Local terms for the categorical trace

TL;DR

This paper develops categorical local term maps for Artin stacks, demonstrating their additivity and compatibility with key operations, and applies these results to prove the equivalence of true and naive local terms for Frobenius and their commutation with pushforwards.

Contribution

It introduces true local term maps for Artin stacks and proves their fundamental properties, extending previous results and providing new proofs of key theorems.

Findings

True local terms are additive and commute with proper pushforwards and smooth pullbacks.

True local terms of Frobenius coincide with naive local terms.

Naive local terms commute with !-pushforwards, generalizing the Grothendieck--Lefschetz trace formula.

Abstract

In this paper we introduce the categorical "true local terms" maps for Artin stacks and show that they are additive and commute with proper pushforwards, smooth pullbacks and specializations. In particular, we generalizing results of [Va2] to this setting. As an application, we supply proofs of two theorems stated in [AGKRRV]. Namely, we show that the "true local terms" of the Frobenius endomorphism coincide with the "naive local terms" and that the "naive local terms" commute with !-pushforwards. The latter result is a categorical version of the classical Grothendieck--Lefschetz trace formula.