Decomposing algebraic vector bundles on the projective line

Abstract:

$\mathcal {V}(R)$ denotes the category of algebraic vector bundles on ${\mathbf {P}}_R^1,\;R$ a commutative, noetherian ring. If $K$ is a field, it is known that any $\mathcal {F} \in \mathcal {V}(K)$ is isomorphic to a (unique) direct sum of line bundles. If $\mathfrak {p} \in \operatorname {Spec} R$ and $K(\mathfrak {p})$ is the quotient field of $R/\mathfrak {p}$, any $\mathcal {F} \in \mathcal {V}(R)$ induces a bundle in $\mathcal {V}(K(\mathfrak {p}))$, and so a decomposition into line bundles. If the decomposition is the same for each $\mathfrak {p},\;\mathcal {F}$ is said to be uniform. It is shown that if $R$ is reduced, uniform vector bundles on ${\mathbf {P}}_R^1$ are sums of tensor products of (pullbacks of) bundles on $\operatorname {Spec} R$ with line bundles on ${\mathbf {P}}_R^1$.