On $\ell$-independence for the étale cohomology of rigid spaces over local fields

Abstract

We investigate the action of the Weil group on the compactly supported $\ell$-adic étale cohomology groups of rigid spaces over a local field. We prove that the alternating sum of the traces of the action is an integer and is independent of $\ell$ when either the rigid space is smooth or the characteristic of the base field is equal to 0. We modify the argument of T. Saito to prove a result on $\ell$-independence for nearby cycle cohomology, which leads to our $\ell$-independence result for smooth rigid spaces. In the general case, we use the finiteness theorem of Huber, which requires the restriction on the characteristic of the base field.