Absolute purity of the sphere spectrum (and algebraic cobordism)

Establish that the sphere spectrum over Spec(Z) is absolutely pure in the sense of motivic bivariant theory: for every smoothable lci morphism f: X → S between regular schemes, prove that the canonical purity transformation makes the sphere spectrum f-pure. Consequently, deduce that the algebraic cobordism spectrum MGL is absolutely pure.

Background

The paper defines f-purity for a spectrum E as the property that the cotrace map (purity transformation) is an isomorphism for smoothable lci morphisms f: X → S. Absolute purity means E is f-pure for all such morphisms between regular schemes. This property underlies duality and Gysin formalism in motivic homotopy.

The authors note that K-theory, rational motivic Eilenberg–MacLane spectra, and the rational sphere and cobordism spectra are known to be absolutely pure. Extending absolute purity to the integral sphere spectrum would settle the status of MGL (algebraic cobordism) as absolutely pure, with broad consequences for Riemann–Roch and bivariant theories in the motivic setting.