The main goal of this paper is to define the so-called Chow weight structure for the category of Beilinson motives over any 'reasonable' base scheme $S$ (this is the version of Voevodsky's motives over $S$ defined by Cisinski and Deglise). We also study the functoriality properties of the Chow weight structure (they are very similar to the well-known functoriality of weights for mixed complexes of sheaves). As shown in a preceding paper, the Chow weight structure automatically yields an exact conservative weight complex functor (with values in $K^b(Chow(S))$). Here $Chow(S)$ is the heart of the Chow weight structure; it is 'generated' by motives of regular schemes that are projective over $S$. Besides, Grothendiek's group of $S$-motives is isomorphic to $K_0(Chow(S))$; we also define a certain 'motivic Euler characteristic' for $S$-schemes. We obtain (Chow)-weight spectral sequences and filtrations for any cohomology of motives; we discuss their relation to Beilinson's 'integral part' of motivic cohomology and to weights of mixed complexes of sheaves. For the study of the latter we introduce a new formalism of relative weight structures.
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 prove formulae for the motives of stacks of coherent sheaves of fixed rank and degree over a smooth projective curve in Voevodsky's triangulated category of mixed motives with rational coefficients.
The goal of this paper is to prove: if certain 'standard' conjectures on motives over algebraically closed fields hold, then over any 'reasonable' $S$ there exists a motivic $t$-structure for the category of Voevodsky's $S$-motives (as constructed by Cisinski and Deglise). If $S$ is 'very reasonable' (for example, 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 semi-simple factors. We also prove a certain 'motivic decomposition theorem' (assuming the conjectures mentioned) and characteri…
We connect two developments aiming at extending Voevodsky's theory of motives over a field in such a way to encompass non-$\mathbf{A}^1$-invariant phenomina. One is theory of reciprocity sheaves introduced by Kahn-Saito-Yamazaki. Another is theory of the triangulated category $\operatorname{\mathbf{logDM}}^{\operatorname{eff}}$ of logarithmic motives launched by Binda, Park and Østvær. We prove that the Nisnevich cohomology of reciprocity sheaves is representable in $\operatorname{\mathbf{logDM}}^{\operatorname{eff}}$.
We study the Tannakian properties of the category of perverse sheaves on elliptic curves endowed with the convolution product. We establish that for certain sheaves with unipotent local monodromy over seven points the corresponding Tannaka group is isomorphic to $G_2$. This monodromy approach generalizes a result of Katz on the existence of $G_2$-motives in the middle cohomology of deformations of Beauville surfaces.
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…
Pure t-motives were introduced by G. Anderson as higher dimensional generalizations of Drinfeld modules, and as the appropriate analogs of abelian varieties in the arithmetic of function fields. In order to construct moduli spaces for pure t-motives the second author has previously introduced the concept of abelian τ-sheaf. In this article we clarify the relation between pure t-motives and abelian τ-sheaves. We obtain an equivalence of the respective quasi-isogeny categories. Furthermore, we develop the elementary theory of both structures regarding morphisms, isogenies, Tate modules, and loca…
The goal of this paper is to extend the work of Voevodsky and Morel on the homotopy $t$-structure on the category of motivic complexes to the context of motives for logarithmic schemes. To do so, we prove an analogue of Morel's connectivity theorem and show a purity statement for $(\mathbf{P}^1, \infty)$-local complexes of sheaves with log transfers. The homotopy $t$-structure on $\mathbf{logDM}^{\textrm{eff}}(k)$ is proved to be compatible with Voevodsky's $t$-structure i.e. we show that the comparison functor $R^{\overline{\square}}ω^*\colon \mathbf{DM}^{\textrm{eff}}(k)\to \mathbf{logDM}^{\…
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 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 …
With representation-theoretic applications in mind, we construct a formalism of reduced motives with integral coefficients. These are motivic sheaves from which the higher motivic cohomology of the base scheme has been removed. We show that reduced stratified Tate motives satisfy favorable properties including weight and t-structures. We also prove that reduced motives on cellular (ind-)schemes unify various approaches to mixed sheaves in representation theory, such as Soergel-Wendt's semisimplified Hodge motives, Achar-Riche's complexes of parity sheaves, as well as Ho-Li's recent category of…
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…
We define a theory of etale 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 rigity theorem of Suslin and Voevodsky over a general base scheme. This can be reformulated by saying that torsion etale motives essentially coincide with the usual complexes of torsion etale sheaves (at least if…
Pure t-motives were introduced by G. Anderson as higher dimensional generalizations of Drinfeld modules, and as the appropriate analogs of abelian varieties in the arithmetic of function fields. In this article we develop their theory regarding morphisms, isogenies, Tate modules, and local shtukas. The later are the analog of p-divisible groups. We investigate which pure t-motives are semisimple, that is, isogenous to direct sums of simple ones. We give examples for pure t-motives which are not semisimple. Over finite fields the semisimplicity is equivalent to the semisimplicity of the endomor…
We associate weight complexes of (homological) motives, and hence Euler characteristics in the Grothendieck group of motives, to arithmetic varieties and Deligne-Mumford stacks; this extends the results in the paper "Descent, Motives and K-theory" in volume 478 of Crelle, where a similar result was proved for varieties over a field of characteristic zero. We use K_0-motives with rational coefficients, rather than Chow motives, because we cannot appeal to resolution of singularities, but rather must use de Jong's results. In addition, for varieties over a field we prove a general result on cont…
For noetherian schemes of finite dimension over a field of characteristic exponent $p$, we study the triangulated categories of $\mathbf{Z}[1/p]$-linear mixed motives obtained from cdh-sheaves with transfers. We prove that these have many of the expected properties. In particular, the formalism of the six operations holds in this context. When we restrict ourselves to regular schemes, we also prove that these categories of motives are equivalent to the more classical triangulated categories of mixed motives constructed in terms of Nisnevich sheaves with transfers. Such a program is achieved by…
The category of framed correspondences $Fr_*(k)$ and framed sheaves were invented by Voevodsky in his unpublished notes [V2]. Based on the theory, framed motives are introduced and studied in [GP1]. These are Nisnivich sheaves of $S^1$-spectra and the major computational tool of [GP1]. The aim of this paper is to show the following result which is essential in proving the main theorem of [GP1]: given an infinite perfect base field $k$, any $k$-smooth scheme $X$ and any $n\geq 1$, the map of simplicial pointed Nisnevich sheaves $(-,\mathbb{A}^1//\mathbb G_m)^{\wedge n}_+\to T^n$ induces a Nisne…
Bloch and Kriz construct an abelian category of mixed Tate motives as the category of comodules over a Hopf algebra obtained by the bar construction of the DGA of cycle complexes. In this paper we generalize their construction to give the definition of a category of mixed elliptic motives, i.e. a Tannakian category of mixed motives generated by an elliptic curve. We introduce the notion of a relative DGA over a reductive group. Then the category of mixed elliptic motives is defined as the category of comodules over the relative bar construction of a certain DGA $A_{EM}$ which is constructed fr…
It is an open conjecture of Orlov that the bounded derived category of coherent sheaves of a smooth projective variety determines its Chow motive with rational coefficients. In this master's thesis we introduce a category of \emph{perfect correspondences}, whose objects are smooth projective varieties and morphisms $X \to Y$ are perfect complexes on $X \times Y$. We show that isomorphism in this category is the same as equivalence of derived categories, and use this to show that the derived category determines the noncommutative Chow motive (in the sense of Tabuada) and, up to Tate twists, the…
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.
In this paper we associate to t-motives with level structures a finite dimensional vector subspace (subspace of uniformizers). We study the particular case of elliptic sheaves finding some relations with some objects from conformal field theory.
We define higher pro-Albanese functors for every effective log motive over a field $k$ of characteristic zero, and we compute them for every smooth log smooth scheme $X=(\underline{X}, \partial X)$. The result involves an inverse system of the coherent cohomology of the underlying scheme as well as a pro-group scheme $\mathrm{Alb}^{\log}(X)$ that extends Serre's semi-abelian Albanese variety of $\underline{X}-|\partial X|$. This generalizes the higher Albanese sheaves of Ayoub, Barbieri-Viale and Kahn.
We give an alternative construction of the Betti realization functor on the derived category of motives of complex algebraic varieties via the category of CW complexes instead of the category of complex analytic spaces. In particular we show that the functor we define via the category of CW complexes coincide with Ayoub's one. We deduce from this construction that Ayoub's realization functor on geometric motives factors trough Nori motives and that the image of this functor on the morphisms between the motive of a point and a shift of a Tate twist of the motive with compact support of a comple…
We study in detail the so-called Chow-weight homology of Voevodsky motivic complexes and relate it to motivic homology. We generalize earlier results and prove that the vanishing of higher motivic homology groups of a motif $M$ implies similar vanishing for its Chow-weight homology along with effectivity properties of the higher terms of its weight complex $t(M)$ and of higher Deligne weight quotients of its cohomology. Applying this statement to motives with compact support we obtain a similar relation between the vanishing of Chow groups and the cohomology with compact support of varieties. …
We will generalize the projective model structure in the category of unbounded complexes of modules over a commutative ring to the category of unbounded complexes of quasi-coherent sheaves over the projective line. Concretely we will define a locally projective model structure in the category of complexes of quasi-coherent sheaves on the projective line. In this model structure the cofibrant objects are the dg-locally projective complexes. We also describe the fibrations of this model structure and show that the model structure is monoidal. We point out that this model structure is necessarily…
For any smooth projective moduli space $M$ of Gieseker stable sheaves on a complex projective K3 surface (or an abelian surface) S, we prove that the Chow motive $\mathfrak{h}(M)$ becomes a direct summand of a motive $\bigoplus \mathfrak{h}(S^{k_i})(n_i)$ with $k_i\leq \dim(M)$. The result implies that finite dimensionality of $\mathfrak{h}(M)$ follows from finite dimensionality of $\mathfrak{h}(S)$. The technique also applies to moduli spaces of twisted sheaves and to moduli spaces of stable objects in ${\rm D}^{\rm b}(S,α)$ for a Brauer class $α\in{\rm Br}(S)$. In a similar vein, we investig…
We obtain a linear algebra data presentation of the category of constructible with respect to perverse triangulation sheaves on a finite simplicial complex. We also establish Koszul duality between the above mentioned category and the category of perverse sheaves constructible with respect to the triangulation
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…
B. Kim and the first author proved a result comparing the virtual fundamental classes of the moduli spaces of stable quasimaps and stable LG-quasimaps by studying localized Chern characters for 2-periodic complexes. In this paper, we study a K-theoretic analogue of the localized Chern character map and show that for a Koszul 2-periodic complex it coincides with the cosection localized Gysin map by Y.-H. Kiem and J. Li. As an application we compare the virtual structure sheaves of the moduli space of stable quasimaps and stable LG-quasimaps.
Fix a scheme $X$ over a field of characteristic zero that is equipped with an action of a reductive algebraic group $G$. We give necessary and sufficient conditions for a $G$-equivariant coherent sheaf on $X$ or a bounded-above complex of $G$-equivariant coherent sheaves on $X$ to descend to a good quotient $X//G$. This gives a description of the coherent derived category of $X//G$ as an admissible subcategory of the equivariant derived category of $X$.
The goal of this paper is to define a certain Chow weight structure for the category of Voevodsky's motivic complexes with integral coefficients (as described by Cisinski and Deglise) over any excellent finite-dimensional separated scheme $S$. Our results are parallel to (though substantially weaker than) the corresponding 'rational coefficient' statements proved by D. Hebert and the author. As an immediate consequence of the existence of 'weights', we obtain certain (Chow)-weight spectral sequences and filtrations for any (co)homology of $S$-motives.
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.
For an abelian or a projective K3 surface $X$ over an algebraically closed field $k$, consider the moduli space $\splcpx_{X/k}\uet$ of the objects $E$ in $D^b(\mathrm{Coh}(X))$ satisfying $\Ext^{-1}_X(E,E)=0$ and $\Hom(E,E)\cong k$. Then we can prove that $\splcpx_{X/k}\uet$ is smooth and has a symplectic structure.
Let $\mathbb{k}$ be a field of characteristic $p$. We introduce a formalism of mixed sheaves with coefficients in $\mathbb{k}$ and showcase its use in representation theory. More precisely, we construct for all quasi-projective schemes $X$ over an algebraic closure of $\mathbb{F}_p$ a $\mathbb{k}$-linear triangulated category of motives on $X$. Using work of Ayoub (2007), Cisinski-Deglise (2012) and Geisser-Levine (2000), we show that this system of categories has a six functors formalism and computes higher Chow groups. Indeed, it behaves similarly to other categories of sheaves that one is u…
We define a tricategory T of length 3 complexes of abelian sheaves, whose hom-bigroupoids consist of weak morphisms of such complexes. We also define a 3-category 2PIC(S) of Picard 2-stacks, whose hom-2-groupoids consist of additive 2-functors. We prove that these categories are triequivalent as tricategories. As a consequence we obtain a generalization of Deligne's analogous result about Picard stacks in SGA4, Exp. XVIII.
In this paper, we try to realize the unbounded derived category of an abelian category as the homotopy category of a Quillen model structure on the category of unbounded chain complexes. We construct such a model structure based on injective resolutions for an arbitrary Grothendieck category, as has apparently also been done by Morel. In particular, this works for sheaves on a ringed space, and for quasi-coherent sheaves on a quasi-compact, quasi-separated scheme. However, this injective model structure is not well suited to studying the derived tensor product, so we investigate other model st…
We establish a theory of complexes of relative correspondences. The theory generalizes the known theory of complexes of correspondences of smooth projective varieties. It will be applied in the sequel of this paper to the construction of the triangulated category of motives over a base variety.
A quasi-coherent ringed scheme is a pair (X,A), where X is a scheme, and A is a noncommutative quasi-coherent O_X-ring. We introduce dualizing complexes over quasi-coherent ringed schemes and study their properties. For a separated differential quasi-coherent ringed scheme of finite type over a field, we prove existence and uniqueness of a rigid dualizing complex. In the proof we use the theory of perverse coherent sheaves in order to glue local pieces of the rigid dualizing complex into a global complex.
This paper is concerned with an interpretation of f-cohomology, a modification of motivic cohomology of motives over number fields, in terms of motives over number rings. Under standard assumptions on mixed motives over finite fields, number fields and number rings, we show that the two extant definitions of f-cohomology of mixed motives $M_η$ over F--one via ramification conditions on $\ell$-adic realizations, another one via the K-theory of proper regular models--both agree with motivic cohomology of $η_{!*} M_η[1]$. Here $η_{!*}$ is constructed by a limiting process in terms of intermediate…
Over a perfect field k, let G be an extension of an abelian variety by the multiplicative group $\G_m$. We compute the motive of G in Voevodsky's category of etale motivic complexes with rational coefficients. The result is a decomposition similar to the classical decomposition of the Chow motive of an abelian variety obtained by Deninger/Murre, Kuennemann, Shermenev and Scholl, and other authors.
In this dissertation, we discuss mainly the corresponding geometric and representation theoretic aspects of relative $p$-adic Hodge theory and $p$-adic motives. To be more precise, we study the corresponding analytic geometry of the corresponding spaces over and attached to period rings in the relative $p$-adic Hodge theory, including derived topological de Rham complexes and derived topological logarithmic de Rham complexes after Bhatt, Gabber, Guo and Illusie which is in some sense equivalent to the derived prismatic cohomology of Bhatt-Scholze as shown in the work of Li-Liu, $\mathcal{O}\ma…
Via the BGG correspondence a simplicial complex Delta on [n] is transformed into a complex of coherent sheaves on P^n-1. We show that this complex reduces to a coherent sheaf F exactly when the Alexander dual Delta^* is Cohen-Macaulay. We then determine when both Delta and Delta^* are Cohen-Macaulay. This corresponds to F being a locally Cohen-Macaulay sheaf. Lastly we conjecture for which range of invariants of such Delta it must be a cone.
A generalization of the usual ideles group is proposed, namely, we construct certain adelic complexes for sheaves of $K$-groups on schemes. More generally, such complexes are defined for any abelian sheaf on a scheme. We focus on the case when the sheaf is associated to the presheaf of a homology theory with certain natural axioms, satisfied by $K$-theory. In this case it is proven that the adelic complex provides a flasque resolution for the above sheaf and that the natural morphism to the Gersten complex is a quasiisomorphism. The main advantage of the new adelic resolution is that it is con…
We generalize the construction of the category of 1-motives with torsion ${}^tM_1$ (introduced by Barbieri-Viale, Rosenschon and Saito) as well as the construction of the category of 1-motivic sheaves ${\rm Shv}_1$ (defined by Barbieri-Viale and Kahn) to perfect fields $k$ (without inverting the exponential characteristic). For $k$ transcendental over the prime field we extend a result of Barbieri-Viale and Kahn, showing that ${}^tM$ and ${\rm Shv}_1$ have equivalent bounded derived categories.
We prove a purely local form of a result of Saito and Yatagawa. They proved that the characteristic cycle of a constructible étale sheaf is determined by wild ramification of the sheaf along the boundary of a compactification. But they had to consider ramification at all the points of the compactification. We give a pointwise result, that is, we prove that the characteristic cycle of a constructible étale sheaf around a point is determined by wild ramification at the point. The key ingredient is to prove that wild ramification of the stalk of the nearby cycle complex of a constructible étale s…
Over a field of characteristic zero, we construct a De Rham motivic complex and generalize the De Rham cohomology of a smooth variety to any Voevodsky motive.
We study certain triangulated categories of $K$-motives $DK(-)$ over a wide class of base schemes, and define certain "weights" for them. We relate the weights of particular $K$-motives to (negative) homotopy invariant $K$-groups (tensored by $\mathbb{Z}[S^{-1}]$ for $S$ being the set of "non-invertible primes") $\mathcal{K}_*(-)$. Our results yield a (new) result on the vanishing of $\mathcal{K}_i(-)$ and of relative $\mathcal{K}$-groups for $i$ being "too negative"; this statement is closely related to a question of Ch. Weibel. We also prove that $\mathcal{K}_i(-)$ for $i<0$ is "supported in…
In the article [GS96], Gillet and Soulé define a weight complex on the category of Voevodsky motives over a field of characteristic 0. In [Bon07], Bondarko generalizes this construction for any f-category with a bounded weight structure, as is the case for Beilinson motives (following Cisinski-Déglise ; [CD09]). The first purpose of this note is to generalize [GS96, thm. 2] in the world of Beilinson motives. This done, we will naturally be led to define the motivic Euler characteristic dual to that considered by Bondarko in [Bon10]. This fact will motivate the second line of this note : provin…
We define a notion of total acyclicity for complexes of flat quasi-coherent sheaves over a semi-separated noetherian scheme, generalising complete flat resolutions over a ring. By studying these complexes as objects of the pure derived category of flat sheaves we extend several results about totally acyclic complexes of projective modules to schemes; for example, we prove that a scheme is Gorenstein if and only if every acyclic complex of flat sheaves is totally acyclic. Our formalism also removes the need for a dualising complex in several known results for rings, including Jorgensen's proof …
In this article, we will give the Deligne 1-motives up to isogeny corresponding to the $\mathbb{Q}$-limiting mixed Hodge structures of semi-stable degenerations of curves, by using logarithmic structures and Steenbrink's cohomological mixed Hodge complexes associated to semi-stable degenerations of curves.
The trivial fiber topology and framed motives over the integers
St. Petersburg State University（圣彼得堡国立大学）；Steklov Mathematical Institute of Russian Academy of Sciences（俄罗斯科学院斯特克洛夫数学研究所）；Western Norway University of Applied Sciences（西挪威应用科学大学）；University of Milan（米兰大学）；University of Oslo（奥斯陆大学）
