Stable 𝔸1-connectivity over a base

Abstract

Abstract Morel’s stable connectivity theorem states the vanishing of the sheaves of the negative motivic homotopy groups π ¯ i s ⁢ ( Y ) {\underline{\pi}^{s}_{i}(Y)} and π ¯ i + j , j s ⁢ ( Y ) {\underline{\pi}^{s}_{i+j,j}(Y)} , i < 0 {i<0} , in the stable motivic homotopy categories 𝐒𝐇 S 1 ⁢ ( k ) {\mathbf{SH}^{S^{1}}(k)} and 𝐒𝐇 ⁢ ( k ) {\mathbf{SH}(k)} for an arbitrary smooth scheme Y over a field k. Originally the same property was conjectured in the relative case over a base scheme S. In view of Ayoub’s conterexamples the modified version of the conjecture states the vanishing of stable motivic homotopy groups π ¯ i s ⁢ ( Y ) {\underline{\pi}^{s}_{i}(Y)} (and π ¯ i + j , j s ⁢ ( Y ) {\underline{\pi}^{s}_{i+j,j}(Y)} ) for i < - d {i<-d} , where d = dim ⁡ S {d=\dim S} is the Krull dimension. The latter version of the conjecture is proven over noetherian domains of finite Krull dimension under the assumption that residue fields of the base scheme are infinite. This is the result by J. Schmidt and F. Strunk for Dedekind schemes case, and the result by N. Deshmukh, A. Hogadi, G. Kulkarni and S. Yadavand for the case of noetherian domains of an arbitrary dimension. In the article, we prove the result for any locally noetharian base scheme of finite Krull dimension without the assumption on the residue fields, in particular for 𝐒𝐇 S 1 ⁢ ( ℤ ) {\mathbf{SH}^{S^{1}}(\mathbb{Z})} and 𝐒𝐇 ⁢ ( ℤ ) {\mathbf{SH}(\mathbb{Z})} . In the appendix, we modify the arguments used for the main result to obtain the independent proof of Gabber’s Presentation Lemma over finite fields.