Projective Decomposition and Intersection Cohomology of Rigid Spaces

We prove a rigid analytic analogue of the Beilinson–Bernstein–Deligne decomposition theorem for projective morphisms over $p$-adic fields, with ${\mathbf{Q}}_{\mathrm{\ell }}$-coefficients for every prime $\mathrm{\ell }$, including $\mathrm{\ell }=p$. As an application, we show that the $p$-adic intersection cohomology of every proper rigid analytic space is de Rham. We also establish an independent criterion for de Rham intersection cohomology with local coefficients in terms of the dimension of the boundary, yielding further results for curves, surfaces, and finite covers. For finite coefficients of characteristic different from $p$, we obtain isotropic analytic upper bounds for singular support and prove that singular support is analytic and Lagrangian for complexes supported in dimension at most one.