  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 Berkovich's 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.
