On singular supports in mixed characteristic

We fix an excellent regular noetherian scheme $S$ over ${\mathbf{Z}}_{(p)}$ satisfying a certain finiteness condition. For a constructible étale sheaf $\mathcal{F}$ on a regular scheme $X$ of finite type over $S$, we introduce a variant of the singular support relatively to $S$ and prove the existence of a saturated relative variant of the singular support by adopting the method of Beilinson using the Radon transform. We may deduce the existence of the singular support itself, if we admit an expected property on the micro support of tensor product and if the scheme $X$ is sufficiently ramified over the base $S$.