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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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