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My impression is that Adams operations are "well known" to act coherently on all levels of the weight spectral sequence for K-theory (of smooth varieties); probably, this fact was established by Gill …

Theorem 1.2 of the paper "The motivic Hopf map solves the homotopy limit problem for K-theory" (see https://elibm.org/article/10011880) says that (under certain assumptions on the base field) the homo …

I will sketch a proof. It suffices to prove that there are only zero morphisms from $M_{\text{gm}}(X)(1)$ into $\mathbb{Z}[q]$ for any smooth $X$ and $q\in \mathbb{Z}$. The latter statement easily f …

The motivic $S^1$-stable homotopy category $SH^{S^1}(k)$ (where $k$ is a field that is often assumed to be perfect; yet one can probably take quite general base schemes here) is "intermediate" betwee …

Morel has defined the motivic Hopf map $\eta$ (in the motivic stable homotopy category $SH(k)$). I suspect that the following facts are valid for it and its topological "cousin"; please correct me if …

I have the following questions on monoidal model structure(s) for the motivic stable homotopy category $SH(k)$ (where $k$ is a field); certainly, I am also interested in general statements concerning …

For a perfect base field $k$ there exists the following collection of "motivic homotopy" categories related to it: (a) the homotopy category of simplicial presheaves (from smooth $k$-varieties); here …

People use several distinct models for the motivic stable homotopy category (so, there are some choices for the underlying category and a collection of available model structures). I would like to ask …

For a perfect field $k$ there is a collection of stable motivic homotopy categories equipped with the corresponding Morel's (homotopy) $t$-structures: $SH^{S^1}(k)$, $SH(k)$, $DA(k)$, and also modules …

I'm not sure that it is possible to compress the big picture into one answer; yet I will try to give a hint. Firstly, one can hardly hope to have a "reasonable" motivic $t$-structure for motives with …

It was proved by Riou in Appendix B of http://arxiv.org/abs/1311.2159 that the spectra of smooth projective varieties do (compactly) generate $SH(k)_{\mathbb{Z}_{(l)}}$ for any $l$ distinct from $\ope …

This is actually an answer to a question you did not ask; sorry. The corresponding diagram $T\wedge T\to T\wedge T$ neither commutes nor anticommutes. An observation of Morel (that was studied in det …

Let $S$ be a (motivic symmetric) ring spectrum (more generally, one can possibly consider a commutative ring object in any symmetric stable model category); let $R$ be an associative commutative unit …

I know of two related applications. For any cohomology theory that factorizes through $H_{A^1}(k)$ one has a certain Gysin long exact sequence $\dots \to H^i(X-Z)\to H^i(X)\to H^i(N_{Z/X}/N_{Z/X}\se …

For a field $k$ there exists the motivic stable homotopy $SH(k)$; it is compactly generated. My question: does there exist a 'reasonable' functor $p$ from $SH(k)$ to a certain triangulated category $S …