η$\eta$‐Periodic motivic stable homotopy theory over Dedekind domains

Abstract

We construct well-behaved extensions of the motivic spectra representing generalized motivic cohomology and connective Balmer–Witt K $K$ -theory (among others) to mixed characteristic Dedekind schemes on which 2 is invertible. As a consequence, we lift the fundamental fiber sequence of η $\eta$ -periodic motivic stable homotopy theory established in Bachmann and Hopkins (2020) from fields to arbitrary base schemes, and use this to determine (among other things) the η $\eta$ -periodized algebraic symplectic and SL ${\rm SL}$ -cobordism groups of mixed characteristic Dedekind schemes containing 1 / 2 $1/2$ .