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.
On the effectivity of spectra representing motivic cohomology theories
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abstract
Let k be an infinite perfect field. We provide a general criterion for a spectrum in the stable homotopy category over k to be effective, i.e. to be in the localizing subcategory generated by the suspension spectra of smooth schemes. As a consequence, we show that two recent versions of generalized motivic cohomology theories coincide.
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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.