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 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.

On the Kottwitz conjecture for local shtuka spaces

“…We adapt here [LZ22] to the setting of v-stacks, but the same language could be used in the world of stacks in the scheme setting. The main point of departure from [LZ22] is that stacks form a 2-category, and so one must keep track of the 2-morphisms witnessing commutativity of diagrams of stacks.…”

“…We adapt here [LZ22] to the setting of v-stacks, but the same language could be used in the world of stacks in the scheme setting. The main point of departure from [LZ22] is that stacks form a 2-category, and so one must keep track of the 2-morphisms witnessing commutativity of diagrams of stacks. This means that the definition of cohomological correspondences we give (Definition 4.3.4) is a little more delicate than its analogue in [LZ22].…”

“…The main result is Theorem 4.3.8 and its Corollary 4.3.9. We very closely follow the approach of [LZ22], putting a suitable symmetric monoidal 2-category of cohomological correspondences at center stage. In both the schematic and perfectoid settings, the finiteness condition required of the object A can be stated in terms of the property of universal local acyclicity (ULA); as noted in [FS21, Theorem IV.2.23], this is precisely the hypothesis necessary to obtain the isomorphism in equation (4.0.1).…”

“…We learned from [LZ22] that the ULA condition is precisely the right hypothesis necessary to prove a Lefschetz–Verdier trace formula applicable to the cohomology of A . This explains the counterexample above: fails to be ULA, where j is the inclusion of the affinoid disc X into its compactification .…”

“…This explains the counterexample above: fails to be ULA, where j is the inclusion of the affinoid disc X into its compactification . In fact, [LZ22] is written in the context of schemes, but their formalism applies equally well in the context of rigid-analytic spaces and diamonds. Indeed, some interesting new phenomena occur in the diamond context.…”