Ivan Panin (St Petersburg/Oslo): Linearization of Voevodsky's conjecture.

We recall the notion of rational framed correspondences in the context of: Conjecture (Voevodsky)  If \(k\) is an infinite perfect field, then for each k-smooth variety X the canonical motivic space morphism         \(\operatorname{Fr}^{rational}(\Delta^{\bullet} \times -, X) \rightarrow \Omega^{\infty}_{\mathbb{P}^1} \ \Sigma^{\infty}_{\mathbb{P}^1}(X_+)\) is a Nisnevich local equivalence.  Using a theorem of Garkusha and Panin this conjecture is equivalent to saying that the motivic space morphism       \(\operatorname{Fr}(\Delta^{\bullet} \times -, X)\rightarrow \operatorname{Fr}^{rational}(\Delta^{\bullet} \times -, X)\). is locally a group completion of the space \(\operatorname{Fr}^{rat}(\Delta^{\bullet} \times -, X)\). A linear version of Voevodsky's conjecture states that for each k-smooth variety X and each field K/k, the inclusion of complexes of abelian groups                 \(\operatorname{ZF}(\Delta^\bullet_K, X) \rightarrow \operatorname{ZF}^{rational}\Delta^\bullet_K, X)\) is a quasi-isomorphism.  Here, \(\operatorname{ZF}\) and \(\operatorname{ZF}^{rational}\) denotes stable linear and stable linear rational framed correspondences, respectively. It is known that the linear version of the conjecture yields the original conjecture.