Birkhoff–Grothendieck theorem

Every algebraic vector bundle on the projective line over a field is isomorphic to a direct sum of line bundles. The summand degrees are uniquely determined up to order.

Connect

Vector bundle: A family of vector spaces parametrized by a space, locally varying like a product. The theorem classifies these objects completely when the base is the projective line.
Projective line: The one-dimensional projective space formed by lines through the origin in a two-dimensional vector space. It is the base curve whose bundles the theorem describes.
Coherent sheaf cohomology: A theory that measures global sections and obstructions for sheaves on algebraic varieties. Its vanishing results on the projective line help establish the bundle decomposition.
Holomorphic vector bundle: A complex vector bundle whose local trivializations and transition maps are holomorphic. The theorem also classifies holomorphic bundles on the complex projective line.
Vector bundles on elliptic curves: The classification of vector bundles on smooth curves of genus one. Bundles on elliptic curves need not split into line bundles, unlike those on the projective line.
Picard group: The group of isomorphism classes of line bundles under tensor product. On the projective line, line bundles are indexed by integer degree, the theorem’s classification data.
Algebraic vector bundle: A locally free sheaf of finite rank on an algebraic variety. The theorem applies to these bundles over the projective line.
Čech cohomology: A cohomology theory computed from local data on an open cover and its overlaps. The two standard affine charts of the projective line give a concrete route to the splitting argument.
Grothendieck splitting type: The ordered list of integer degrees of line-bundle summands in a bundle on the projective line. It packages the theorem’s classification into a computable invariant.
Harder–Narasimhan filtration: A canonical filtration of a vector bundle whose successive quotients have decreasing slopes. For general curves, a filtration replaces the direct-sum classification available on the projective line.