Is there any description of a subgroup of the algebraic $K$-groups of a curve that its generators lie in the subcategory that its objects are direct sums of $\mathcal{O}(n)$'s (for possibly different $n$). I know this subcategory does not form an exact category but let's assume we have a curve that every vector bundle is in the aforementioned form (I think this implies the curve is $\mathbb{P}^1$). Is there a way to calculate higher $K$-groups based on this? (Or is it possible to give a description of $K$-groups of $\mathbb{P}^1$ based only on the fact every vector bundle splits into direct sum of $\mathcal{O}(n)$'s.)
Edit: Another question, is it possible to give a resolution of a vector bundle on $\mathbb{P}^1$ by vector bundles of the form $\mathcal{O}(n)^{\oplus k}$?
