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Trace maps in motivic homotopy and local terms

Fangzhou Jin

Abstract

We define a trace map for every cohomological correspondence in the motivic stable homotopy category over a general base scheme, which takes values in the twisted bivariant groups. Local contributions to the trace map give rise to quadratic refinements of the classical local terms, and some $\mathbb{A}^1$-enumerative invariants, such as the local $\mathbb{A}^1$-Brouwer degree and the Euler class with support, can be interpreted as local terms. We prove an analogue of a theorem of Varshavsky, which states that for a contracting correspondence, the local terms agree with the naive local terms.

Trace maps in motivic homotopy and local terms

Abstract

We define a trace map for every cohomological correspondence in the motivic stable homotopy category over a general base scheme, which takes values in the twisted bivariant groups. Local contributions to the trace map give rise to quadratic refinements of the classical local terms, and some -enumerative invariants, such as the local -Brouwer degree and the Euler class with support, can be interpreted as local terms. We prove an analogue of a theorem of Varshavsky, which states that for a contracting correspondence, the local terms agree with the naive local terms.

Paper Structure

This paper contains 25 sections, 33 theorems, 118 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations and conventions
  4. Bivariant theory in motivic homotopy
  5. Base change
  6. Proper push-forward
  7. (Refined) pullback
  8. Specialization
  9. On the generators of motivic categories
  10. Correspondences and the trace
  11. Relative Künneth formulas
  12. Correspondences and the trace
  13. Operations on correspondences
  14. Base change
  15. Push-forward
  16. ...and 10 more sections

Key Result

Theorem 1.0.1

Let $C\xrightarrow{(c_1,c_2)}X\times_SX$ be a morphism of schemes, let $K\in\mathcal{SH}_c(X)$ be a constructible motivic spectrum over $X$ which is universally strongly locally acyclic over $S$ (see Definition def:locacy), and let $v$ be a virtual vector bundle on $C$. Then for a cohomological corr where $Fix(c)$ is given by the fiber product

Theorems & Definitions (50)

  • Theorem 1.0.1: see § \ref{['sec:corrtr']}
  • Theorem 1.0.2: see Theorem \ref{['th:LTnaive']}
  • Lemma 2.2.6
  • Lemma 2.2.8
  • Lemma 2.3.2: ILO
  • Lemma 2.3.4: EK
  • Lemma 2.3.6
  • Remark 2.3.7
  • Definition 3.1.1: JY
  • Lemma 3.1.2
  • ...and 40 more