Cohomology of integral structure sheaves for open varieties

We study the cohomology of integral structure sheaves for two types of open rigid analytic varieties: complements of divisors in quasi-compact varieties, and partially proper spaces. As applications, we prove a strengthened form of the Hebbarkeitssatz (Riemann’s extension theorem), involving the almost vanishing of $R^1\lim$, and study pro-étale $G$-torsors on open varieties.