A Weight Structure on Rigid Analytic Motives over a Field

Abstract

In this paper, we construct a monoidal weight structure on the stable $\infty$-category of rigid analytic motives over a local field $K$ via Galois descent. This extends the weight structure on the full subcategory of rigid analytic motives with good reduction, which is defined by Binda-Gallauer-Vezzani. As an application, we show that the Hyodo-Kato realization factors through the weight complex functor studied by Bondarko and Sosnilo. In particular, the weight complex yields a spectral sequence converging to the Hyodo-Kato cohomology of smooth quasi-compact $K$-rigid analytic spaces, thereby inducing a weight filtration on it.