Bachmann, Tom (2016): Invertible objects in motivic homotopy theory. Dissertation, LMU München: Faculty of Mathematics, Computer Science and Statistics
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### Abstract

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.