Direct summands of successive extensions of line bundles

Question

On a projective variety any direct summand of a direct sum of line bundles is going to be direct sum of a subset of initial line bundles. This follows from the Krull-Schmidt property of vector bundles on projective varieties. Now my question is a little bit similar to this. Let's assume the vector bundle is constructed by taking successive extensions of a set of line bundles in the form $\mathcal{O}(n)$. Is any direct summand of this vector bundle in the similar form?

Answer 1

Of course not. Let $X$ be a smooth curve and take any line bundle $\mathcal{L}$. Then for $n \gg 0$ the bundle $(\mathcal{L} \oplus \mathcal{L}^{-1}) \otimes \mathcal{O}(n)$ is globally generated. Therefore, a general morphism $$ \mathcal{O}(-n) \to \mathcal{L} \oplus \mathcal{L}^{-1} $$ is a fiberwise monomorphism, hence its cokernel is a line bundle, which by computing the determinant can be identified with $\mathcal{O}(n)$. Thus, we have an exact sequence $$ 0 \to \mathcal{O}(-n) \to \mathcal{L} \oplus \mathcal{L}^{-1} \to \mathcal{O}(n) \to 0 $$ which shows that any line bundle on $X$ is a direct summand of an extension of $\mathcal{O}(n)$.