The local model of a totally ramified extension is defined in this paper as follow:
What does the writer means by the fact that $\mathcal{F}_i$ are locally direct summand of $\mathcal{O}_S$ modules?
What does the writer means by $\mathrm{det}_{\mathcal{O}_S}(T-\Pi|\mathcal{F}_i)$? I know for sure that each $\mathcal{F}_i$ is stable under $\pi\otimes 1$ but i can't see how he defines the determinant over $\mathcal{O}_S$.