I am reading the proof of Castelnuovo's contractibility criterion ( Hartshorne's Algebraic geometry). In the proof, he claim that there must be a very ample divisor $H$ on a projective surface $X$ such that $H^1(X,\mathscr L(H))=0$.
By Serre's theorem
Let $X$ be a projective scheme over a noetherian ring $A$, $\mathcal O_X(1)$ a very ample invertible sheaf on $X$ over $Spec A$ , and $\mathscr F$ a coherent sheaf on $X$. Then there is an integer $n_0$ such that for each $ i>0 $ and each $n\geqslant n_0$, $H^i(X,\mathscr F(n))=0$
Here $X$ is a projective surface over an algebraically closed field $\mathrm k$, a divisor $H$ corresponds to an invertible sheaf $\mathscr L(H)$, so, does he use $\mathscr L(H)\otimes_{\mathcal O_X}\mathcal O_X(n)\cong \mathscr L(nH)$ to give that claim ? I am not sure. But why we have this isomorphism ? Thank you!