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  "body": "\n## Statement\n\nLet $k$ be a field and let $\\mathcal{E}$ be a [[vector bundle|P4DZM975BV]] (a locally free sheaf) of rank $r \\ge 1$ on the [[projective line|ZT9A6JK8QF]] $\\mathbb{P}^1_k$.\nThen there are integers $a_1 \\ge a_2 \\ge \\dots \\ge a_r$ such that\n$$\\mathcal{E} \\cong \\mathcal{O}(a_1) \\oplus \\mathcal{O}(a_2) \\oplus \\dots \\oplus \\mathcal{O}(a_r),$$\nwhere $\\mathcal{O}(a)$ is the $a$-th power of [[Serre's twisting sheaf|V93Y0XYPZD]].\nThe integers $a_1, \\dots, a_r$ are uniquely determined by $\\mathcal{E}$.\n\n## Remarks\n\nThe idea of the proof: induct on $r$.\nChoose the largest $a$ with $H^0(\\mathcal{E}(-a)) \\ne 0$; a section of $\\mathcal{E}(-a)$ gives a subbundle $\\mathcal{O}(a) \\subseteq \\mathcal{E}$ whose quotient is a vector bundle of rank $r - 1$, which splits by induction as $\\bigoplus \\mathcal{O}(b_i)$.\nThe maximality of $a$ forces $b_i \\le a$, so the extension class lies in $\\bigoplus H^1(\\mathbb{P}^1, \\mathcal{O}(a - b_i)) = 0$ and the sequence splits.\nUniqueness follows from the dimensions $h^0(\\mathcal{E}(n))$ for all $n$.\n\nExample: the tangent bundle of $\\mathbb{P}^1_k$ is $\\mathcal{O}(2)$.\nThe bundles $\\mathcal{O}(1) \\oplus \\mathcal{O}(1)$ and $\\mathcal{O} \\oplus \\mathcal{O}(2)$ both have rank $2$ and degree $2$, but they are not isomorphic.\nIndeed, after twisting by $\\mathcal{O}(-2)$, the first has no nonzero global sections and the second has a one-dimensional space of them.\n\nThe statement fails on other curves and in higher dimension.\nOn an elliptic curve there are indecomposable vector bundles of rank $2$, and the tangent bundle of $\\mathbb{P}^2$ is not a direct sum of line bundles.\n\n## Explanations\n\n### L1 human\n\nOn the projective line, every vector bundle breaks into a sum of line bundles, and line bundles there are classified by one integer, the degree.\nSo a vector bundle on $\\mathbb{P}^1$ is described by a finite list of integers.\nThis makes $\\mathbb{P}^1$ the simplest possible base for vector bundles.\n\n### L2 human\n\nThe integers $a_i$ are called the splitting type of $\\mathcal{E}$.\nThe theorem is used to study vector bundles on $\\mathbb{P}^n$ by restricting them to lines, where they split, and seeing how the splitting type varies.\nIn the analytic setting it says that every holomorphic vector bundle on the Riemann sphere is a sum of line bundles.\nThis is equivalent to a factorization theorem for matrices of functions on a circle.\n",
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