= Birkhoff–Grothendieck theorem
{c}
{title2=$E\cong\bigoplus_i\mathcal O(a_i)$}
{wiki}
Every <vector bundle> on the <projective line> over a <field> splits as a <direct sum> of <line bundles>. Choose the largest integer $a$ with $H^0(E(-a))\ne0$. A corresponding map $\mathcal O(a)\to E$ is saturated, since saturation otherwise increases its degree. Inductively split its quotient as $\bigoplus_i\mathcal O(b_i)$. Twisting the exact sequence by $-a-1$ and using $H^1(\mathcal O(-1))=0$ gives $b_i\leq a$. Hence all extension groups $H^1(\mathcal O(a-b_i))$ vanish, and the sequence splits. The multiset of degrees is determined by the dimensions of all twisted global-section spaces.
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