Categorical traces and a relative Lefschetz-Verdier formula

TL;DR

This paper establishes a relative Lefschetz-Verdier theorem for locally acyclic objects using duals and traces in a symmetric monoidal 2-category, with applications to nearby cycle functors and duality preservation.

Contribution

It introduces a new approach to the Lefschetz-Verdier theorem via cohomological correspondences and proves duality preserves local acyclicity.

Findings

Local acyclicity is equivalent to dualizability.

Duality preserves local acyclicity.

Nearby cycle functor over Henselian rings preserves duals.

Abstract

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 dualizability 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, generalizing a theorem of Gabber.