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.
Best Answer
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 $x$.