A new A1-invariant motivic cohomology for all qcqs schemes is constructed from the slice filtration of KGL, with a spectral sequence to homotopy K-theory and etale/syntomic comparisons.
$\eta$-periodic motivic stable homotopy theory over Dedekind domains
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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.
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$\mathbb{A}^1$-invariant motivic cohomology of schemes
A new A1-invariant motivic cohomology for all qcqs schemes is constructed from the slice filtration of KGL, with a spectral sequence to homotopy K-theory and etale/syntomic comparisons.