I believe to have found a typo in Griffiths & Harris.


In the chapter on surfaces, section Rational Surfaces 1, I am trying to read the result that a holomorphic vector bundle over $\mathbb{P}^1\_{\mathbb{C}}$ is a sum of invertible bundles.


What is the exact sequence that shows up at the start of his argument? Mine only has 2 terms and involves the fibres of E and H, which doesn't really make sense to me.


Any help would be appreciated.



Help with Griffiths & Harris, Surfaces
Let $H$ be the divisor corresponding to the point $x\in \mathbb{P}^1$. Tensoring the exact sequence 
$$
0\to O\_{\mathbb{P}^1}(-H)\to O\_{\mathbb{P}^1}\to O\_x\to 0.
$$ with $E\otimes H^k$, gives
$$
0\to O\_{\mathbb{P}^1}(E\otimes H^{k-1})\to O\_{\mathbb{P}^1}(E\otimes H^{k})\to E\_x\otimes H\_x^{k}\to 0.
$$Here GH writes $E\_x\otimes H\_x^{k}$ for $E\otimes H^k\otimes O\_x$, or what is the same, the fiber of $E\otimes O(H^{k})$ over $x$.


