20 Marzec 2024

Neeraj Deshmukh

Title: Stable A^1-connectivity theorem after Morel

Abstract: The stable connectivity theorem of Morel in motivic homotopy theory asserts that, over a field, the A^1-localisation functor for S^1-spectra preserves connectivity. Roughly speaking this means that no homotopy groups are gained in the "wrong direction", after applying the A^1-localisation functor. Such a theorem fails for higher dimensional bases as shown by a counterexample of Joseph Ayoub (in dimension 2). However, it is still possible to quantify this failure of connectivity after A^1-localisation: if S is a base scheme of Krull dimension d, then the A^1-localisation of a connective S^1-spectrum is (-d)-connective. This is the (shifted) stable connectivity theorem over a base. In this talk, we will discuss these results and the ingredients that go into proving them.