In (Katz's '$p$-adic properties of modular forms and modular schemes', 1972) Theorem 1.7.1., we use the following fact:
To show $K\otimes H^0(M_n,\omega^{\otimes k})\cong H^0(M_n,\omega^{\otimes k}\otimes K)$, it suffices to show $H^1(M_n,\omega^{\otimes k})=0$. (Assuming $K$ is a $\mathbb{Z}[\frac1{N}]$-module, and $(M_n,\omega)$ is the level $n$ moduli scheme and its canonical differential.)
I am lost in this fact, and I think the main obstruction is that $H^0$ is left exact while $-\otimes K$ is right exact, hence I don't know how to apply methods from resolution or spectral sequence...
Any help will be appreciated.