Filtrations on the globalization of twisted D-modules over Dedekind schemes

Fabian Januszewski and the author established the theory of twisted D-modules over general base schemes. In this short note, we construct a $K$-invariant positive exhaustive filtration on the globalization of the twisted D-module on a smooth quasi-compact $K$-scheme over a Dedekind scheme $S$ obtained by the direct image of a $K$-equivariant twisted integrable connection along a $K$-equivariant closed immersion from a smooth proper $K$-scheme $Y$ with $K$ a smooth $S$-affine group scheme, whose $p$th associated graded $\mathcal{O}_S$-module is locally free of finite rank for every integer $p$. In particular, the $k$-module of its global sections is projective if $S$ is affine with coordinate ring $k$.