Slice Filtration Research Articles

Intermediate Jacobians and the slice filtration

For every $n$-dimensional smooth projective variety $X$ over ℂ, the motive $M(X)$ is expected to admit a Chow-Künneth decomposition $M_0(X)\oplus \cdots \oplus M_{2n}(X)$. Inspired by the slice filtration of $M(X)$ we propose the definitions of $M_2(X)$ and $M_{2n-2}(X)$. In our construction we use intermediate Jacobians.

Voevodsky's slice conjectures via Hilbert schemes

Using recent development in motivic infinite loop space theory, we offer short and conceptual reproofs of some conjectures of Voevodsky's on the slice filtration using the birational geometry of Hilbert schemes. The original proofs were due to Marc Levine using very different methods, namely, the homotopy coniveau tower.

Norms in motivic homotopy theory

If $f : S' \to S$ is a finite locally free morphism of schemes, we construct a symmetric monoidal "norm" functor $f_\otimes : \mathcal{H}_{\bullet}(S')\to \mathcal{H}_{\bullet}(S)$, where $\mathcal{H}_\bullet(S)$ is the pointed unstable motivic homotopy category over $S$. If $f$ is finite étale, we show that it stabilizes to a functor $f_\otimes : \mathcal{S}\mathcal{H}(S') \to \mathcal{S}\mathcal{H}(S)$, where $\mathcal{S}\mathcal{H}(S)$ is the $\mathbb{P}^1$-stable motivic homotopy category over $S$. Using these norm functors, we define the notion of a normed motivic spectrum, which is an enhancement of a motivic $E_\infty$-ring spectrum. The main content of this text is a detailed study of the norm functors and of normed motivic spectra, and the construction of examples. In particular: we investigate the interaction of norms with Grothendieck's Galois theory, with Betti realization, and with Voevodsky's slice filtration; we prove that the norm functors categorify Rost's multiplicative transfers on Grothendieck-Witt rings; and we construct normed spectrum structures on the motivic cohomology spectrum $H\mathbb{Z}$, the homotopy $K$-theory spectrum $KGL$, and the algebraic cobordism spectrum $MGL$. The normed spectrum structure on $H\mathbb{Z}$ is a common refinement of Fulton and MacPherson's mutliplicative transfers on Chow groups and of Voevodsky's power operations in motivic cohomology.

On Infinite Effectivity of Motivic Spectra and the Vanishing of their Motives

This paper is dedicated to the study of the kernel of the motivization functor $M_{k}^c:SH^c(k)\to DM^c(k)$ (i.e., we try to describe those compact objects of $SH(k)$ whose associated motives vanish. Moreover, we study the question when the $m$-connectivity of $M^c_{k}(E)$ ensures the $m$-connectivity of $E$ itself (with respect to the corresponding homotopy t-structures). We prove that the kernel of $M_{k}^c$ vanishes and the corresponding connectivity detection statement is also valid if and only if $k$ is a non-orderable field; this is an easy consequence of the corresponding results of T. Bachmann (who considered the case where the $2$-adic cohomological dimension of $k$ is finite). We also sketch a deduction of these statements from the slice-convergence results of M. Levine. Moreover, for a general $k$ we prove that this kernel does not contain any $2$-torsion; the author also suspects that all its elements are odd torsion. Besides we prove that the kernel in question consists exactly of infinitely effective (in the sense of Voevodsky's slice filtration) objects of $SH^c(k)$ (assuming that the exponential characteristic of $k$ is inverted in the coefficient ring). These result allow (following another idea of Bachmann) to carry over his results on the tensor invertibility of certain motives of affine quadrics to the corresponding motivic spectra whenever $k$ is non-orderable. We also generalize a theorem of A. Asok.

On equivariant and motivic slices

Let $k$ be a field with a real embedding. We compare the motivic slice filtration of a motivic spectrum over $Spec(k)$ with the $C_2$-equivariant slice filtration of its equivariant Betti realization, giving conditions under which realization induces an equivalence between the associated slice towers. In particular, we show that, up to reindexing, the towers agree for all spectra obtained from localized quotients of $MGL$ and $MR$, and for motivic Landweber exact spectra and their realizations. As a consequence, we deduce that equivariant spectra obtained from localized quotients of $MR$ are even in the sense of Hill--Meier, and give a computation of the slice spectral sequence converging to $\pi_{*,*}BP\langle n \rangle/2$ for $1 \le n \le \infty$.

The slice spectral sequence for singular schemes and applications

We examine the slice spectral sequence for the cohomology of singular schemes with respect to various motivic T-spectra, especially the motivic cobordism spectrum. When the base field k admits resolution of singularities and X is a scheme of finite type over k, we show that Voevodsky's slice filtration leads to a spectral sequence for MGL(X) whose terms are the motivic cohomology groups of X defined using the cdh-hypercohomology. As a consequence, we establish an isomorphism between certain geometric parts of the motivic cobordism and motivic cohomology of X. A similar spectral sequence for the connective K-theory leads to a cycle class map from the motivic cohomology to the homotopy invariant K-theory of X. We show that this cycle class map is injective for projective schemes. We also deduce applications to the torsion in the motivic cohomology of singular schemes.

The Motivic Hopf Map Solves the Homotopy Limit Problem for $K$-Theory

We solve affirmatively the homotopy limit problem for (effective) K -theory over fields of finite virtual cohomological dimension. Our solution uses the motivic slice filtration and the first motivic Hopf map.

A new formulation of the equivariant slice filtration with applications to $C_p$-slices

A new formulation of the equivariant slice filtration with applications to $C_p$-slices

The slices of $S^n \wedge H \underline{\mathbb{Z}}$ for cyclic $p$-groups

The slice filtration is a filtration of equivariant spectra. While the tower is analogous to the Postnikov tower in the nonequivariant setting, complete slice towers are known for relatively few $G$-spectra. In this paper, we determine the slice tower for all $G$-spectra of the form $S^n \wedge H\underline{\mathbb{Z}}$ where $n\geq 0$ and $G$ is a cyclic $p$-group for $p$ an odd prime.

The slice spectral sequence for certain $$RO({C_{p^{n}}} )$$ R O ( C p n ) -graded suspensions of $$H{\underline{\mathbf{Z}}}$$ H Z ̲

We study the slice filtration and associated spectral sequence for a family of \(RO(C_{p^{n}})\)-graded suspensions of the Eilenberg–MacLane spectrum for the constant Mackey functor \(\underline{\mathbb Z}\). Since \(H\underline{\mathbb Z}\) is the zero slice of the sphere spectrum, this begins an analysis of how one can describe the slices of a suspension in terms of the original slices.

The unstable slice filtration

The main goal of this paper is to construct an analogue of Vo- evodsky's slice filtration in the motivic unstable homotopy category. The construction is done via birational invariants, this is motivated by the exis- tence of an equivalence of categories between the orthogonal components for Voevodsky's slice filtration and the birational motivic stable homotopy cat- egories constructed in (9). Another advantage of this approach is that the slices appear naturally as homotopy fibres (and not as in the stable setting, where they are defined as homotopy cofibres) which behave much better in the unstable setting.

On the slice spectral sequence

We introduce a variant of the slice spectral sequence which uses only regular slice cells, and state the precise relationship between the two spectral sequences. We analyze how the slice filtration of an equivariant spectrum that is concentrated over a normal subgroup is related to the slice filtration of its geometric fixed points, and use this to prove a conjecture of Hill on the slice filtration of an Eilenberg MacLane spectrum. We also show how the (co)connectivity of a spectrum results in the (co)connectivity of its slice tower, demonstrating the "efficiency" of the slice spectral sequence.

On the Functoriality of the Slice Filtration

AbstractLet k be a field with resolution of singularities, and X a separated k-scheme of finite type with structure map g. We show that the slice filtration in the motivic stable homotopy category commutes with pullback along g. Restricting the field further to the case of characteristic zero, we are able to compute the slices of Weibel's homotopy invariant K-theory [24] extending the result of Levine [10], and also the zero slice of the sphere spectrum extending the result of Levine [10] and Voevodsky [23]. We also show that the zero slice of the sphere spectrum is a strict cofibrant ring spectrum HZXsf which is stable under pullback and that all the slices have a canonical structure of strict modules over HZXsf. If we consider rational coefficients and assume that X is geometrically unibranch then relying on the work of Cisinski and Déglise [4], we deduce that the zero slice of the sphere spectrum is given by Voevodsky's rational motivic cohomology spectrum HZX ⊗ ℚ and that the slices have transfers. This proves several conjectures of Voevodsky [22, conjectures 1, 7, 10, 11] in characteristic zero.

Birational motivic homotopy theories and the slice filtration

We show that there is an equivalence of categories between the orthogonal components for the slice filtration and the birational motivic stable homotopy categories which are constructed in this paper. Relying on this equivalence, we are able to describe the slices for projective spaces (including \ P ^{\infty} ), Thom spaces and blow ups.

Motivic slices and coloured operads

Colored operads were introduced in the 1970's for the purpose of studying homotopy invariant algebraic structures on topological spaces. In this paper we introduce colored operads in motivic stable homotopy theory. Our main motivation is to uncover hitherto unknown highly structured properties of the slice filtration. The latter decomposes every motivic spectrum into its slices, which are motives, and one may ask to what extend the slice filtration preserves highly structured objects such as algebras and modules. We use colored operads to give a precise solution to this problem. Our approach makes use of axiomatic setups which specialize to classical and motivic stable homotopy theory. Accessible t-structures are central to the development of the general theory. Concise introductions to colored operads and Bousfield (co)localizations are given in separate appendices.

The equivariant slice filtration: a primer

We present an introduction to the equivariant slice filtration. After reviewing the definitions and basic properties, we determine the slice-connectivity of various families of naturally arising spectra. This leads to an analysis of pullbacks of slices defined on quotient groups, producing new collections of slices. Building on this, we determine the slice tower for the EilenbergMac Lane spectrum associated to a Mackey functor for a cyclic p-group. We then relate the Postnikov tower to the slice tower for various spectra. Finally, we pose a few conjectures about the nature of slices and pullbacks.

The Slice Filtration and Grothendieck-Witt Groups

Let k be a perfect field of characteristic different from two. We show that the filtration on the Grothendieck-Witt group GW(k) induced by the slice filtration for the sphere spectrum in the motivic stable homotopy category is the I-adic filtration, where I is the augmentation ideal in GW(k).

On the orientability of the slice filtration

Let $X$ be a Noetherian separated scheme of finite Krull dimension. We show that the layers of the slice filtration in the motivic stable homotopy category $\mathcal{SH}$ are strict modules over Voevodsky’s algebraic cobordism spectrum. We also show that the zero slice of any commutative ring spectrum in $\mathcal{SH}$ is an oriented ring spectrum in the sense of Morel, and that its associated formal group law is additive. As a consequence, we deduce that with rational coefficients the slices are in fact motives in the sense of Cisinski-Deglise and have transfers if the base scheme is excellent. This proves a conjecture of Voevodsky.

Motives of Azumaya algebras

AbstractWe study the slice filtration for theK-theory of a sheaf of Azumaya algebrasA, and for the motive of a Severi-Brauer variety, the latter in the case of a central simple algebra of prime degree over a field. Using the Beilinson–Lichtenbaum conjecture, we apply our results to show the vanishing ofSK2(A) for a central simple algebraAof square-free index (prime to the characteristic). This proves a conjecture of Merkurjev.

Mixed motives and the slice filtration

We construct several Quillen model structures in Jardine's category Spt of motivic symmetric T-spectra [J.F. Jardine, Motivic symmetric spectra, Doc. Math. 5 (2000) 445–553], such that their associated homotopy categories are naturally isomorphic to Voevodsky's slice filtration [V. Voevodsky, Open problems in the motivic stable homotopy theory. I, in: Motives, Polylogarithms and Hodge Theory, Part I, Int. Press Lect. Ser., Irvine, CA, 1998]. We prove a conjecture of Voevodsky [V. Voevodsky, Open problems in the motivic stable homotopy theory. I, in: Motives, Polylogarithms and Hodge Theory, Part I, Int. Press Lect. Ser., Irvine, CA, 1998], which says that over a perfect field all the slices s q have a canonical structure of modules in Spt over the motivic Eilenberg–MacLane spectrum H Z . Restricting the field even further to the case of characteristic zero, we get that the slices s q may be interpreted as big motives in the sense of Voevodsky. We also show that the smash product in Spt induces pairings in the motivic Atiyah–Hirzebruch spectral sequence. To cite this article: P. Pelaez, C. R. Acad. Sci. Paris, Ser. I 347 (2009).