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On the functoriality of the slice filtration. (English) Zbl 1319.14029
This interesting paper studies the compatibility between the slice filtration and the structure maps of varieties over a field $$k$$ satisfying resolution of singularities.
Let $$X$$ be a Noetherian separated scheme of finite Krull dimension, and let $$\mathcal{M}_X$$ be the category of pointed simplicial presheaves in the smooth Nisnevich site $$Sm_X$$ over $$X$$ equipped with the motivic Quillen model structure. Let $$T_X=S^1\wedge \mathbb{G}_m$$. Let $$Spt(\mathcal{ M})$$ be Jardine’s category of symmetric $$T_X$$-spectra on $$\mathcal{M}_X$$ equipped with the motivic model structure, and let $$\mathcal{SH}_X$$ be its homotopy category. Let $$\mathcal{SH}^\bot_X(q)$$ be the triangulated full subcategory of $$\mathcal{SH}_X$$ consisting of the symmetric $$T_X$$-spectra which are $$q$$-orthogonal with respect to the slice filtration in $$\mathcal{SH}_X$$. Let $$s_q$$ be the $$q$$-th slice in the slice filtration, and let $$g:X\to Y$$ be a map of Noetherian, separated and of finite Krull dimensional schemes. The main result of the paper shows that the natural transformations $$\beta_q:\mathbf{L}g^*\circ s_q\to s_q\circ \mathbf{L}g^*$$ is an isomorphism, provided that $$\mathbf{L}g^*(s_qE)\in \mathcal{SH}^\bot_X(q+1)$$ for any symmetric $$T_Y$$-spectrum $$E\in \mathcal{SH}_Y$$ and $$q\in \mathbb{Z}$$. Here $$\mathbf{L}g^*$$ is the triangulated functor induced by $$g$$. As a corollary, it shows that when the field $$k$$ satisfies resolution of singularities, the structure map $$g$$ of a variety $$X$$ is compatible with the pullback along $$g$$.
The paper then provides some applications in computations of the slice filtration. For example, it computes the slices of Weibel’s homotopy invariant $$K$$-theory, and the zero slice of the sphere spectrum. It also shows that the zero slice of the sphere spectrum is a strict cofibrant ring spectrum $$\mathbf{HZ}^{sf}_X$$.

MSC:
 14F42 Motivic cohomology; motivic homotopy theory
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References:
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