Trace maps in motivic homotopy and local terms

TL;DR

This paper introduces a trace map in motivic homotopy theory, linking local contributions to quadratic refinements and $A^1$-enumerative invariants, and proves a theorem relating local terms for contracting correspondences.

Contribution

It defines a new trace map in the motivic stable homotopy category and relates local terms to quadratic refinements and enumerative invariants, extending classical results.

Findings

Trace map defined for cohomological correspondences in motivic homotopy

Local contributions yield quadratic refinements of local terms

For contracting correspondences, local terms match naive 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 $\mathbb{A}^1$-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.