Classfication of vector bundles on projective line over a local ring

Question

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?

Answer 1

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.

Answer 2

1. 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.]

2. 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.

3. 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.

4. 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 $Spec(R)$.]