Over characteristic 0 fields, the unit map from the motivic sphere spectrum to the special linear cobordism spectrum MSL is an isomorphism on homotopy modules, proven by comparing framed and SL-oriented framed correspondences.
Cdh descent in equivariant homotopy K-theory
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abstract
We construct geometric models for classifying spaces of linear algebraic groups in G-equivariant motivic homotopy theory, where G is a tame group scheme. As a consequence, we show that the equivariant motivic spectrum representing the homotopy K-theory of G-schemes (which we construct as an E-infinity-ring) is stable under arbitrary base change, and we deduce that homotopy K-theory of G-schemes satisfies cdh descent.
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The unit map of the algebraic special linear cobordism spectrum
Over characteristic 0 fields, the unit map from the motivic sphere spectrum to the special linear cobordism spectrum MSL is an isomorphism on homotopy modules, proven by comparing framed and SL-oriented framed correspondences.