Yoichi Mieda

On

We investigate the action of the Weil group on the compactly supported l-adic etale 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 l 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 l-independence for nearby cycle cohomology, which leads to our l-independence result for smooth rigid spaces. In the general case, we use the finiteness theorem of R. Huber, which requires the restriction on the characteristic of the base field.

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In this paper, we introduce variants of formal nearby cycles for a locally noetherian formal scheme over a complete discrete valuation ring. If the formal scheme is locally algebraizable, then our nearby cycle gives a generalization of Berkovichs formal nearby cycle. Our construction is entirely scheme-theoretic and does not require rigid geometry. Our theory is intended for applications to the local study of the cohomology of Rapoport-Zink spaces.

-independence for the étale cohomology of rigid spaces over local fields

We investigate the action of the Weil group on the compactly supported l-adic cohomology groups of rigid spaces over local fields. We prove that every eigenvalue of the action is a Weil number when either a rigid space is smooth or the characteristic of the base field is equal to 0. Since a smooth rigid space is locally isomorphic to the Raynaud generic fiber of an algebraizable formal scheme, we can use the techniques of alterations and monodromy spectral sequences in the smooth case. In the general case, we use the continuity theorem of Huber, which requires the restriction on the characteristic of the base field.