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!



Localization, Slice Tower, and Motivic Spectra
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.


