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

TL;DR

This paper extends $\\eta$-periodic motivic stable homotopy theory to Dedekind schemes with mixed characteristic, constructing new spectra and lifting fundamental fiber sequences from fields to schemes, enabling computation of cobordism groups.

Contribution

It develops well-behaved extensions of motivic spectra over Dedekind schemes and lifts key fiber sequences from fields to schemes, advancing the understanding of motivic homotopy theory in mixed characteristic.

Findings

Constructed extensions of motivic spectra over Dedekind schemes.

Lifted the fundamental fiber sequence from fields to schemes.

Determined $\\eta$-periodized algebraic cobordism groups in mixed characteristic.

Abstract

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.