scholarly journals NON-COMMUTATIVE LOCALIZATIONS OF ADDITIVE CATEGORIES AND WEIGHT STRUCTURES

2016 ◽  
Vol 17 (4) ◽  
pp. 785-821 ◽  
Author(s):  
Mikhail V. Bondarko ◽  
Vladimir A. Sosnilo
Keyword(s):  
Triangulated Category ◽  
Perfect Field ◽  
Efficient Tool ◽  
Direct Sums ◽  
Base Scheme ◽  
General Base ◽  
Additive Category ◽  
Weight Structure ◽  
W Prime ◽  
Weight Structures

In this paper we demonstrate thatnon-commutative localizationsof arbitrary additive categories (generalizing those defined by Cohn in the setting of rings) are closely (and naturally) related to weight structures. Localizing an arbitrary triangulated category$\text{}\underline{C}$by a set$S$of morphisms in the heart$\text{}\underline{Hw}$of a weight structure$w$on it one obtains a triangulated category endowed with a weight structure$w^{\prime }$. The heart of$w^{\prime }$is a certain version of the Karoubi envelope of the non-commutative localization$\text{}\underline{Hw}[S^{-1}]_{\mathit{add}}$(of$\text{}\underline{Hw}$by$S$). The functor$\text{}\underline{Hw}\rightarrow \text{}\underline{Hw}[S^{-1}]_{\mathit{add}}$is the natural categorical version of Cohn’s localization of a ring, i.e., it is universal among additive functors that make all elements of$S$invertible. For any additive category$\text{}\underline{A}$, taking$\text{}\underline{C}=K^{b}(\text{}\underline{A})$we obtain a very efficient tool for computing$\text{}\underline{A}[S^{-1}]_{\mathit{add}}$; using it, we generalize the calculations of Gerasimov and Malcolmson (made for rings only). We also prove that$\text{}\underline{A}[S^{-1}]_{\mathit{add}}$coincides with the ‘abstract’ localization$\text{}\underline{A}[S^{-1}]$(as constructed by Gabriel and Zisman) if$S$contains all identity morphisms of$\text{}\underline{A}$and is closed with respect to direct sums. We apply our results to certain categories of birational motives$DM_{gm}^{o}(U)$(generalizing those defined by Kahn and Sujatha). We define$DM_{gm}^{o}(U)$for an arbitrary$U$as a certain localization of$K^{b}(Cor(U))$and obtain a weight structure for it. When$U$is the spectrum of a perfect field, the weight structure obtained this way is compatible with the corresponding Chow and Gersten weight structures defined by the first author in previous papers. For a general$U$the result is completely new. The existence of the correspondingadjacent$t$-structure is also a new result over a general base scheme; its heart is a certain category of birational sheaves with transfers over$U$.

scholarly journals ℤ[1/p]-motivic resolution of singularities

2011 ◽  
Vol 147 (5) ◽  
pp. 1434-1446 ◽  
Author(s):  
M. V. Bondarko
Keyword(s):  
Preceding Paper ◽  
Resolution Of Singularities ◽  
Cohomology Theory ◽  
Spectral Sequences ◽  
Perfect Field ◽  
Chow Motives ◽  
Unramified Cohomology ◽  
Weight Structure ◽  
Weight Structures

AbstractThe main goal of this paper is to deduce (from a recent resolution of singularities result of Gabber) the following fact: (effective) Chow motives with ℤ[1/p]-coefficients over a perfect field k of characteristic p generate the category DMeffgm[1/p] (of effective geometric Voevodsky’s motives with ℤ[1/p]-coefficients). It follows that DMeffgm[1/p] can be endowed with a Chow weight structure wChow whose heart is Choweff[1/p] (weight structures were introduced in a preceding paper, where the existence of wChow for DMeffgmℚ was also proved). As shown in previous papers, this statement immediately yields the existence of a conservative weight complex functor DMeffgm[1/p]→Kb (Choweff [1/p]) (which induces an isomorphism on K0-groups), as well as the existence of canonical and functorial (Chow)-weight spectral sequences and weight filtrations for any cohomology theory on DMeffgm[1/p] . We also mention a certain Chow t-structure for DMeff−[1/p] and relate it with unramified cohomology.

scholarly journals Chow motives without projectivity

2009 ◽  
Vol 145 (5) ◽  
pp. 1196-1226 ◽  
Author(s):  
Jörg Wildeshaus
Keyword(s):  
Shimura Varieties ◽  
Spectral Sequences ◽  
Perfect Field ◽  
Basic Properties ◽  
Structure Building ◽  
Princeton University ◽  
Chow Motives ◽  
Weight Structure ◽  
Compositio Math ◽  
Weight Structures

AbstractIn a recent paper, Bondarko [Weight structures vs. t-structures; weight filtrations, spectral sequences, and complexes (for motives and in general), Preprint (2007), 0704.4003] defined the notion of weight structure, and proved that the category DMgm(k) of geometrical motives over a perfect field k, as defined and studied by Voevodsky, Suslin and Friedlander [Cycles, transfers, and motivic homology theories, Annals of Mathematics Studies, vol. 143 (Princeton University Press, Princeton, NJ, 2000)], is canonically equipped with such a structure. Building on this result, and under a condition on the weights avoided by the boundary motive [J. Wildeshaus, The boundary motive: definition and basic properties, Compositio Math. 142 (2006), 631–656], we describe a method to construct intrinsically in DMgm(k) a motivic version of interior cohomology of smooth, but possibly non-projective schemes. In a sequel to this work [J. Wildeshaus, On the interior motive of certain Shimura varieties: the case of Hilbert–Blumenthal varieties, Preprint (2009), 0906.4239], this method will be applied to Shimura varieties.

Cohomologically triangulated categories II

2008 ◽  
Vol 3 (1) ◽  
pp. 1-52 ◽  
Author(s):  
H.-J. Baues ◽  
F. Muro
Keyword(s):  
Cohomology Class ◽  
Triangulated Category ◽  
Triangulated Categories ◽  
Additive Category

AbstractA cohomologically triangulated category is an additive categoryAtogether with a translation functortand a cohomology class Δ ∈H3(A,t) such that any good translation track category representing Δ is a triangulated track category. In this paper we give purely cohomological conditions on Δ which imply that (A,t,Δ) is a cohomologically triangulated category, avoiding the use of track categories. This yields a purely cohomological characterization of triangulated cohomology classes.

scholarly journals Weight structures vs.t-structures; weight filtrations, spectral sequences, and complexes (for motives and in general)

2010 ◽  
Vol 6 (3) ◽  
pp. 387-504 ◽  
Author(s):  
M.V. Bondarko
Keyword(s):  
Preceding Paper ◽  
Spectral Sequences ◽  
Motivic Cohomology ◽  
Adjacent Structures ◽  
Unramified Cohomology ◽  
Homological Functor ◽  
Stable Homotopy Category ◽  
Weight Structure ◽  
The One ◽  
Weight Structures

AbstractIn this paper we introduce a new notion ofweight structure (w)for a triangulated categoryC; this notion is an important natural counterpart of the notion oft-structure. It allows extending several results of the preceding paper [Bon09] to a large class of triangulated categories and functors.Theheartofwis an additive categoryHw⊂C. We prove that a weight structure yields Postnikov towers for anyX∈ObjC(whose 'factors'Xi∈ObjHw). For any (co)homological functorH:C→A(Ais abelian) such a tower yields aweight spectral sequenceT : H(Xi[j]) ⇒H(X[i + j]); Tis canonical and functorial inXstarting fromE2.Tspecializes to the usual (Deligne) weight spectral sequences for 'classical' realizations of Voevodsky's motivesDMeffgm(if we considerw = wChowwithHw=Choweff) and to Atiyah-Hirzebruch spectral sequences in topology.We prove that there often exists an exact conservative weight complex functorC→K(Hw). This is a generalization of the functort:DMeffgm→Kb(Choweff) constructed in [Bon09] (which is an extension of the weight complex of Gillet and Soulé). We prove thatK0(C) ≅K0(Hw) under certain restrictions.We also introduce the concept of adjacent structures: at-structure isadjacenttowif their negative parts coincide. This is the case for the Postnikovt-structure for the stable homotopy categorySH(of topological spectra) and a certain weight structure for it that corresponds to the cellular filtration. We also define a new (Chow)t-structuretChowforDMeff_⊃DMeffgmwhich is adjacent to the Chow weight structure. We haveHtChow≅ AddFun(Choweffop,Ab);tChowis related to unramified cohomology. Functors left adjoint to those that aret-exact with respect to somet-structures are weight-exact with respect to the corresponding adjacent weight structures, and vice versa. Adjacent structures identify two spectral sequences converging toC(X,Y): the one that comes from weight truncations ofXwith the one coming fromt-truncations ofY(forX,Y∈ObjC). Moreover, the philosophy of adjacent structures allows expressing torsion motivic cohomology of certain motives in terms of the étale cohomology of their 'submotives'. This is an extension of the calculation of E2of coniveau spectral sequences (by Bloch and Ogus).

scholarly journals Tensor Structure on Smooth Motives

2011 ◽  
Vol 9 (1) ◽  
pp. 57-101 ◽  
Author(s):  
Anandam Banerjee
Keyword(s):  
Characteristic Zero ◽  
Full Subcategory ◽  
Resolution Of Singularities ◽  
Homotopy Category ◽  
Tensor Structure ◽  
Perfect Field ◽  
Base Scheme

AbstractRecently, Bondarko constructed a DG category of motives, whose homotopy category is equivalent to Voevodsky's category of effective geometric motives, assuming resolution of singularities. Soon after, Levine extended this idea to construct a DG category whose homotopy category is equivalent to the full subcategory of motives over a base-scheme S generated by the motives of smooth projective S-schemes, assuming that S is itself smooth over a perfect field. In both constructions, the tensor structure requires ℚ-coefficients. In this article, it is shown how to provide a tensor structure on the homotopy category mentioned above, when S is semi-local and essentially smooth over a field of characteristic zero. This is done by defining a pseudo-tensor structure on the DG category of motives constructed by Levine, which induces a tensor structure on its homotopy category.

scholarly journals Mixed motivic sheaves (and weights for them) exist if ‘ordinary’ mixed motives do

2015 ◽  
Vol 151 (5) ◽  
pp. 917-956 ◽  
Author(s):  
Mikhail V. Bondarko
Keyword(s):  
Finite Type ◽  
Main Tool ◽  
Decomposition Theorem ◽  
Weight Filtration ◽  
Closed Fields ◽  
Weight Structure ◽  
Residue Fields ◽  
Algebraically Closed Fields ◽  
Mixed Motives ◽  
Weight Structures

The goal of this paper is to prove that if certain ‘standard’ conjectures on motives over algebraically closed fields hold, then over any ‘reasonable’ scheme $S$ there exists a motivic$t$-structure for the category $\text{DM}_{c}(S)$ of relative Voevodsky’s motives (to be more precise, for the Beilinson motives described by Cisinski and Deglise). If $S$ is of finite type over a field, then the heart of this $t$-structure (the category of mixed motivic sheaves over $S$) is endowed with a weight filtration with semisimple factors. We also prove a certain ‘motivic decomposition theorem’ (assuming the conjectures mentioned) and characterize semisimple motivic sheaves over $S$ in terms of those over its residue fields. Our main tool is the theory of weight structures. We actually prove somewhat more than the existence of a weight filtration for mixed motivic sheaves: we prove that the motivic $t$-structure is transversal to the Chow weight structure for $\text{DM}_{c}(S)$ (that was introduced previously by Hébert and the author). We also deduce several properties of mixed motivic sheaves from this fact. Our reasoning relies on the degeneration of Chow weight spectral sequences for ‘perverse étale homology’ (which we prove unconditionally); this statement also yields the existence of the Chow weight filtration for such (co)homology that is strictly restricted by (‘motivic’) morphisms.

The triangulated category of K-motives

2014 ◽  
Vol 14 (1) ◽  
pp. 103-137 ◽  
Author(s):  
Grigory Garkusha ◽  
Ivan Panin
Keyword(s):  
Triangulated Category ◽  
Perfect Field ◽  
Image Position

AbstractFor any perfect field k a triangulated category of K-motives is constructed in the style of Voevodsky's construction of the category . To each smooth k-variety X the K-motive is associated in the category andwhere pt = Spec(k) and K(X) is Quillen's K-theory of X.

scholarly journals On the Category of Weakly U-Complexes

2020 ◽  
Vol 13 (2) ◽  
pp. 323-345
Author(s):  
Gustina Elfiyanti ◽  
Intan Muchtadi-Alamsyah ◽  
Fajar Yuliawan ◽  
Dellavitha Nasution
Keyword(s):  
Homological Algebra ◽  
Abelian Category ◽  
Universal Property ◽  
Homotopy Category ◽  
Triangulated Category ◽  
Additive Category

Motivated by a study of Davvaz and Shabbani which introduced the concept of U-complexes and proposed a generalization on some results in homological algebra, we study thecategory of U-complexes and the homotopy category of U-complexes. In [8] we said that the category of U-complexes is an abelian category. Here, we show that the object that we claimed to be the kernel of a morphism of U-omplexes does not satisfy the universal property of the kernel, hence wecan not conclude that the category of U-complexes is an abelian category. The homotopy category of U-complexes is an additive category. In this paper, we propose a weakly chain U-complex by changing the second condition of the chain U-complex. We prove that the homotopy category ofweakly U-complexes is a triangulated category.

scholarly journals Étale motives

2015 ◽  
Vol 152 (3) ◽  
pp. 556-666 ◽  
Author(s):  
Denis-Charles Cisinski ◽  
Frédéric Déglise
Keyword(s):  
Rigidity Theorem ◽  
Triangulated Categories ◽  
Noetherian Scheme ◽  
Base Scheme ◽  
General Base ◽  
Rational Part

We define a theory of étale motives over a noetherian scheme. This provides a system of categories of complexes of motivic sheaves with integral coefficients which is closed under the six operations of Grothendieck. The rational part of these categories coincides with the triangulated categories of Beilinson motives (and is thus strongly related to algebraic $K$-theory). We extend the rigidity theorem of Suslin and Voevodsky over a general base scheme. This can be reformulated by saying that torsion étale motives essentially coincide with the usual complexes of torsion étale sheaves (at least if we restrict ourselves to torsion prime to the residue characteristics). As a consequence, we obtain the expected results of absolute purity, of finiteness, and of Grothendieck duality for étale motives with integral coefficients, by putting together their counterparts for Beilinson motives and for torsion étale sheaves. Following Thomason’s insights, this also provides a conceptual and convenient construction of the $\ell$-adic realization of motives, as the homotopy $\ell$-completion functor.

Mixed motives over k[t]/(tm+1)

2012 ◽  
Vol 11 (3) ◽  
pp. 611-657 ◽  
Author(s):  
Amalendu Krishna ◽  
Jinhyun Park
Keyword(s):  
Polynomial Ring ◽  
Chow Groups ◽  
Triangulated Category ◽  
Perfect Field ◽  
Tensor Triangulated Category ◽  
Mixed Motives ◽  
Higher Chow Groups

AbstractFor a perfect field k, we use the techniques of Bondal-Kapranov and Hanamura to construct a tensor triangulated category of mixed motives over the truncated polynomial ring k[t]/(tm+1). The extension groups in this category are given by Bloch's higher Chow groups and the additive higher Chow groups. The main new ingredient is the moving lemma for additive higher Chow groups by the authors and its refinements.

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