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A motivic Segal theorem for pairs (announcement). (English. Russian original) Zbl 1455.14044

J. Math. Sci., New York 252, No. 6, 860-872 (2021); translation from Zap. Nauchn. Semin. POMI 484, 165-184 (2019).
Summary: In order to provide a new, more computation-friendly, construction of the stable motivic category \(SH (k)\), V. Voevodsyky laid the foundation of delooping motivic spaces. G. Garkusha and I. Panin [“Framed motives of algebraic varieties (after V. Voevodsky)”, Preprint, arXiv:1409.4372] based on joint works with G. Garkusha et al. [“Framed motives of relative motivic spheres”, Preprint, arXiv:1604.02732], A. Ananyevskiy et al. [“Cancellation theorem for framed motives of algebraic varieties”, Preprint, arXiv:1601.06642], A. Druzhinin and I. Panin [“Surjectivity of the etale excision map for homotopy invariant framed presheaves”, Preprint, arXiv:1808.07765] made that project a reality. In particular, G. Garkusha and I. Panin proved that for an infinite perfect field \(k\) and any \(k\)-smooth scheme \(X\), the canonical morphism of motivic spaces \({C}_{\ast } Fr(X)\to{\Omega}_{{\mathbb{P}}^1}^{\infty }{\sum}_{{\mathbb{P}}^1}^{\infty}\left({X}_+\right)\) is a Nisnevich locally group-completion. In the present paper, a generalization of that theorem is established to the case of smooth open pairs \((X,U)\), where \(X\) is a \(k\)-smooth scheme and \(U\) is its open subscheme intersecting each component of \(X\) in a nonempty subscheme. It is claimed that in this case the motivic space \(C_* Fr ((X,U))\) is a Nisnevich locally connected, and the motivic space morphism \({C}_{\ast } Fr\left(\left(X,U\right)\right)\to{\Omega}_{{\mathbb{P}}^1}^{\infty }{\sum}_{{\mathbb{P}}^1}^{\infty}\left(X/U\right)\) is Nisnevich locally weak equivalence. Moreover, it is proved that if the codimension of \(S = X -U\) in each component of \(X\) is greater than \(r \geq 0\), then the simplicial sheaf \(C_* Fr ((X,U))\) is locally \(r\)-connected.

MSC:

14F42 Motivic cohomology; motivic homotopy theory
18G99 Homological algebra in category theory, derived categories and functors
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References:

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