In p-adic properties of modular schemes and modular forms Katz formulates the following base change theorem as Theorem 1.7.1
Let $n\geq 3$ and $\overline{\mathcal{M}}_n$ be the compactified moduli scheme of elliptic curves with level-$n$-structure over $\mathbb{Z}[\frac1n]$. Let $K$ be any $\mathbb{Z}[\frac1n]$-module. Then the morphism $$K\otimes H^0(\overline{\mathcal{M}}_n; \omega^{\otimes k}) \to H^0(\overline{\mathcal{M}}_n; K\otimes \omega^{\otimes k})$$ is an isomorphism for $k\geq 2$.
The key step is to show that $H^1(\overline{\mathcal{M}}_n; \omega^{\otimes k})$ vanishes for $k\geq 2$. He claims that this follows from $\omega^{\otimes k}$ having degree strictly greater than $2g-2$ on every connected component of $\overline{\mathcal{M}}_n \otimes \mathbb{Z}[\frac1n, \zeta_n]$ (where $g$ is the genus of such a connected component) by Riemann-Roch.
I am familiar with this kind of Riemann-Roch argument for projective curves over a field, where it can be found in Hartshorne in nice cases and in more general cases in Liu's book. But Katz uses it in the case that the base is $\mathbb{Z}[\frac1n]$. What kind of Riemann-Roch argument works here?
