I have queries relating to Hartshorne Exercise II.9.1. Take a connected, non-singular, positive-dimensional subvariety $i: Y \to X = \mathbb{P}_k^n$, where $k$ is algebraically closed, with ideal sheaf $\mathcal{I}$.
- In (a) we need to derive an inclusion $\mathcal{I}/\mathcal{I}^2 \hookrightarrow \mathcal{O}_Y(-1)^{n+1}$ using exact sequences (8.17) and (8.13): $$0 \to \mathcal{I}/\mathcal{I}^2 \to \Omega_{X/k} \otimes \mathcal{O}_Y \to \Omega_{Y/k} \to 0 \quad \text{and} \quad 0 \to \Omega_{X/k} \to \mathcal{O}_X(-1)^{n+1} \to \mathcal{O}_X \to 0.$$ Viewing the middle term of the first exact sequence as $i^{*}(\Omega_{X/k})$, my first instict was to apply $i^*$ to the second exact sequence. Indeed, $i^* \mathcal{O}_X = \mathcal{O}_Y$, but there's no reason for $i^*$ to be exact. On closer inspection of the map $i^* \Omega_{X/k} \to i^* \mathcal{O}_X(-1)^{n+1}$, I can't see why it's injective.
- In (b) we need to prove that $\Gamma(Y,\mathcal{I}^r/\mathcal{I}^{r+1}) = 0$ for all $r \ge 1$. I can see how the base case follows from (a); does the general case too?
Thanks for any advice! Hints/nudges welcomed.