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A classical proof that the algebraic homotopy class of a rational function is the residue pairing. (English) Zbl 1437.14030
The authors of this paper have a substantial research project based around the concept of “degree” in \(\mathbb{A}^1\)-homotopy theory. This paper is a smaller companion of a larger paper of theirs [Duke Math. J. 168, No. 3, 429–469 (2019; Zbl 1412.14014)]. Fix a field \(k\). The main theorem of the major paper identifies the local \(\mathbb{A}^1\)-degree at \(0\) of a map \(f: \mathbb{A}_k^n \to \mathbb{A}_k^n\) having an isolated zero at the origin with the class in the Grothendieck-Witt group \(GW(k)\) of a specific bilinear form depending on \(f\).
This paper concerns maps \(f/g: \mathbb{P}_k^1 \to \mathbb{P}_k^1\). A construction by Hurwitz from 1895 calculates the topological degree of a rational map \(f/g: S^1=\mathbb{R}P^1 \to \mathbb{R}P^1=S^1\) as the signature of an explicitly-associated bilinear form. For general \(k\), the variety \(\mathbb{P}_k^1\) is an \(\mathbb{A}^1\)-homotopy sphere, and the set of stable \(\mathbb{A}^1\)-homotopy self-maps of \(\mathbb{P}_k^1\) is \(GW(k)\), by virtue of work of F. Morel [in: Proceedings of the international congress of mathematicians (ICM), Madrid, Spain, August 22–30, 2006. Volume II: Invited lectures. Zürich: European Mathematical Society (EMS). 1035–1059 (2006; Zbl 1097.14014)].
A paper of C. Cazanave [C. R., Math., Acad. Sci. Paris 346, No. 3–4, 129–133 (2008; Zbl 1151.14016)] gives an explicit description of the unstable \(\mathbb{A}^1\)-homotopy classes of self-maps of \(\mathbb{P}^1_k\), and in particular, given any rational map \(f/g\), one can produce an associated stable-degree class in \(GW(k)\), the class of a completely explicit “Bézout matrix”.
Fix a rational map \(f/g : \mathbb{P}^1_k \to \mathbb{P}^1_k\). This paper uses the authors’ work [Duke Math. J. 168, No. 3, 429–469 (2019; Zbl 1412.14014)] to produce Cazanave’s class for \(f/g\) by the classical topological method of summation over local degrees. The result is somewhat weaker than can be obtained by Cazanave’s methods, because the invariants considered here are only in the stable \(\mathbb{A}^1\)-homotopy groups, not the unstable. Nonetheless, it is extremely pleasant to see that the machinery of the \(\mathbb{A}^1\)-degree producing a completely explicit, calculable invariant of self-maps of \(\mathbb{P}^1_k\) by classical homotopy-theoretic methods.
The paper is short, and much of it is exposition. This is sure to be of great help to anyone who wishes to learn about the \(\mathbb{A}^1\)-degree, which might be daunting to the newcomer. Most of what is not exposition is concerned with the details required to make the \(\mathbb{A}^1\)-degree machinery work in this context. It is highly recommended to anyone wanting to see how this theory actually works in practice. What remains is some brief results required to show certain matrices are congruent to one another, and a final Section 3 that discusses the connection between the work at hand and the work of D. Eisenbud and H. I. Levine [Ann. Math. (2) 106, 19–44 (1977; Zbl 0398.57020)] and G. N. Khimshiashvili [Soobshch. Akad. Nauk Gruz. SSR 85, 309–312 (1977; Zbl 0346.55008); Proc. A. Razmadze Math. Inst. 125, 1–121 (2001; Zbl 1059.58027)]. The final section may function as motivation and background for the longer paper: [loc. cit.].

14F42 Motivic cohomology; motivic homotopy theory
14B05 Singularities in algebraic geometry
55M25 Degree, winding number
Full Text: DOI
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