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The stable \(\mathbb{A}^1\)-connectivity theorems. (English) Zbl 1117.14023
A sheaf of \(S^1\)-spectra \(E\) on the category of smooth schemes over \(S\) in the Nisnevich topology is said to be \((-1)\)-connected if all its negative homotopy sheaves vanish. One says that the stable \(\mathbb A^1\)-connectivity property holds over \(S\) if the \(\mathbb A^1\)-localization functor preserves \((-1)\)-connected sheaves. The main result of the paper is that the stable \(\mathbb A^1\)-connectivity property holds when the base \(S\) is the spectrum of a field. The proof essentially uses Gabber’s presentation lemma over infinite fields. As a consequence the author proves that for any sheaf of spectra \(E\) defined over a field its \(\mathbb A^1\)-homotopy sheaves are strictly \(\mathbb A^1\)-invariant. In particular, it holds for the sheaf of Balmer-Witt groups. In the language of stable homotopy categories it also implies that there is a \(t\)-structure on the stable \(\mathbb A^1\)-homotopy category of \(S^1\)-spectra whose heart consists of strictly \(\mathbb A^1\)-invariant sheaves. This \(t\)-structure can be viewed as a direct analogue in the stable \(\mathbb A^1\)-homotopy theory of Voevodsky’s homotopy \(t\)-structure for the triangulated category \(\text{DM}^{\text{eff}}\) over a perfect field. As an important application the author proves the Gersten conjecture for pure sheaves over a field, e.g., strictly \(\mathbb A^1\)-homotopy invariant sheaves. He also discusses the finitness properties of \(\mathbb A^1\)-homotopy groups.

14F35 Homotopy theory and fundamental groups in algebraic geometry
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