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In this note, we provide an axiomatic framework that characterizes the stable $infty$-categories that are module categories over a motivic spectrum. This is done by invoking Luries $infty$-categorical version of the Barr--Beck theorem. As an application, this gives an alternative approach to Rondigs and O stvae rs theorem relating Voevodskys motives with modules over motivic cohomology, and to Garkushas extension of Rondigs and O stvae rs result to general correspondence categories, including the category of Milnor-Witt correspondences in the sense of Calm`es and Fasel. We also extend these comparison results to regular Noetherian schemes over a field (after inverting the residue characteristic), following the methods of Cisinski and Deglise.
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We show that Shipleys detection functor for symmetric spectra generalizes to motivic symmetric spectra. As an application, we construct motivic strict ring spectra representing morphic cohomology, semi-topological $K$-theory, and semi-topological cobordism for complex varieties. As a further application to semi-topological cobordism, we show that it is related to semi-topological $K$-theory via a Conner-Floyd type isomorphism and that after inverting a lift of the Friedlander-Mazur $s$-element in morphic cohomology, semi-topological cobordism becomes isomorphic to periodic complex cobordism.
Recently an algebra of smooth valuations was attached to any smooth manifold. Roughly put, a smooth valuation is finitely additive measure on compact submanifolds with corners which satisfies some extra properties. In this note we initiate a study of modules over smooth valuations. More specifically we study finitely generated projective modules in analogy to the study of vector bundles on a manifold. In particular it is shown that on a compact manifold there exists a canonical isomorphism between the $K$-ring constructed out of finitely generated projective modules over valuations and the classical topological $K^0$-ring constructed out of vector bundles.
We study various triangulated motivic categories and introduce a vast family of aisles (these are certain classes of objects) in them. These aisles are defined in terms of the corresponding motives (or motivic spectra) of smooth varieties in them; we relate them to the corresponding homotopy t-structures. We describe our aisles in terms of stalks at function fields and prove that they widely generalize the ones corresponding to slice filtrations. Further, the filtrations on the homotopy hearts $Ht_{hom}^{eff}$ of the corresponding effective subcategories that are induced by these aisles can be described in terms of (Nisnevich) sheaf cohomology as well as in terms of the Voevodsky contractions $-_{-1}$. Respectively, we express the condition for an object of $Ht_{hom}^{eff}$ to be weakly birational (i.e., that its $n+1$th contraction is trivial or, equivalently, the Nisnevich cohomology vanishes in degrees $>n$ for some $nge 0$) in terms of these aisles; this statement generalizes well-known results of Kahn and Sujatha. Next, these classes define weight structures $w_{Smooth}^{s}$ (where $s=(s_{j})$ are non-decreasing sequences parameterizing our aisles) that vastly generalize the Chow weight structures $w_{Chow}$ defined earlier. Using general abstract nonsense we also construct the corresponding adjacent $t-$structures $t_{Smooth}^{s}$ and prove that they give the birationality filtrations on $Ht^{eff}_{hom}$. Moreover, some of these weight structures induce weight structures on the corresponding $n-$birational motivic categories (these are the localizations by the levels of the slice filtrations). Our results also yield some new unramified cohomology calculations.
We prove a topological invariance statement for the Morel-Voevodsky motivic homotopy category, up to inverting exponential characteristics of residue fields. This implies in particular that SH[1/p] of characteristic p>0 schemes is invariant under passing to perfections. Among other applications we prove Grothendieck-Verdier duality in this context.
We construct a period regulator for motivic cohomology of an algebraic scheme over a subfield of the complex numbers. For the field of algebraic numbers we formulate a period conjecture for motivic cohomology by saying that this period regulator is surjective. Showing that a suitable Betti--de Rham realization of 1-motives is fully faithful we can verify this period conjecture in several cases. The divisibility properties of motivic cohomology imply that our conjecture is a neat generalization of the classical Grothendieck period conjecture for algebraic cycles on smooth and proper schemes. These divisibility properties are treated in an appendix by B. Kahn (extending previous work of Bloch and Colliot-Thel`ene--Raskind).
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