The Birkhoff–Grothendieck theorem is a fundamental result in the field of lattice theory and universal algebra. It characterizes the representability of certain types of categories, especially in the context of complete lattice structures. **Statement of the theorem:** The Birkhoff–Grothendieck theorem states that a distributive lattice can be represented as the lattice of open sets of some topological space if and only if it is generated by its finitely generated ideals.
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Every vector bundle on the projective line over a field splits as a direct sum of line bundles. Choose the largest integer with . A corresponding map is saturated, since saturation otherwise increases its degree. Inductively split its quotient as . Twisting the exact sequence by and using gives . Hence all extension groups vanish, and the sequence splits. The multiset of degrees is determined by the dimensions of all twisted global-section spaces.