A general descendability criterion for topological six-functor formalisms yields h-descent for étale motivic spectra and rational motivic cohomology, plus arc-descent in weights ≤1.
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abstract
Over any field of characteristic not 2, we establish a 2-term resolution of the $\eta$-periodic, 2-local motivic sphere spectrum by shifts of the connective 2-local Witt K-theory spectrum. This is curiously similar to the resolution of the K(1)-local sphere in classical stable homotopy theory. As applications we determine the $\eta$-periodized motivic stable stems and the $\eta$-periodized algebraic symplectic and SL-cobordism groups. Along the way we construct Adams operations on the motivic spectrum representing Hermitian K-theory and establish new completeness results for certain motivic spectra over fields of finite virtual 2-cohomological dimension. In an appendix, we supply a new proof of the homotopy fixed point theorem for the Hermitian K-theory of fields.
representative citing papers
Geometric formal ternary laws plus framed involutions classify Sp-orientations of MSp[η−1] injectively and become isomorphisms after inverting 2, with residual 2-primary data left open.
A new fibre sequence relates classical Grothendieck-Witt groups to K-theory orbits and symmetric L-theory, enabling removal of the 2-unit assumption and resolution of multiple open problems for Dedekind rings and number rings.
citing papers explorer
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Descendability and descent in topological weaves
A general descendability criterion for topological six-functor formalisms yields h-descent for étale motivic spectra and rational motivic cohomology, plus arc-descent in weights ≤1.
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On $\eta$-periodic Formal Ternary Laws
Geometric formal ternary laws plus framed involutions classify Sp-orientations of MSp[η−1] injectively and become isomorphisms after inverting 2, with residual 2-primary data left open.
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Hermitian K-theory for stable $\infty$-categories III: Grothendieck-Witt groups of rings
A new fibre sequence relates classical Grothendieck-Witt groups to K-theory orbits and symmetric L-theory, enabling removal of the 2-unit assumption and resolution of multiple open problems for Dedekind rings and number rings.