Invertible objects in motivic homotopy theory

If $X$ is a (reasonable) base scheme then there are the categories of interest in stable motivic homotopy theory \SH(X) and \DM(X), constructed by Morel-Voevodsky and others. These should be thought of as generalisations respectively of the stable homotopy category \SH and the derived category of abelian groups $D(Ab)$, which are studied in classical topology, to the ``world of smooth schemes over $X$''. Just like in topology, the categories \SH(X), \DM(X) are symmetric monoidal: there is a bifunctor $(E, F) \mapsto E \otimes F$ satisfying certain properties; in particular there is a \emph{unit} \tunit satisfying E \otimes \tunit \wequi \tunit \otimes E \wequi E for all $E$. In any symmetric monoidal category $\mathcal{C}$ an object $E$ is called \emph{invertible} if there is an object $F$ such that E \otimes F \wequi \tunit. Modulo set theoretic problems (which do not occur in practice) the isomorphism classes of invertible objects of a symmetric monoidal category $\mathcal{C}$ form an abelian group $Pic(\mathcal{C})$ called the \emph{Picard group of $\mathcal{C}$}. The aim of this work is to study Pic(\SH(X)), Pic(\DM(X)) and Relations between these various groups. A complete computation seems out of reach at the moment. We can show that (in good cases) the natural homomorphism Pic(\SH(X)) \to \prod_{x \in X} Pic(\SH(x)) coming from pull back to points has as kernel the \emph{locally trivial invertible spectra} Pic^0(\SH(X)). (For \DM, this homomorphism is injective.) Moreover if $x = Spec(k)$ is a point, then (again in good cases) the homomorphism Pic(\SH(x)) \to Pic(\DM(x)) is injective. This reduces (in some sense) the study of Pic(\SH(X)) to the study of the Picard groups of \DM over fields, and the latter category is much better understood. We then show that, for example, the reduced motive of a smooth affine quadric is invertible in \DM(k). By the previous results, it follows that affine quadric bundles over $X$ yield (in good cases) invertible objects in \SH(X). This is related to a conjecture of Po Hu

η\eta-periodic motivic stable homotopy theory over Dedekind domains

We construct well-behaved extensions of the motivic spectra representing generalized motivic cohomology and connective Balmer--Witt K-theory (among others) to mixed characteristic Dedekind schemes on which 2 is invertible. As a consequence we lift the fundamental fiber sequence of $\eta$-periodic motivic stable homotopy theory established in [arxiv:2005.06778] from fields to arbitrary base schemes, and use this to determine (among other things) the $\eta$-periodized algebraic symplectic and SL-cobordism groups of mixed characteristic Dedekind schemes containing 1/2.Comment: 14 page