Duality and nearby cycles over general bases

TL;DR

This paper investigates the behavior of the sliced nearby cycle functor and its interaction with duality over various base schemes, providing new proofs and extending known results in the context of algebraic geometry.

Contribution

It proves the commutation of the sliced nearby cycle functor with duality over general bases, confirming Deligne's prediction and extending results to excellent schemes.

Findings

Confirmed commutation over Henselian discrete valuation rings

Provided a new proof of Beilinson's theorem on vanishing cycles

Extended duality preservation of local acyclicity to excellent regular bases

Abstract

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.