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.

We prove a relative Lefschetz–Verdier theorem for locally acyclic objects over a Noetherian base scheme. This is done by studying duals and traces in the symmetric monoidal $2$ -category of cohomological correspondences. We show that local acyclicity is equivalent to dualisability and deduce that duality preserves local acyclicity. As another application of the category of cohomological correspondences, we show that the nearby cycle functor over a Henselian valuation ring preserves duals, generalising a theorem of Gabber.

We show that compatible systems of ℓ-adic sheaves on a scheme of finite type over the ring of integers of a local field are compatible along the boundary up to stratification. This extends a theorem of Deligne on curves over a finite field. As an application, we deduce the equicharacteristic case of classical conjectures on ℓ-independence for proper smooth varieties over complete discrete valuation fields. Moreover, we show that compatible systems have compatible ramification. We also prove an analogue for integrality along the boundary.

Let F q be a finite field of q elements. For multiplicative characters χ 1 , . . . , χ m of F × q , we let J(χ 1 , . . . , χ m ) denote the Jacobi sum. Nicholas Katz and Zhiyong Zheng showed that for m = 2, the normalized Jacobi sum q −1/2 J(χ 1 , χ 2 ) (χ 1 χ 2 nontrivial) is asymptotically equidistributed on the unit circle as q → ∞, when χ 1 and χ 2 run through all nontrivial multiplicative characters of F × q . In this paper, we show a similar property for m ≥ 2. More generally, we show that the normalized Jacobi sum q −(m−1)/2 J(χ 1 , . . . , χ m ) (χ 1 · · · χ m nontrivial) is asymptotically equidistributed on the unit circle, when χ 1 , . . . , χ m run through arbitrary sets of nontrivial multiplicative characters of F × q with two of the sets being sufficiently large. The case m = 2 answers a question of Shparlinski.