An alternative approach to the classical Morel-Voevodsky stable motivic homotopy theory $SH(k)$ is suggested. The triangulated category of framed bispectra $SH_{nis}^{fr}(k)$ and effective framed bispectra $SH_{nis}^{fr,eff}(k)$ are introduced in the paper. Both triangulated categories only use Nisnevich local equivalences and have nothing to do with any kind of motivic equivalences. It is shown that $SH_{nis}^{fr}(k)$ and $SH_{nis}^{fr,eff}(k)$ recover the classical Morel-Voevodsky triangulated categories of bispectra $SH(k)$ and effective bispectra $SH^{eff}(k)$ respectively.
We also recover $SH(k)$ and $SH^{eff}(k)$ as the triangulated category of framed motivic spectral functors $SH_{S^1}^{fr}[\mathcal Fr_0(k)]$ and the triangulated category of framed motives $\mathcal {SH}^{fr}(k)$ respectively constructed in the paper.
For any perfect field k a triangulated category of K-motives DK_(k) is constructed in the style of Voevodsky's construction of the category DM_(k). To each smooth k-variety X the K-motive is associated in the category DK_(k). Also, it is shown that K_n(X)=DK_(k)(M_K(X)[n],M_K(pt)), where K(X) is Quillen's K-theory of X.
Using the localization property, we construct a triangulated category of motives over quasi-projective T-schemes for any coefficient where T is a noetherian separated scheme, and we prove the Grothendieck six operations formalism. We also construct integral étale realization of motives.
We explain why every non-trivial exact tensor functor on the triangulated category of mixed motives over a field F has zero kernel, if one assumes "all" motivic conjectures. In other words, every non-zero motive generates the whole category up to the tensor triangulated structure. Under the same assumptions, we also give a complete classification of triangulated étale motives over F with integral coefficients, up to the tensor triangulated structure, in terms of the characteristic and the orderings of F.
We introduce the notion of log motivic triangulated categories, which is the theoretical framework for understanding the motivic aspect of cohomology theories for fs log schemes. Then we study the Grothendieck six operations formalism for log motivic triangulated categories.
We investigate certain categorical aspects of Voevodsky's triangulated categories of motives. For this, various recollements for Grothendieck categories of enriched functors and their derived categories are established. In order to extend these recollements further with respect to Serre's localization, the concept of the (strict) Voevodsky property for Serre localizing subcategories is introduced. This concept is inspired by the celebrated Voevodsky theorem on homotopy invariant presheaves with transfers. As an application, it is shown that Voevodsky's triangulated categories of motives fit in…
We construct and study a candidate for the standard motivic t-structure on the triangulated category of relative cohomological 1-motives with rational coefficients over a noetherian finite dimensional scheme S. This t-structure is defined as a generated t-structure, and we show it is non-degenerate. We relate its heart MM^1(S) with Deligne 1-motives over S; in particular, when S is regular, the category of Deligne 1-motives embeds in MM^1(S) fully faithfully. We also study the inclusion of DA^1(S) into the larger category DA^{coh}(S) of relative cohomological motives on S, and prove that its r…
Using the theory of framed correspondences developed by Voevodsky, we introduce and study framed motives of algebraic varieties. They are the major computational tool for constructing an explicit quasi-fibrant motivic replacement of the suspension $\mathbb P^1$-spectrum of any smooth scheme $X\in Sm/k$. Moreover, it is shown that the bispectrum $$(M_{fr}(X),M_{fr}(X)(1),M_{fr}(X)(2),\ldots),$$ each term of which is a twisted framed motive of $X$, has motivic homotopy type of the suspension bispectrum of $X$. Furthermore, an explicit computation of infinite $\mathbb P^1$-loop motivic spaces is …
We prove 2-out-of-3 property for rationality of motivic zeta function in distinguished triangles in Voevodsky's category DM. As an application, we show rationality of motivic zeta functions for all varieties whose motives are in the thick triangulated monoidal subcategory generated by motives of quasi-projective curves in DM. Joint with a result of P.O'Sullivan it also gives an example of a variety whose motive is not finite-dimensional while the motivic zeta function is rational.
We study certain 'weights' for triangulated categories endowed with $t$-structures. Our results axiomatize and describe in detail the relations between the Chow weight structure (introduced in a preceding paper), the (conjectural) motivic t-structure for Voevodsky's motives, and the conjectural weight filtration for them. This picture becomes non-conjectural when restricted to the derived categories of 1-motives (over a smooth base) and of Artin-Tate motives over number fields; other examples include Beilinson's derived category of graded polarizable mixed Hodge complexes and Saito's derived c…
In this work we develop a theory of motives for logarithmic schemes over fields in the sense of Fontaine, Illusie, and Kato. Our construction is based on the notion of finite log correspondences, the dividing Nisnevich topology on log schemes, and the basic idea of parameterizing homotopies by $\overline{\square}$, i.e. the projective line with respect to its compactifying logarithmic structure at infinity. We show that Hodge cohomology of log schemes is a $\overline{\square}$-invariant theory that is representable in the category of logarithmic motives. Our category is closely related to Voev…
This book discusses the construction of triangulated categories of mixed motives over a noetherian scheme of finite dimension, extending Voevodsky's definition of motives over a field. In particular, it is shown that motives with rational coefficients satisfy the formalism of the six operations of Grothendieck. This is achieved by studying descent properties of motives, as well as by comparing different presentations of these categories, following and extending insights and constructions of Deligne, Beilinson, Bloch, Thomason, Gabber, Levine, Morel, Voevodsky, Ayoub, Spitzweck, Röndigs, Østvær…
We first study the weight structure on the triangulated category of Artin-Tate motives over a perfect base field k, building on results of Bondarko's. We then study the t-structure on the triangulated category of Artin-Tate motives, when k is algebraic over the rationals, generalizing a result of Levine's. We finally study the interaction of the weight structure and the t-structure. When k is a number field, this will give a useful criterion identifying the weight structure via realizations.
This is a survey of author's results on weight structures and Voevodsky's motives. Weight structures are natural counterparts of t-structures (for triangulated categories) introduced by the author. They allow to construct weight complexes, weight filtrations, and weight spectral sequences for various cohomology theories. Partial cases of the latter are: 'classical' weight spectral sequences (for singular and etale cohomology), coniveau spectral sequences, and Atiyah-Hirzebruch spectral sequences. All of those are mentioned in the current paper. The details, proofs, and several more results cou…
The category of framed correspondences $Fr_*(k)$ was invented by Voevodsky in his notes in order to give another framework for SH(k) more amenable to explicit calculations. Based on that notes and on their JAMS paper Garkusha and the author introduced in a very recent paper a triangulated category of framed bispectra $\text{SH}_{nis}^{fr}(k)$. It is shown in the latter paper that $\text{SH}_{nis}^{fr}(k)$ recovers classical Morel-Voevodsky triangulated category of bispectra $\text{SH}(k)$. For any infinite perfect field $k$ a triangulated category of $\mathbb{F}\text{r}$-motives $\text{D}\math…
A kind of motivic stable homotopy theory of algebras is developed. Explicit fibrant replacements for the $S^1$-spectrum and $(S^1,\mathbb G)$-bispectrum of an algebra are constructed. As an application, unstable, Morita stable and stable universal bivariant theories are recovered. These are shown to be embedded by means of contravariant equivalences as full triangulated subcategories of compact generators of some compactly generated triangulated categories. Another application is the introduction and study of the symmetric monoidal compactly generated triangulated category of $K$-motives. It i…
Let k be a perfect field. In this paper we prove that biextensions of 1-motives define multilinear morphisms between 1-motives in Voevodsky's triangulated category of effective geometrical motives over k with rational coefficients.
We prove that the embedding of the derived category of 1-motives into the triangulated category of effective Voevodsky motives, as well as its left adjoint functor $LAlb$, commute with the Hodge realization.
Grothendieck first defined the notion of a "motif" as a way of finding a universal cohomology theory for algebraic varieties. Although this program has not been realized, Voevodsky has constructed a triangulated category of geometric motives over a perfect field, which has many of the properties expected of the derived category of the conjectural abelian category of motives. The construction of the triangulated category of motives has been extended by Cisinski-Déglise to a triangulated category of motives over a base-scheme $S$. Recently, Bondarko constructed a DG category of motives, whose ho…
We develop a general theory which enables the computation of the Picard group of a symmetric monoidal triangulated category, equipped with a weight structure, in terms of the Picard group of the associated heart. As an application, we compute the Picard group of several categories of motivic nature - mixed Artin motives, mixed Artin-Tate motives, motivic spectra, noncommutative mixed Artin motives, noncommutative mixed motives of central simple algebras, noncommutative mixed motives of separable algebras - as well as the Picard group of the derived categories of symmetric ring spectra.
This survey covers some of the recent developments on noncommutative motives and their applications. Among other topics, we compute the additive invariants of relative cellular spaces and orbifolds; prove Kontsevich's semi-simplicity conjecture; prove a far-reaching noncommutative generalization of the Weil conjectures; prove Grothendieck's standard conjectures of type C+ and D, Voevodsky's nilpotence conjecture, and Tate's conjecture, in several new cases; embed the (cohomological) Brauer group into secondary K-theory; construct a noncommutative motivic Gysin triangle; compute the localizing …
Given a perfect field of exponential characteristic $e$ and a functor $f:\mathcal A\to\mathcal B$ between symmetric monoidal strict $V$-categories of correspondences satisfying the cancellation property such that the induced morphisms of complexes of Nisnevich sheaves $$f_*:\mathbb Z_{\mathcal A}(q)[1/e]\to\mathbb Z_{\mathcal B}(q)[1/e],\quad q\geq 0,$$ are quasi-isomorphisms, it is proved that for every $k$-smooth algebraic variety $X$ the morphisms of twisted motives of $X$ with $\mathbb Z[1/e]$-coefficients $$M_{\mathcal A}(X)(q)\otimes\mathbb Z[1/e]\to M_{\mathcal B}(X)(q)\otimes\mathbb Z[…
We show that the reduced motive of a smooth affine quadric is invertible as an object of the triangulated category of motives DM(k, ZZ[1/e]) (where k is a perfect field of exponential characteristic e). We also establish a motivic version of the conjectures of Po Hu on products of certain affine Pfister quadrics. Both of these results are obtained by studying a novel conservative functor on (a subcategory of) DM(k, ZZ[1/e]), the construction of which constitutes the main part of this work.
A kind of motivic algebra of spectral categories and modules over them is developed to introduce K-motives of algebraic varieties. As an application, bivariant algebraic K-theory as well as bivariant motivic kohomology groups are defined and studied. We use Grayson's machinery to produce the Grayson motivic spectral sequence connecting bivariant K-theory to bivariant motivic kohomology. It is shown that the spectral sequence is naturally realized in the triangulated category of K-motives constructed in the paper. It is also shown that ordinary algebraic K-theory is represented by the K-motive …
We introduce a tower of localizing subcategories in Voevodsky's big (closed under infinite coproducts) triangulated category of motives. We show that the tower induces an interesting finite filtration on the motivic cohomology groups of smooth schemes over a perfect field. With rational coefficients, this finite filtration satisfies several of the properties of the still conjectural Bloch-Beilinson-Murre filtration.
We construct and study a triangulated category of motives with modulus $\mathbf{MDM}_{\mathrm{gm}}^{\mathrm{eff}}$ over a field $k$ that extends Voevodsky's category $\mathbf{DM}_{\mathrm{gm}}^{\mathrm{eff}}$ in such a way as to encompass non-homotopy invariant phenomena. In a similar way as $\mathbf{DM}_{\mathrm{gm}}^{\mathrm{eff}}$ is constructed out of smooth $k$-varieties, $\mathbf{MDM}_{\mathrm{gm}}^{\mathrm{eff}}$ is constructed out of proper modulus pairs, introduced in Part I of this work. To such a modulus pair we associate its motive in $\mathbf{MDM}_{\mathrm{gm}}^{\mathrm{eff}}$. In…
This thesis presents a way to apply this theorem of Gabber to a large portion of Voevodsky's work in order to lift the assumption that resolution of singularities holds. This gives unconditional versions of many of his and others' theorems provided we work Z[1/p] linearly, where p is the exponential characteristic of the base field. One example of the many applications we give is a partial answer to a 1980 conjecture of Weibel. Another is the removal of the hypothesis of resolution of singularities from a result of Suslin that compares Bloch's higher Chow groups and etale cohomology. Voevodsky…
We study Grothedieck groups of triangulated categories using weight structures, weight complexes, and the corresponding pure (co)homological functors. We prove some general statements on $K_0$ of weighted categories and apply it to Voevodsky motives endowed with so-called Chow weight structures. We obtain certain "motivic substitutes" for smooth compactifications of smooth varieties over arbitrary perfect fields; this enables us to make certain unramified cohomology and Euler characteristic calculations that are closely related to results of T. Ekedahl and B. Kahn.
Let X and Y be smooth complex projective varieties. Orlov conjectured that if X and Y are derived equivalent then their motives M(X) and M(Y) are isomorphic in Voevodsky's triangulated category of geometrical motives with rational coefficients. In this paper we prove the conjecture in the case X is a K3 surface admitting an elliptic fibration (a case that always occurs if the Picard rank of X is at least 5) with finite-dimensional Chow motive. We also relate this result with a conjecture by Huybrechts showing that, for a K3 surface with a symplectic involution, the finite-dimensionality of its…
We start by developing a theory of noncommutative (=NC) mixed motives with coefficients in any commutative ring. In particular, we construct a symmetric monoidal triangulated category of NC mixed motives, over a base field k, and a full subcategory of NC mixed Artin motives. Making use of Hochschild homology, we then apply Ayoub's weak Tannakian formalism to these motivic categories. In the case of NC mixed motives, we obtain a motivic Hopf dg algebra, which we describe explicitly in terms of Hochschild homology and complexes of exact cubes. In the case of NC mixed Artin motives, we compute th…
In this paper we demonstrate that 'non-commutative localizations' of arbitrary additive categories (generalizing those defined by Cohn for rings) are closely (and naturally) related with weight structures. Localizing an arbitrary triangulated $C$ by a set $S$ of morphisms in the heart of a weight structure $w$ for it one obtains a triangulated category endowed with a weight structure $w'$. The heart of $w'$ is a certain idempotent completion of the non-commutative localization of the heart of $w$ by $S$. The latter is the natural categorical version of Cohn's localizations of rings i.e. the fu…
The goal of this paper is to give an explicit description of the triangulated categories of Tate and Artin-Tate motives with finite coefficients Z/m over a field K containing a primitive m-root of unity as the derived categories of exact categories of filtered modules over the absolute Galois group of K with certain restrictions on the successive quotients. This description is conditional upon (and its validity is equivalent to) certain Koszulity hypotheses about the Milnor K-theory/Galois cohomology of K. This paper also purports to explain what it means for an arbitrary nonnegatively graded …
We introduce the notion of a quasi DG category, generalizing that of a DG category. To a quasi DG category satisfying certain additional conditions, we associate another quasi DG category, the quasi DG category of $C$-diagrams. We then show the homotopy category of the quasi DG category of $C$-diagrams has the structure of a triangulated category. This procedure is then applied to produce a triangulated category of mixed motives over a base variety.
Considering a (co)homology theory $\mathbb{T}$ on a base category $\mathcal{C}$ as a fragment of a first-order logical theory we here construct an abelian category $\mathcal{A}[\mathbb{T}]$ which is universal with respect to models of $\mathbb{T}$ in abelian categories. Under mild conditions on the base category $\mathcal{C}$, e.g. for the category of algebraic schemes, we get a functor from $\mathcal{C}$ to ${\rm Ch}({\rm Ind}(\mathcal{A}[\mathbb{T}]))$ the category of chain complexes of ind-objects of $\mathcal{A}[\mathbb{T}]$. This functor lifts Nori's motivic functor for algebraic schemes …
In this article we introduce the categories of noncommutative (mixed) Artin motives. In the pure world, we start by proving that the classical category AM(k) of Artin motives (over a base field k) can be characterized as the largest category inside Chow motives which fully-embeds into noncommutative Chow motives. Making use of a refined bridge between pure motives and noncommutative pure motives we then show that the image of this full embedding, which we call the category NAM(k) of noncommutative Artin motives, is invariant under the different equivalence relations and modification of the sym…
We construct and study a triangulated category of motives with modulus $\mathbf{MDM}_{\mathrm{gm}}^{\mathrm{eff}}$ over a field $k$ that extends Voevodsky's category $\mathbf{DM}_{\mathrm{gm}}^{\mathrm{eff}}$ in such a way as to encompass non-homotopy invariant phenomena. In a similar way as $\mathbf{DM}_{\mathrm{gm}}^{\mathrm{eff}}$ is constructed out of smooth $k$-varieties, $\mathbf{MDM}_{\mathrm{gm}}^{\mathrm{eff}}$ is constructed out of \emph{proper modulus pairs}, that is, pairs of a proper $k$-variety $X$ and an effective divisor $D$ on $X$ such that $X \setminus |D|$ is smooth. To a mo…
We prove an additivity for evenly (oddly) finite dimensional objects in distinguished triangles in a triangulated monoidal category structured by an underlying model monoidal category. In particular, the result holds in the Q-localized motivic stable homotopy category of spectra and in Q-localized Voevodsky's category of motives over a field, char=0. As an application, we show that the motives of schemes of dimension one (separated and of finite type over a field, char=0) are finite dimensional.
As we proved earlier, for a triangulated category $\underline{C}$ endowed with a weight structure $w$ and a triangulated subcategory $\underline{D}$ of $\underline{C}$ (strongly) generated by cones of a set of morphisms $S$ in the heart $\underline{Hw}$ of $w$ there exists a weight structure $w'$ on the Verdier quotient $\underline{C}'=\underline{C}/\underline{D}$ such that the localization functor $\underline{C} \to \underline{C}'$ is weight-exact (i.e., "respects weights"). The goal of this paper is to find conditions ensuring that for any object of $\underline{C}'$ of non-negative (resp. no…
We describe the Voevodsky's category $DM^{eff}_{gm}$ of motives in terms of Suslin complexes of smooth projective varieties. This shows that Voeovodsky's $DM_{gm}$ is anti-equivalent to Hanamura's one. We give a description of any triangulated subcategory of $DM^{eff}_{gm}$ (including the category of effective mixed Tate motives). We descibe 'truncation' functors $t_N$ for $N>0$. $t=t_0$ generalizes the weight complex of Soule and Gillet; its target is $K^b(Chow_{eff})$; it calculates $K_0(DM^{eff}_{gm})$, and checks whether a motive is a mixed Tate one. $t_N$ give a weight filtration and a 'm…
In this paper we give a construction of phantom categories, i.e. admissible triangulated subcategories in bounded derived categories of coherent sheaves on smooth projective varieties that have trivial Hochschild homology and trivial Grothendieck group. We also prove that these phantom categories are phantoms in a stronger sense, namely, they have trivial K-motives and, hence, all their higher K-groups are trivial too.
We describe a new method for constructing a weight structure $w$ on a triangulated category $C$. For a given $C$ and $w$ it allow us to give a fairly comprehensive (and new) description of those triangulated categories consisting of retracts of objects of $C$ (i.e., of subcategories of the Karoubi envelope of $C$ that contain $C$; we call them idempotent extensions of $C$) such that $w$ extends to them. In particular, any bounded above or below $w$ extends to any idempotent extension of $C$. We also discuss the applications of our results to certain triangulated categories of ("relative") moti…
This paper is dedicated to triangulated categories endowed with weight structures (a new notion; D. Pauksztello has independently introduced them as co-t-structures). This axiomatizes the properties of stupid truncations of complexes in $K(B)$. We also construct weight structures for Voevodsky's categories of motives and for various categories of spectra. A weight structure $w$ defines Postnikov towers of objects; these towers are canonical and functorial 'up to morphisms that are zero on cohomology'. For $Hw$ being the heart of $w$ (in $DM_{gm}$ we have $Hw=Chow$) we define a canonical conser…
The main goal of this paper is to prove the following: for a triangulated category $ \underline{C}$ and $E\subset \operatorname{Obj} \underline{C}$ there exists a cohomological functor $F$ (with values in some abelian category) such that $E$ is its set of zeros if (and only if) $E$ is closed with respect to retracts and extensions (so, we obtain a certain Nullstellensatz for functors of this type). Moreover, for $ \underline{C}$ being an $R$-linear category (where $R$ is a commutative ring) this is also equivalent to the existence of an $R$-linear $F: \underline{C}^{op}\to R-\operatorname{mod}…
We present a research programme aimed at constructing classifying toposes of Weil-type cohomology theories and associated categories of motives, and introduce a number of notions and preliminary results already obtained in this direction. In order to analyze the properties of Weil-type cohomology theories and their relations, we propose a framework based on atomic two-valued toposes and homogeneous models. Lastly, we construct a syntactic triangulated category whose dual maps to the derived categories of all the usual cohomology theories.
The purpose of this article is to study conservativity in the context of triangulated categories equipped with a weight structure. As application, we establish (weight) conservativity for the restriction of the (generic) l-adic realization to the category of motives of Abelian type of characteristic zero.
We propose a construction of a tensor exact category F_X^m of Artin-Tate motivic sheaves with finite coefficients Z/m over an algebraic variety X (over a field K of characteristic prime to m) in terms of etale sheaves of Z/m-modules over X. Among the objects of F_X^m, in addition to the Tate motives Z/m(j), there are the cohomological relative motives with compact support M_cc^m(Y/X) of varieties Y quasi-finite over X. Exact functors of inverse image with respect to morphisms of algebraic varieties and direct image with compact supports with respect to quasi-finite morphisms of varieties Y\to …
This paper is part of an endeavor to define an analogue of the slice filtration in the unstable motivic homotopy category. Our approach was inspired by the fact that the triangulated structures do not play a relevant role for the construction of birational homotopy categories as well as by the work of Kahn-Sujatha \cite{K-theory/0596} on birational motives, where the existence of a connection between the layers of the slice filtration and birational invariants is explicitly suggested. Our main result, shows that there is an equivalence of categories between the orthogonal components for the sl…
We study various triangulated motivic categories and introduce a vast family of aisles (these are certain classes of objects) in them. These aisles are defined in terms of the corresponding "motives" (or motivic spectra) of smooth varieties in them; we relate them to the corresponding homotopy t-structures. We describe our aisles in terms of stalks at function fields and prove that they widely generalize the ones corresponding to slice filtrations. Further, the filtrations on the "homotopy hearts" $Ht_{hom}^{eff}$ of the corresponding effective subcategories that are induced by these aisles ca…
