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η-periodic motivic stable homotopy theory over fields

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arxiv 2005.06778 v4 pith:KH4KYJPT submitted 2020-05-14 math.KT math.AGmath.AT

η-periodic motivic stable homotopy theory over fields

Tom Bachmann , Michael J. Hopkins This is my paper
classification math.KT math.AGmath.AT
keywords motivicfieldshomotopyk-theorylocalspectrumstableestablish
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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.

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Cited by 5 Pith papers

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