Categorical traces and a relative Lefschetz-Verdier formula

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.

arXiv comment27 pages. v4: minor improvements. To appear in Forum of Mathematics, Sigma

References in corpus (7)

• The characteristic cycle and the singular support of a constructible sheaf
• Cycles on Shimura varieties via geometric Satake
• Geometrization of the local Langlands correspondence
• Modifications et cycles évanescents sur une base de dimension supérieure à un
• Duality and nearby cycles over general bases
• Six operations and Lefschetz-Verdier formula for Deligne-Mumford stacks
• The Lefschetz-Lunts formula for deformation quantization modules

Cited by in corpus (3)

• The singular support of an $\ell$-adic sheaf
• Cohomological Milnor formula and Saito's conjecture on characteristic classes
• Trace maps in motivic homotopy and local terms