*$, $\mathcal O(k)=\mathcal O(1)^{\otimes k}$ for $k>0$, $\mathcal O(0)$ to be trivial, and $\mathcal O(k)=\mathcal O(-1)^{\otimes(-k)}$ for $k<0$. These are the <tensor powers of the hyperplane line bundle>.
On the two given charts, use the <holomorphic local frames> $e_0(w)=(1,w)$ and $e_1(z)=(z,1)$ of $\mathcal O(-1)$. The maps $([1:w],a)\mapsto([1:w],a(1,w))$ and $([z:1],b)\mapsto([z:1],b(z,1))$ are its trivializations. On the overlap $z=1/w$,
$$
e_1=z e_0=w^{-1}e_0,
\qquad a e_0=b e_1\iff b=wa.
$$
Thus the frame-transition factor from $e_0$ to $e_1$ is $w^{-1}$, while the fibre-coordinate transition from chart 0 to chart 1 is $w$. Specifying both avoids an inverse-convention ambiguity.
For $\mathcal O(k)$, let $e_i^{(k)}$ be the induced frames; they satisfy $e_1^{(k)}=w^k e_0^{(k)}$. Identifying overlap sections using $e_0^{(k)}$, the <Čech cochain groups> and <Čech coboundary> for $k\geq0$ are
$$
\boxed{C^0=\mathcal O(\mathbb C)\oplus\mathcal O(\mathbb C),\qquad
C^1=\mathcal O(\mathbb C^*),\qquad
\delta(a,b)(w)=w^k b(1/w)-a(w).}
$$
Here $\mathcal O(\mathbb C)$ means entire functions in the coordinate of the corresponding chart. Global sections are $\ker\delta$, so $a(w)=w^k b(1/w)$, or $b(z)=z^k a(1/z)$. Expanding the entire function $a(w)=\sum_{j\geq0}a_jw^j$ shows that $b$ is holomorphic at zero exactly when $a_j=0$ for $j>k$. Thus a global section is determined by a polynomial of degree at most $k$, with basis $1,w,\ldots,w^k$ in the chart-0 frame. Therefore \b[$\boxed{\dim H^0(\mathbb P^1,\mathcal O(k))=k+1\quad(k\geq0)}$].
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