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Gabber’s presentation lemma over Noetherian domains. (English) Zbl 1454.14056

Morel’s famous stable \(\mathbb{A}^1\)-connectivity theorem [F. Morel, K-Theory 35, No. 1–2, 1–68 (2005; Zbl 1117.14023)] says that, over an infinite field \(K\), the \(\mathbb{A}^1\)-localization of a connected spectrum on the smooth Nisnevich site over \(K\) is still connected. It is a natural question to ask to what extend this result can be generalized to more general base schemes. However it turns out that this fails if one just replaces \(K\) by an arbitrary scheme, even in the regular case, as shown in [J. Ayoub, C. R., Math., Acad. Sci. Paris 342, No. 12, 943-948 (2006; Zbl 1103.14009)]. Instead the more natural property to ask for is the so called shifted stable \(\mathbb{A}^1\)-connectivity property, introduced in [J. Schmidt and F. Strunk, Ann. K-Theory 3, No. 2, 331–367 (2018; Zbl 1423.14158)] which is said to be satisfied by a scheme \(S\) of dimension \(d\), if the \(\mathbb{A}^1\)-localization of a \(d\)-connected spectrum on the smooth Nisnevich site of \(S\) is connected.
In the present article, the authors show that the spectrum of any noetherian domain of dimension \(d\) with infinite residue fields satisfies the shifted stable connectivity property. This is a vast generalization of the result proven by Schmidt and Strunk in their previously mentioned paper, where this was shown for Dedekind schemes with infinite residue fields. The main novelty is Theorem 1.1, which is a version of O. Gabbers presentation Lemma (see [O. Gabber, Manuscr. Math. 85, No. 3–4, 323–343 (1994, Zbl 0827.19002)]) over a general noetherian domain with infinite residue fields. Armed with this, the proof of the stable connectivity result is the same as in the paper of Schmidt and Strunk. A sketch of this proof is given in the fourth Chapter of the article under review.
In order to prove Theorem 1.1, the authors observe that the arguments given by Schmidt and Strunk can be adapted to the more general situation. The main technical crux is Theorem 2.1, which is a weaker version of a result of K. Wataru (see Theorem 4.1 in [K. Wataru, “A moving lemma for algebraic cycles with modulus and contravariance”, Preprint, arXiv:1507.07619]) but over a more general base, which is still sufficient to generalize the arguments given by Schmidt and Strunk.

MSC:

14F20 Étale and other Grothendieck topologies and (co)homologies
14F42 Motivic cohomology; motivic homotopy theory
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[1] Colliot-Thélene, Jean-Louis; Hoobler, Raymond T.; Kahn, Bruno, The Bloch-Ogus-Gabber theorem, Algebraic K-Theory. Algebraic K-Theory, Toronto, ON, 1996, 16, 31-94 (1997) · Zbl 0911.14004
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