  This paper studies the sliced nearby cycle functor and its commutation with
duality. Over a Henselian discrete valuation ring, we show that this
commutation holds, confirming a prediction of Deligne. As an application we
give a new proof of Beilinson's theorem that the vanishing cycle functor
commutes with duality up to twist. Over an excellent base scheme, we show that
the sliced nearby cycle functor commutes with duality up to modification of the
base. We deduce that duality preserves universal local acyclicity over an
excellent regular base. We also present Gabber's theorem that local acyclicity
implies universal local acyclicity over a Noetherian base.
