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Categorical traces and a relative Lefschetz-Verdier formula

Qing Lu, Weizhe Zheng

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.

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 -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.

Paper Structure

This paper contains 13 sections, 30 theorems, 40 equations.

Table of Contents

  1. Pairings in symmetric monoidal $2$-categories
  2. Pairings
  3. Functoriality of pairings
  4. The Grothendieck construction
  5. A relative Lefschetz-Verdier formula
  6. Relative Künneth formulas
  7. The category of cohomological correspondences
  8. Dualizable objects
  9. The relative Lefschetz-Verdier pairing
  10. Base change and duals
  11. Nearby cycles over Henselian valuation rings
  12. Künneth formulas and duals
  13. Pushforward and fixed points

Key Result

Theorem 1

Let $S$ be a Noetherian scheme and let $\Lambda$ be a Noetherian commutative ring with $m\Lambda=0$ for some $m$ invertible on $S$. Let \xymatrix{X\ar[d]_{f} & C\ar[l]_{\overleftarrow{c}}\ar[d]^{p}\ar[r]^{\overrightarrow{c}} & Y\ar[d]^{g} & D\ar[l]_{\overleftarrow{d}}\ar[r]^{\overrightarrow{d}}\ar[d

Theorems & Definitions (68)

  • Theorem 1
  • Remark 2
  • Definition 1.1: dual
  • Remark 1.2
  • Remark 1.3
  • Lemma 1.4
  • proof
  • Remark 1.9
  • Lemma 1.11
  • proof
  • ...and 58 more