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Local terms for the categorical trace

Dennis Gaitsgory, Yakov Varshavsky

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.

Local terms for the categorical trace

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.

Paper Structure

This paper contains 12 sections, 29 theorems, 272 equations.

Table of Contents

  1. Sheaves on Artin stacks
  2. Generalized base change morphisms
  3. Functoriality of correspondences and traces
  4. True local terms maps
  5. Functoriality of true local terms
  6. Local terms and the Grothendieck--Lefschetz trace formula
  7. Deformation to the normal cone, and proof of Theorem \ref{['T:contracting']}
  8. Proof of Propositions \ref{['P:compatpf']} and \ref{['P:compatpb']}
  9. Completion of proofs, I
  10. Completion of proofs, II
  11. Proof of Proposition \ref{['P:safe']} and Corollary \ref{['C:safe']}
  12. Quasi-smooth morphisms

Key Result

Theorem 4

(a) Every commutative diagram (Eq:morph of corr) such that morphisms $f$ and $g$ are proper and safeSee Section E:safe what safe morphism means. gives rise to a homotopy commutative diagram \begin{CD} \Tr(\Shv(X)^{\ren},[c])@>\LT_{c}^{\true}>> \Gm(\Fix(c),\om_{\Fix(c)})\\ @V\Tr([f_!]) VV @VV(g_{\Dt

Theorems & Definitions (55)

  • Theorem 4
  • Theorem 8
  • Corollary 9
  • Proposition 1.4
  • Corollary 1.5
  • Lemma 1.9
  • proof
  • Corollary 1.10
  • proof
  • Proposition 3.5
  • ...and 45 more