We give a geometric proof of Horrocks' theorem, which states that a vector bundle on the affine line over a local ring, extending to the projective line, is trivial. Unlike Horrocks' original approach via formal cohomology, our method uses semicontinuity and cohomology and base-change to construct a nowhere-vanishing section.

We thank Prof. Aravind Asok and the anonymous referee for their valuable comments and suggestions that improved this article.