Another proof of Grothendieck's theorem on the splitting of vector bundles on the projective line

TL;DR

This paper presents an alternative proof of Grothendieck's theorem on the splitting of vector bundles on the projective line, using classical lattice and valuation methods over a field.

Contribution

It offers a new proof of Grothendieck's theorem formulated entirely in classical lattice and valuation terms, providing a different perspective on the splitting of vector bundles.

Findings

Proof formulated using lattice and valuation methods.

Reinforces the classical understanding of vector bundle splitting.

Provides an alternative approach to Grothendieck's theorem.

Abstract

This note contains another proof of Grothendieck`s theorem on the splitting of vector bundles on the projective line over a field $k$. Actually the proof is formulated entirely in the classical terms of a lattice $\Lambda \cong k[T]^d$, discretely embedded into the vector space $V \cong K_\infty^d$, where $K_\infty \cong k((1/T))$ is the completion of the field of rational functions $k(T)$ at the place $\infty$ with the usual valuation.