= Solution
On the standard charts $D_+(x)$ and $D_+(y)$ of $\mathbb P_k^1$, a section of the <twisting sheaf on projective space> $\mathcal O(d)$ is represented by a degree-zero element of the corresponding localization of $k[x,y](d)$. A global section is therefore a homogeneous polynomial of degree $d$ when $d\geq0$. There are no nonzero global sections for $d<0$. Hence
$$
\boxed{\dim_kH^0(\mathbb P_k^1,\mathcal O(d))=
\begin{cases}
d+1,&d\geq0,\\
0,&d<0.
\end{cases}}
$$
Local isomorphism on every member of a fixed affine cover does not imply a global isomorphism: the local identifications may have different transition functions. For example, $\mathcal O$ and $\mathcal O(1)$ are both trivial on the two standard affine charts of $\mathbb P^1$, but they are not isomorphic because their spaces of global sections have dimensions one and two.
An injective map between <line bundle>[line bundles] need not be an isomorphism. Multiplication by a nonzero section gives
$$
\mathcal O(-1)\hookrightarrow\mathcal O
$$
on $\mathbb P^1$; its cokernel is a nonzero <skyscraper sheaf> supported at the zero of the section.
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