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Suppose $k$ is an algebraically closed field of characteristic $p>0$. There is an $\infty$-category of motivic spectra over $k$, denoted $\mathcal{S}pt(k)$. As in algebraic topology, there are motivic Eilenberg-Maclane spectra $\mathbf{EM}(A)$ for each abelian group $A$. I have a few related questions:

1) Do $\mathbf{EM}(\mathbb{Z}[1/p])\wedge(-)$ and $\mathbf{EM}(\mathbb{Z}_{\ell})\wedge(-)$, for $\ell\neq p$ preserve cofiber sequences?

2) Do these operations commute with the formation of slices?

3) What is a good theory of inverting $p$ or completing at $\ell\neq p$ in $\mathcal{S}pt(k)$ or its homotopy category $\mathcal{SH}(k)$ that behaves well with respect to the formation of slices and cofiber sequences?

Thank you!

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    $\begingroup$ (2) is false because motivic eilenberg maclane spectra have only one slice whereas the sphere has many. For (1), the derived smash product is exact in both variables, so preserves distinguished triangles (on the homotopy category). I'm not sure if this answers your question. $\endgroup$
    – Tom Bachmann
    Commented Nov 7, 2016 at 9:38

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The stable motivic category is a presentable symmetric monoidal ∞-category so smashing with any motivic spectrum preserves all homotopy colimits. On the other hand smashing with a spectrum need not to preserve the slice filtration exactly for the reasons that Tom Bachmann mentioned in the comments: the single slice categories are not ⊗-ideals (even if you smash two slices together they do not necessarily stay slices). For example smashing with S¹ (the simplicial sphere) shifts the slices by one.

You can certainly complete or localize at a prime in the stable motivic category (after all you do have Eilenberg-MacLane spectra) and this will play reasonably well with cofiber sequences, but I ignore if it is known what this will do to the slice tower.

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