A DG guide to Voevodsky's motives

Voevodsky motives 和 motives with modulus 在正特征情况下

Voevodsky motives and motives with modulus in positive characteristic

本文在不假设解析性的情况下，证明了带有模的动机三角范畴与Voevodsky的动机三角范畴在有理系数下等价，展示了正特征下模的乘数在动机范畴中变得不可见。

Explicit generators for (conjectural) mixed motives (in Voevodsky's $\dmge$). The Kunneth decomposition of pure (numerical) motives

In this note we describe very explicitly a rich family of mixed motives that generates Voevodsky's $DM^{eff}_{gm}{\mathbb{Q}}$ (as a triangulated category). They "should be" mixed since they have only one non-zero Betti cohomology group. Our method also allows to define a family of direct summands of the numerical motif of any smooth projective variety $P$. Modulo certain standard conjectures, this construction yields the Kunneth decomposition of the diagonal of $P$.

The Voevodsky motive of a rank one semiabelian variety

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.

A note on relative duality for Voevodsky motives

Let X be an n-dimensional smooth proper variety over a field admitting resolution of singularities, and Y,Z two disjoint closed subsets of X. We establish an isomorphism M(X-Z,Y) isomorphic to M(X-Y,Z)^*(n)[2n] in Voevodsky's triangulated category of geometric motives. Here, M(X-Z,Y) is the motive of X -Z relative to its closed subset Y.

Voevodsky's motives and Weil reciprocity

We describe Somekawa's K-group associated to a finite collection of semi-abelian varieties (or more general sheaves) in terms of the tensor product in Voevodsky's category of motives. While Somekawa's definition is based on Weil reciprocity, Voevodsky's category is based on homotopy invariance. We apply this to explicit descriptions of certain algebraic cycles.

相对 Nori 与 Voevodsky 动机中的 Artin 动机

Artin motives in relative Nori and Voevodsky motives

本文证明特征零域上有限型概形的相对 Artin 动机在 Nori 与 Voevodsky 范畴中等价，并应用于对偶性及动机伽罗瓦群的正合序列类比。

Pure motives, mixed motives and extensions of motives associated to singular surfaces

We first recall the construction of the Chow motive modelling intersection cohomology of a proper surface and study its fundamental properties. Using Voevodsky's category of effective geometrical motives, we then study the motive of the exceptional divisor in a non-singular blow-up. If all geometric irreducible components of the divisor are of genus zero, then Voevodsky's formalism allows us to construct certain one-extensions of Chow motives, as canonical sub-quotients of the motive with compact support of the smooth part of the surface. Specializing to Hilbert--Blumenthal surfaces, we recove…

Framed motives of algebraic varieties (after V. Voevodsky)

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 …

Differential graded motives: weight complex, weight filtrations and spectral sequences for realizations; Voevodsky vs. Hanamura

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…

On the Voevodsky motive of the moduli space of Higgs bundles on a curve

We study the motive of the moduli space of semistable Higgs bundles of coprime rank and degree on a smooth projective curve C over a field k under the assumption that C has a rational point. We show this motive is contained in the thick tensor subcategory of Voevodsky's triangulated category of motives with rational coefficients generated by the motive of C. Moreover, over a field of characteristic zero, we prove a motivic non-abelian Hodge correspondence: the integral motives of the Higgs and de Rham moduli spaces are isomorphic.

Voedodsky动机的巴勒默谱和纯符号

The Balmer spectrum of Voevodsky motives and pure symbols

本文引入了巴勒默谱中Voevodsky动机范畴的不变量，其值为MilnorK理论中纯符号模块的'轻Rost循环子模块'。证明了各向同性点的闭合性，并引入了边界类型的点概念，表明该类包含各向同性点但不包含微分型点。

关于Voevodsky动机的Nori实现与Hodge实现

On the Nori and Hodge realisations of Voevodsky motives

本文通过证明反常Nori动机与混合Hodge模导出范畴的可构造心性质，构造了六运算的∞-范畴提升，建立了Voevodsky étale动机到两者的实现函子，并证明了Artin动机范畴等价及motivic t-结构存在时的等价性。

Integration of Voevodsky motives

In this paper, we construct four different theories of integration, two that are for Voevodsky motives, one for mixed $\ell$-adic sheaves, and a fourth theory of integration for rational mixed Hodge structures. We then show that they circumvent some of the complications of classical motivic integration, leading to new arithmetic and geometric results concerning K-equivalent $k$-varieties. For example, in addition to recovering known results regarding K-equivalent smooth projective complex varieties, we show that K-equivalent smooth projective $\mathbb{F}_q$-varieties have isomorphic rational $…

On the Voevodsky motive of the moduli stack of vector bundles on a curve

We define and study the motive of the moduli stack of vector bundles of fixed rank and degree over a smooth projective curve in Voevodsky's category of motives. We prove that this motive can be written as a homotopy colimit of motives of smooth projective Quot schemes of torsion quotients of sums of line bundles on the curve. When working with rational coefficients, we prove that the motive of the stack of bundles lies in the localising tensor subcategory generated by the motive of the curve, using Bialynicki-Birula decompositions of these Quot schemes. We conjecture a formula for the motive o…

与对数概形相关的 Voevodsky 动机

A Voevodsky motive associated to a log scheme

本文为域上 fs 对数概形构造几何 Voevodsky 动机，证明复数域上其 Betti 实现等于对数 Betti 上同调，并应用于管状邻域、极限动机和单值滤子。

Réalisation de de Rham des motifs de Voevodsky

After reviewing some basic facts about pure and mixed motives, we explain, following Deligne-Goncharov, how to construct a de Rham realisation functor from the category of geometric mixed motives to the category of bifiltered vector spaces.

Hodge realizations of 1-motives and the derived Albanese

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.

Comparison of the Categories of Motives defined by Voevodsky and Nori

In this thesis we compare V. Voevodsky's geometric motives to the derived category of M. Nori's abelian category of mixed motives by constructing a triangulated tensor functor between them. It will be compatible with the Betti realizations on both sides. We allow an arbitrary noetherian ring of coefficients, but require it to be a field or a Dedekind domain for the tensor structure on derived Nori motives to exist. There are three key ingredients: we present a theory of Nisnevich covers on finite acyclic diagrams of finite correspondences, explain, following D. Rydh, how to interpret finite co…

Motives for an elliptic curve

In this paper we describe the category of motives for an elliptic curve in the sense of Voevodsky as a derived category of dg modules over a commutative differential graded algebra in the category of representations over some reductive group.

Recent developments on noncommutative motives

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 …

通过丰富函子构造大动机的三角范畴

Triangulated Categories of Big Motives via Enriched Functors

本文利用丰富函子的Grothendieck范畴同调代数，为Voevodsky的合理对应大动机范畴构造了两个模型。

Voevodsky 动机的Zeta函数与L函数

Zeta and L functions of Voevodsky motives

本研究为整体域上的几何Voevodsky动机定义了具有欧拉乘积的近L函数，明确了其与Serre zeta函数的差异、乘性及函数域上的有理性与函数方程，核心工具为Ayoub的六/七函子运算。

Mixed Motives and Motivic Birational Covers

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.

Smooth blowup square for motives with modulus

In this self-contained paper we prove that Voevodsky's smooth blowup triangle of motives generalises to a smooth blowup triangle of motives with modulus.

带模的动机，I：非真模对的带转移模层

Motives with modulus, I: Modulus sheaves with transfers for non-proper modulus pairs

本研究构建了推广Voevodsky带转移层理论的带转移模层理论，为[KMSY20]中的带模动机理论奠定基础。

Algebraic cycles and the triangulated category of mixed motives

In this paper, we shall give a candidate for the t-structure on the triangulated category of mixed motives due to Voevodsky.

Triangulated categories of motives in positive characteristic

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…

Groups of order $p^3$ are mixed Tate

A natural place to study the Chow ring of the classifying space $BG$, for $G$ a linear algebraic group, is Voevodsky's triangulated category of motives, inside which Morel and Voevodsky, and Totaro have defined motives $M(BG)$ and $M^c(BG)$, respectively. We show that, for any group $G$ of order $p^3$ over a field of characteristic not $p$ which contains a primitive $p^2$-th root of unity, the motive $M(BG)$ is a mixed Tate motive. We also show that, for a finite group $G$ over a field of characteristic zero, $M(BG)$ is a mixed Tate motive if and only $M^c(BG)$ is a mixed Tate motive.

有理系数下的动机比较

Comparison of motives with rational coefficients

本文证明在准优秀基方案上，Voevodsky的有理动机DM(S, Q)与Morel/Beilinson/Ayoub的有理动机等价，并部分验证了Voevodsky的猜想。

关于代数栈的动机同伦类型

On the Motivic Homotopy Type of Algebraic Stacks

本文构造了在Morel-Voevodsky A1-同伦范畴中为局部满射的代数栈的光滑展示，并证明光滑栈的动机具有与光滑概形动机相似的许多性质。

Cancellation theorem for framed motives of algebraic varieties

The machinery of framed (pre)sheaves was developed by Voevodsky [V1]. Based on the theory, framed motives of algebraic varieties are introduced and studied in [GP1]. An analog of Voevodsky's Cancellation Theorem [V1] is proved in this paper for framed motives stating that a natural map of framed $S^1$-spectra $$M_{fr}(X)(n)\to\underline{\textrm{Hom}}(\mathbb G,M_{fr}(X)(n+1)),\quad n\geq 0,$$ is a schemewise stable equivalence, where $M_{fr}(X)(n)$ is the $n$th twisted framed motive of $X$. This result is also necessary for the proof of the main theorem of [GP1] computing fibrant resolutions o…

Birational motives, II: Triangulated birational motives

We develop birational versions of Voevodsky's triangulated categories of motives over a field, and relate them with the pure birational motives studied in arXiv:0902.4902 [math.AG]. We also get an interpretation of unramified cohomology in this framework, leading to "higher derived functors of unramified cohomology".

motivic与经典同伦理论的全景图

Big pictures of motivic and classical homotopy theories

本文通过与经典对应物比较，探讨motivic同伦理论的全景及其与Voevodsky motives的联系。

Some applications of weight structures of Bondarko

We construct a functor from the triangulated category of Voevodsky motives to a certain derived category of mixed Hodge structures enriched with integral weight filtration. We use this construction to prove a strong integral version of the Deligne conjecture on 1-motives which was known previously only up to isogeny.

Adjoint functors on the derived category of motives

Voevodsky's derived category of motives is the main arena today for the study of algebraic cycles and motivic cohomology. In this paper we study whether the inclusions of three important subcategories of motives have a left or right adjoint. These adjoint functors are useful constructions when they exist, describing the best approximation to an arbitrary motive by a motive in a given subcategory. We find a fairly complete picture: some adjoint functors exist, including a few which were previously unexplored, while others do not exist because of the failure of finite generation for Chow groups …

Nisnevich topology with modulus

In Voevodsky's theory of motives, the Nisnevich topology on smooth schemes is used as an important building block. In this paper, we introduce a Grothendieck topology on proper modulus pairs, which will be used to construct a non-homotopy invariant generalization of motives. We also prove that the topology satisfies similar properties to the Nisnevich topology.

The motive of a classifying space

We give the first examples of finite groups G such that the Chow ring of the classifying space BG depends on the base field, even for fields containing the algebraic closure of Q. As a tool, we give several characterizations of the varieties which satisfy Kunneth properties for Chow groups or motivic homology. We define the (compactly supported) motive of a quotient stack in Voevodsky's derived category of motives. This makes it possible to ask when the motive of BG is mixed Tate, which is equivalent to the motivic Kunneth property. We prove that BG is mixed Tate for various "well-behaved" fin…

关于Chow对应的一些评论

Some remarks on Chow correspondences

本文在Voevodsky的三角动机范畴中研究了Chow环中几种适当的等价关系及其性质。

Framed motives of smooth affine pairs

The theory of framed motives by Garkusha and Panin gives computations in the stable motivic homotopy category $\mathbf{SH}(k)$ in terms of Voevodsky's framed correspondences. In particular the motivically fibrant $Ω$-resolution in positive degrees of the motivic suspension spectrum $Σ_{\mathbb P^1}^\infty X_+$, where $X_+=X\amalg *$, for a smooth scheme $X\in \mathrm{Sm}_k$ over an infinite perfect field $k$, is computed. The computation by Garkusha, Neshitov and Panin of the framed motives of relative motivic spheres $(\mathbb A^l\times X,(\mathbb A^l-0)\times X)$, $X\in \mathrm{Sm}_k$, is on…