We prove that the following statement follows from the Open Colouring Axiom (OCA): if X is locally compact σ-compact but not compact and if itš Cech-Stone remainder X * is a continuous image of ω * then X is the union of ω and a compact set. It follows that the remainders of familiar spaces like the real line or the sum of countably many Cantor sets need not be continuous images of ω * .

We study Deligne’s conjecture on the monodromy weight filtration on the nearby cycles in the mixed characteristic case, and reduce it to the nondegeneracy of certain pairings in the semistable case. We also prove a related conjecture of Rapoport and Zink which uses only the image of the Cech restriction morphism, if Deligne’s conjecture holds for a general hyperplane section. In general we show...