I'm currently reading through Kleiman's proof that $H^n(X,\mathcal F)$ vanishes when $\mathcal F$ is a coherent sheaf on an $n$-dimensional non-proper (reduced and irredicible) variety and got stuck in the proof of Lemma 2, p. 943. The points that are unclear to me are the following:
- Kleiman casually chooses an irreducible hyperplane section $Y''$. I can see why such a thing exists if we work over an infinite field $k$, but this is nowhere mentioned in the text. Why does this still work?
- I don't see why $Z+mY''$ is very ample for $m\gg 0$, nor why this implies $X-(Z\cup Y'')$ is affine.
I would be grateful if anyone could clarify this for me.