Classfication of vector bundles on projective line over a local ring Let $R$ be a local ring. Let $\mathbb{P}^1_R=\rm{Proj}~R[x_0, x_1]$ be the Projective line over $R$.
Is there a classification of vector bundles of rank $n$ on  $\mathbb{P}^1_R$ in terms of splitting into line bundles?
In the case $R$ is a field, this is the famous Grothendieck's classification of vector bundles on projective line in terms of the line bundles $\mathcal{O}(n)$ for $n\in\mathbb{Z}$.
In how much generality does the description of splitting of vector bundles on $\mathbb{P}^1_R$ extend?
 A: It doesn't. Here is a classical example: let $R$ be a DVR, with uniformizing parameter $\pi $. On $\mathbb{P}:=\mathbb{P}^1_R$, consider the extension
$$0\rightarrow \mathscr{O}_{\mathbb{P}}(-1) \rightarrow E \rightarrow \mathscr{O}_{\mathbb{P}}(1)\rightarrow 0 $$given by the extension class $\pi \in \operatorname{Ext}^1_{\mathbb{P}}(\mathscr{O}_{\mathbb{P}}(1),\mathscr{O}_{\mathbb{P}}(-1)  )\cong R $. The vector bundle $E$ restricts to $\mathscr{O}_{\mathbb{P}}^2$ on the generic fiber and to $\mathscr{O}_{\mathbb{P}}(-1) \oplus \mathscr{O}_{\mathbb{P}}(1) $ on the special fiber, hence it cannot be a direct sum of line bundles.
A: *

*If you think of the projective line as two affine lines --- namely $Spec(R[t])$ and $Spec(R[t^{-1}])$ --- patched together over $Spec(R[t,t^{-1}])$, then Horrocks's Theorem tells you that any vector bundle on the projective line must become trivial when restricted to either of the two affine lines.  [I am using your assumption that $R$ is local.]


*Therefore a vector bundle on the projective line is determined by the way those two trivial bundles are patched together on the overlap.  Another way to say this is that a rank $n$ vector bundle on the projective line is given by an element of $GL_n(R[t,t^{-1}])$ modulo the images of $GL_n(R[t])$ and $GL_n(R[t^{-1}])$.  This classifies vector bundles on the projective line, though not in a way that makes it immediately obvious whether the splitting principle holds.


*But if $R$ is not a field --- even if $R$ is local, and in fact even if $R$ is a $DVR$ --- it turns out that there are in general vector bundles that don't split as sums of line bundles.


*We have this theorem of Hubl and Sun (I am quoting almost verbatim from their 1999 paper in Communications in Algebra):

If $R$ is a $DVR$, then a vector bundle $E$ on ${\mathbb P}^1_R$ splits as a direct sum of line bundles if and only if $R^1f_*E(m)$ is projective as an $R$-module for all integers $m$.  [Here $f$ is the map from ${\mathbb P}^1_R$ to $Spec(R)$.]

