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    Vector bundles on the projective line split

    T-BIRKGROTH

    Every vector bundle on $\PP^1$ splits uniquely into an ordered direct sum of line bundles $\OO(a_i)$.Suggest a slogan

    (Birkhoff--Grothendieck)

    Let \(k\) be a field. Every locally free sheaf of finite rank on \(\PP^1_k\) is isomorphic to \(\OO(a_1) \oplus \cdots \oplus \OO(a_r)\) for integers \(a_1 \geq \cdots \geq a_r\), and these integers are uniquely determined. [@Har10a, Exercise V.2.6]

    The tangent bundle of \(\PP^2\) restricted to a line \(\ell\) is \(\OO_\ell(2) \oplus \OO_\ell(1)\): the tangent bundle of the line is the first summand and the normal bundle \(\OO_\ell(1)\) the second.

    What is the Birkhoff--Grothendieck theorem?

    Dependencies

    • T-MODVBLocally free sheaves are vector bundlesuses
    • D-CB9XSThe twisting sheaves $\OO(n)$, and $\Pic(\PP^n) = \ZZ$uses

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