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A counter-example to the \(\mathbb A^1\)-connectivity conjecture of F. Morel. (Un contre-exemple à la conjecture de \(\mathbb A^1\)-connexité de F. Morel.) (French) Zbl 1103.14009

Let \(X\) be a noetherian scheme and let \(\text{DM}_{\text{eff}}(X)\) be the subcategory of \(D(\text{Sh}\,{\nu}^{\text{tr}}_{\text{Nis}}(\text{Sm}/X))\) consisting of \({\mathbb A}^{1}\)-local complexes of Nisnevich sheaves with transfers. The inclusion functor has a left adjoint [cf. F. Morel, V. Voevodsky, Publ. Math., Inst. Hautes Étud. Sci. 90, 45–143 (1999; Zbl 0983.14007)] – the \({\mathbb A}^{1}\)-localization functor: \[ \text{Loc}_{{\mathbb A}^{1}} : D(\text{Sh}\,{\nu}^{\text{tr}}_{\text{Nis}} (\text{Sm}/X))\rightarrow \text{DM}_{\text{eff}}(X). \] The following \({\mathbb A}^{1}\)-connectivity conjecture was formulated by F. Morel [\(K\)-Theory 35, No. 1-2, 1–68 (2005; Zbl 1117.14023)].
Conjecture. If \(X\) is regular, the functor \(\text{Loc}_{{\mathbb A}^{1}}\) preserves \((-1)\)-connected complexes.
The author constructs a normal surface \(X\), which is an object of \(\text{DM}_{\text{eff}}(X)\) whose homology sheaves are not strictly \({\mathbb A}^{1}\)-invariant. This disproves the conjecture in general, although it might be (as the author points out) still true for curves.

MSC:

14F35 Homotopy theory and fundamental groups in algebraic geometry
14F99 (Co)homology theory in algebraic geometry
55N30 Sheaf cohomology in algebraic topology
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