Finiteness theorems for vanishing cycles of formal schemes

Abstract

Let k be a non-Archimedean field with nontrivial valuation, and kpo its ring of integers. In this paper we prove constructibility of vanishing cycles sheaves for arbitrary formal schemes locally finitely presented over kpo as well as special formal schemes over kpo (for discretely valued k). This allows us to extend continuity results, established earlier for locally algebraic formal schemes, to the whole classes of formal schemes.