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The objects of study in stable $A^1$-homotopy theory are $A^1$-invariant Nisnevich sheaves of (topological) spectra. Given a Nisnevich sheaf of spectra, there exists a universal process of making it $A^1$-invariant, called {$A^1$-localization}. Unfortunately, this is not a stalkwise process and the property of being stalkwise a connected spectrum may be destroyed. However, the $A^1$-connectivity theorem of Morel shows that this is not the case when working over a field.
We report on joint work with Johannes Schmidt and sketch our approach towards the follwing theorem: Over a regular scheme of dimension one with all residue fields infinite, $A^1$-localization decreases the stalkwise connectivity by at most one. As in Morel's case, we use a strong geometric input which is a Nisnevich-local version of Gabber's geometric presentation result over a henselian discrete valuation ring with infinite residue field.
For most parts of the talk, no prior knowledge on $A^1$-homotopy theory is needed.*

Freitag, den 8. Juli 2016 um 13:30 Uhr, in INF205, SR C Freitag, den 8. Juli 2016 at 13:30, in INF205, SR C

Der Vortrag folgt der Einladung von The lecture takes place at invitation by Dr. Johannes Schmidt