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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.

##### MSC:
 14F35 Homotopy theory and fundamental groups in algebraic geometry
##### Keywords:
strictly homotopy invariant sheaf
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##### References:
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