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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 $d\ge 0$. There are no nonzero global sections for $d<0$. Hence

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 bundles need not be an isomorphism. Multiplication by a nonzero section giveson ${\mathbb{P}}^1$; its cokernel is a nonzero skyscraper sheaf supported at the zero of the section.