Cancellation theorem for framed motives of algebraic varieties

On modules over motivic ring spectra

“…[20]. For the ∞-category of framed motivic spaces, cancellation holds by [12,Theorem 3.5.8], which in turn relies on the cancellation theorem of Ananyevskiy, Garkusha and Panin [1]. Moreover, for any base scheme S, the subcategory SH(S) eff ⊆ SH(S) of effective motivic spectra is T-prestable.…”

“…cit. 1 In the present paper, we aim to provide a general axiomatic approach to the above results. More precisely, by making use of Lurie's ∞-categorical version of the Barr-Beck theorem we derive a characterization of those stable ∞-categories that are equivalent to a module category over a motivic spectrum.…”

The Unit Map of the Algebraic Special Linear Cobordism Spectrum

“…In algebraic topology, an important resource for analyzing the stable homotopy groups of spheres is given by the unit map of the complex cobordism spectrum MU. This map has at least two features: (1) it induces an isomorphism on π 0 = Z; (2) it detects nilpotence, giving rise to the field of chromatic homotopy theory.…”

“…On the other hand, the abelian group π 0 of a spectrum is replaced by a richer invariant in motivic settings. For a motivic P 1 -spectrum E, one considers a sequence of Nisnevich sheaves of abelian groups {π 0 (E) l } l∈Z , called a homotopy module. One may ask an analogous question: does the unit map of a motivic cobordism spectrum induce an isomorphism of homotopy modules?…”

“…In a series of papers, Ananyevskiy, Druzhinin, Garkusha, Neshitov and Panin developed a theory of framed motives [1,13,[18][19][20]. As one of their main results, they computed infinite P 1 -loop spaces of P 1 -suspension spectra in terms of framed correspondences, when k is a perfect field.…”