Euler class groups and motivic stable cohomotopy (with an appendix by Mrinal Kanti Das). (English) Zbl 1487.14053

Asok, Aravind; Fasel, Jean

J. Eur. Math. Soc. (JEMS) 24, No. 8, 2775-2822 (2022).

There are several ways, with very different flavours, to attach an Euler class to a rank \(d\) vector bundle on an algebraic variety of dimension \(d\). This paper compares some of them, with extensive explanations.
Summary: “We study maps from a smooth scheme to a motivic sphere in the Morel-Voevodsky \(\mathbb{A}^1\)-homotopy category, i.e., motivic cohomotopy sets. Following Borsuk, we show that, in the presence of suitable hypotheses on the dimension of the source, motivic cohomotopy sets can be equipped with functorial abelian group structures. We then explore links between motivic cohomotopy groups, Euler class groups à la Nori-Bhatwadekar-Sridharan and Chow-Witt groups. We show that, again under suitable hypotheses on the base field \({k} \), if \({X}\) is a smooth affine \(k\)-variety of dimension \(d\), then the Euler class group of codimension \(d\) cycles coincides with the codimension \(d\) Chow-Witt group; the identification proceeds by comparing both groups with a suitable motivic cohomotopy group. As a byproduct, we describe the Chow group of zero cycles on a smooth affine \(k\)-scheme as the quotient of the free abelian group on zero cycles by the subgroup generated by reduced complete intersection ideals; this answers a question of S. Bhatwadekar and R. Sridharan.”
The authors make heavy use of the smooth quadric \(Q_{2n}\subset \mathbb A^{2n+1}_{\mathbb Z}\) with equation \[ \sum_{i=1}^n x_i y_i = z (1- z) . \]
One knows that \(Q_{2n}\) is a sphere from the standpoint of the Morel–Voevodsky \(\mathbb A^1\)-homotopy theory. “As a consequence, if \(X\) is a smooth scheme, by analogy with the ideas of Borsuk, we will call the set \([X,Q_{2n}]_{\mathbb A^1}\) of morphisms in the \(\mathbb A^1\)-homotopy category a motivic cohomotopy set (see Definition 1.3.1). Paralleling our discussion of cohomotopy above, our goals in this paper are
(i) to equip the set of free \(\mathbb A^1\)-homotopy classes of maps \([X,Q_{2n}]_{\mathbb A^1}\) with a functorial abelian group structure,
(ii) to study algebro-geometric analogs of the Hurewicz homomorphism and Hopf classification theorem and
(iii) to describe a presentation of \([X,Q_{2n}]_{\mathbb A^1}\) with explicit generators and relations.”
There is an appendix by Mrinal Kanti Das, that “attempts to streamline (e.g., clarify necessary hypotheses) the presentation of some results on Euler class groups used in the main body of the paper.”

Reviewer: Wilberd van der Kallen (Utrecht)

MSC:

14F42	Motivic cohomology; motivic homotopy theory
13C10	Projective and free modules and ideals in commutative rings
19A13	Stability for projective modules
55Q55	Cohomotopy groups

Keywords:

motivic homotopy; Euler class; projective modules

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