On perverse homotopy tt-structures, coniveau spectral sequences, cycle modules, and relative Gersten weight structures

We study the category $DM(S)$ of Beilinson motives (as described by Cisinski and Deglise) over a more or less general base scheme $S$, and establish several nice properties for a version $t_{hom}(S)$ of the perverse homotopy $t$-structure (essentially defined by Ayoub) for it. $t_{hom}(S)$ is characterized in terms of certain stalks of an $S$-motif $H$ and its Tate twists at fields over $S$; it is closely related to certain coniveau spectral sequences for the cohomology of (the Borel-Moore motives of) arbitrary finite type $S$-schemes. We conjecture that the heart of $t_{hom}(S)$ is given by cycle modules over $S$ (as defined by Rost); for varieties over characteristic $0$ fields this conjecture was recently proved by Deglise. Our definition of $t_{hom}(S)$ is closely related to a new effectivity filtration for $DM(S)$ (and for the subcategory of Chow $S$-motives in it). We also sketch the construction of a certain Gersten weight structure for the category of $S$-comotives; this weight structure yields one more description of $t_{hom}(S)$ and its heart.Comment: Several corrections made. In particular, we introduce a certain dimension function \delta\ for our schemes; \delta(-) is a certain "regularization" of the Krull dimension function. This allows to formulate (and prove) our results for not necessarily Jacobson scheme

Intersecting the dimension filtration with the slice one for (relative) motivic categories

In this paper we prove that the intersections of the levels of the dimension filtration on Voevodsky's motivic complexes over a field $k$ with the levels of the slice one are "as small as possible", i.e., that $Obj d_{\le m}DM^{eff}_{-,R} \cap Obj DM^{eff}_{-,R} (i)=Obj d_{\le m-i} DM^{eff}_{-,R} (i)$ (for $m,i\ge 0$ and $R$ being any coefficient ring in which the exponential characteristic of $k$ invertible). This statement is applied to prove that a conjecture of J. Ayoub is equivalent to a certain orthogonality assumption. We also establish a vast generalization of our intersection result to relative motivic categories (that are required to fulfil a certain list of "axioms"). In the process we prove several new properties of relative motives and of the so-called Chow weight structures for them.Comment: A few minor corrections made. A shorter version of this paper (without section 3) will probably appear in "Homology, Homotopy and Applications" under the name "Intersecting the dimension and slice filtrations for motives

On infinite effectivity of motivic spectra and the vanishing of their motives

This paper is dedicated to the study of the kernel of the "compact 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.Comment: A few typos were corrected; exposition improved; Theorem 3.1.1 was extended. The paper is submitted to Documenta Mathematic

On weight complexes, pure functors, and detecting weights

This paper is dedicated to the study of weight complexes (defined on triangulated categories endowed with weight structures) and their applications. We introduce pure (co)homological functors that "ignore all non-zero weights"; these have a nice description in terms of weight complexes. For the weight structure $w^G$ generated by the orbit category in the $G$-equivariant stable homotopy category $SH(G)$ the corresponding pure cohomological functors into abelian groups are the Bredon cohomology associated to Mackey functors ones; pure functors related to motivic weight structures are also quite useful. Our results also give some (more) new weight structures. Moreover, we prove that certain exact functors are conservative and "detect weights".Comment: Several minor corrections made. Appendices (closely related to weight complexes) were extende

A 'fat hyperplane section' weak Lefschetz (in arbitrary characteristic), and Barth-type theorems

We prove a certain 'fat hyperplane section' Weak Lefschetz-type theorem for etale cohomology of non-projective varieties, similar to a result of Goresky and MacPherson (over complex numbers). This statement easily yields certain (vast) generalizations of the 'ordinary' Weak Lefschetz and Barth's theorems in arbitrary characteristic (that do not require any stratified Morse theory for their proof).Comment: Several minor corrections mad

Gersten weight structures for motivic homotopy categories; direct summands of cohomology of function fields and coniveau spectral sequences

For any cohomology theory $H$ that can be factorized through (the Morel-Voevodsky's triangulated motivic homotopy category) $SH^{S^1}(k)$ (or through $SH(k)$) we establish the $SH^{S^1}(k)$-functorialty (resp. $SH(k)$-one) of coniveau spectral sequences for $H$. We also prove: for any affine essentially smooth semi-local $S$ the Cousin complex for $H^*(S)$ splits; if $H$ also factorizes through $SH^+(k)$ or $SH^{MGL}(k)$, then this is also true for any primitive $S$. Moreover, the cohomology of such an $S$ is a direct summand of the cohomology of any its open dense subscheme. This is a vast generalization of the results of a previous paper. In order to prove these results we consider certain categories of motivic pro-spectra, and introduce Gersten weight structures for them. Our results rely on several interesting statements on weight structures in cocompactly cogenerated triangulated categories and on the '$SH^+(k)$-acyclity' of primitive schemes. .Comment: Several minor corrections made; this includes the localization by a set of primes issue (see Remarks 2.2.7 and 4.4.2

Conservativity of realizations implies that numerical motives are Kimura-finite and motivic zeta functions are rational

We prove: if the (\'etale or de Rham) realization functor is conservative on the category $DM_{gm}$ of Voevodsky motives with rational coefficients then motivic zeta functions of arbitrary varieties are rational and numerical motives are Kimura-finite. The latter statement immediately implies that the category of numerical motives is (essentially) Tannakian. This observation becomes actual due to the recent announcement of J. Ayoub that the De Rham cohomology realization is conservative on $DM_{gm}(k)$ whenever $\operatorname{char} k=0$. We apply this statement to exterior powers of motives coming from generic hyperplane sections of smooth affine varieties.Comment: A collection of minor corrections mad

On morphisms killing weights and Hurewicz-type theorems

We study "canonical weight decompositions" slightly generalizing that defined by J. Wildeshaus. For an triangulated category $C$, any integer $n$, and a weight structure $w$ on $C$ a triangle $LM\to M\to RM\to LM[1]$, where $LM$ is of weights at most $m-1$ and $RM$ is of weights at least $n+1$ for some $m\le n$, is determined by $M$ if exists. This happens if and only if the weight complex $t(M)\in Obj K(Hw)$ ($Hw$ is the heart of $w$) is homotopy equivalent to a complex with zero terms in degrees $-n,\dots, -m$; hence this condition can be "detected" via pure functors. One can also take $m=-\infty$ or $n=+\infty$ to obtain that the weight complex functor is "conservative and detects weights up to objects of infinitely small and infinitely large weights"; this is a significant improvement over previously known bounded conservativity results. Applying this statement we "calculate intersections of purely generated subcategories" and prove that certain weight-exact functors are conservative up to weight-degenerate objects. The main tool is the new interesting notion of morphisms killing weights $m,\dots, n$ that we study in detail as well. We apply general results to equivariant stable homotopy categories and spherical weight structures for them (as introduced in the previous paper) and obtain a certain converse to the (equivariant) stable Hurewicz theorem. In particular, the singular homology of a spectrum $E$ vanishes in negative degrees if and only if $E$ is an extension of a connective spectrum by an acyclic one.Comment: Several minor corrections mad

On Chow weight structures for cdhcdh-motives with integral coefficients

The main goal of this paper is to define a certain Chow weight structure $w_{Chow}$ on the category $DM_c(S)$ of (constructible) $cdh$-motives over an equicharacteristic scheme $S$. In contrast to the previous papers of D. H\'ebert and the first author on weights for relative motives (with rational coefficients), we can achieve our goal for motives with integral coefficients (if $\operatorname{char}S=0$; if $\operatorname{char}S=p>0$ then we consider motives with $\mathbb{Z}[\frac{1}{p}]$-coefficients). We prove that the properties of the Chow weight structures that were previously established for $\mathbb{Q}$-linear motives can be carried over to this "integral" context (and we generalize some of them using certain new methods). In this paper we mostly study the version of $w_{Chow}$ defined via "gluing from strata"; this enables us to define Chow weight structures for a wide class of base schemes. As a consequence, we certainly obtain certain (Chow)-weight spectral sequences and filtrations for any (co)homology of motives.Comment: To appear in Algebra i Analiz (St. Petersburg Math Journal). arXiv admin note: substantial text overlap with arXiv:1007.454

On purely generated α\alpha-smashing weight structures and weight-exact localizations

This paper is dedicated to new methods of constructing weight structures and weight-exact localizations; our arguments generalize their bounded versions considered in previous papers of the authors. We start from a class of objects $P$ of triangulated category $C$ that satisfies a certain negativity condition (there are no $C$-extensions of positive degrees between elements of $P$; we actually need a somewhat stronger condition of this sort) to obtain a weight structure both "halves" of which are closed either with respect to $C$-coproducts of less than $\alpha$ objects (for $\alpha$ being a fixed regular cardinal) or with respect to all coproducts (provided that $C$ is closed with respect to coproducts of this sort). This construction gives all "reasonable" weight structures satisfying the latter condition. In particular, we obtain certain weight structures on spectra (in $SH$) consisting of less than $\alpha$ cells and on certain localizations of $SH$; these results are new.Comment: Several minor corrections mad