Daniel Dugger, Daniel C. Isaksen
An object in motivic homotopy theory is called cellular if it can be built out of motivic spheres using homotopy colimit constructions. We explore some examples and consequences of cellularity. We explain why the algebraic K-theory and algebraic cobordism spectra are both cellular, and prove some Kunneth theorems for cellular objects.
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Motivic stable stems and Galois approximations of cellular motivic categories
本文证明小上同调维数域上的胞腔动机谱范畴可由绝对 Galois 群刻画，并据此确定几乎所有域在特征外的动机稳定茎。
On P^1-stabilization in unstable motivic homotopy theory
本文研究了不稳定动机同调理论中P^1稳定化问题，提出了双连通性概念并改进了Whitehead定理，同时解决了Murthy猜想并计算了新的动机同调。
Construction of the motivic cellular spectrum $\mathbf{KO}^{geo}$ over $Spec(\mathbb{Z})$
本文在$Spec(\mathbb{Z})$上利用Grassmann簇的闭子概形构造周期动机谱$\mathbf{KO}^{geo}$，证明其在2可逆基上恢复Panin–Walter的$HP^1$-谱，并说明该谱具有胞腔性。
We define the Chow $t$-structure on the $\infty$-category of motivic spectra $SH(k)$ over an arbitrary base field $k$. We identify the heart of this $t$-structure $SH(k)^{c\heartsuit}$ when the exponential characteristic of $k$ is inverted. Restricting to the cellular subcategory, we identify the Chow heart $SH(k)^{cell, c\heartsuit}$ as the category of even graded $MU_{2*}MU$-comodules. Furthermore, we show that the $\infty$-category of modules over the Chow truncated sphere spectrum is algebraic. Our results generalize the ones in Gheorghe--Wang--Xu in three aspects: To integral results; To …
We exhibit a relationship between motivic homotopy theory and spectral algebraic geometry, based on the motivic $τ$-deformation picture of Gheorghe, Isaksen, Wang, Xu. More precisely, we identify cellular motivic spectra over $\mathbf C$ with ind-coherent sheaves (in a slightly non-standard sense) on a certain spectral stack $τ_{\ge 0}(\mathcal M_\mathrm{FG}^\mathrm{or})$. The latter is the connective cover of the non-connective spectral stack $\mathcal M_\mathrm{FG}^\mathrm{or}$, the moduli stack of oriented formal groups, which we have introduced previously and studied in connection with chr…
To an Adams-type homology theory we associate a notion of a synthetic spectrum, this is a product-preserving sheaf on the site of finite spectra with projective $E$-homology. We prove that the $\infty$-category $Syn_{E}$ of synthetic spectra based on $E$ is in a precise sense a deformation of the $\infty$-category of spectra into quasi-coherent sheaves over a certain algebraic stack, and show that this deformation encodes the $E$-based Adams spectral sequence. We describe a symmetric monoidal functor from cellular motivic spectra over the complex numbers into an even variant of synthetic spect…
Cellular objects in isotropic motivic categories
该研究刻画了柔性域上的等向胞腔谱范畴，证明其等价于经典拓扑Steenrod代数对偶上左余模的导出范畴，还研究了相关模范畴、构建了两类谱序列，并计算了等向Tate动机范畴中的同态集。
In this note, we describe motivic cell structures arising from the Bialynicki-Birula decomposition. This provides a description of the stable A^1-homotopy types of smooth projective G_m-varieties where the G_m-action has isolated fixed points.
We give a method for computing the C_2-equivariant homotopy groups of the Betti realization of a p-complete cellular motivic spectrum over R in terms of its motivic homotopy groups. More generally, we show that Betti realization presents the C_2-equivariant p-complete stable homotopy category as a localization of the p-complete cellular real motivic stable homotopy category.
Real isotropic motivic spectra
本文研究了实等价的动机谱，并揭示了其与F₂-合成谱的关系，以及实数上的动机同调理论与F₂-合成同调理论之间的联系。
Cellular $\mathbb A^1$-homology and the motivic version of Matsumoto's theorem
本文定义了适用于域上光滑概形的胞腔 $\mathbb A^1$-同调，通过显式计算确定了任意域上分裂约化群的 $\mathbb A^1$-基本群，从而获得了 Matsumoto 定理的动形版本，并统一推广了相关的中心扩张与同伦群结果。
Hermitian K-theory and Witt-theory are cellular in the sense of stable motivic homotopy theory over any base scheme without points of characteristic two.
We prove a motivic version of Landweber's exact functor theorem from topology. The main result is that the assignment given by a Landweber-type formula using the MGL-homology of a motivic spectrum defines a homology theory on the stable motivic homotopy category and is representable by a Tate-like (or cellular) spectrum. Using the universal coefficient spectral sequence of Dugger-Isaksen we deduce formulas for operations of motivic Landweber spectra of a certain type including homotopy algebraic K-theory. Finally we construct a Chern character as a map between motivic spectra.
We show that the cellular objects in the module category over a motivic E infinity ring spectrum E can be described as the module category over a graded topological spectrum if E is strongly periodizable in our language. A similar statement is proven for triangulated categories of motives. Since MGL is strongly periodizable we obtain topological incarnations of motivic Landweber spectra. Under some categorical assumptions the unit object of the model category for triangulated motives is as well strongly periodizable giving motivic cochains whose module category models integral triangulated cat…
We construct a topological model for cellular, 2-complete, stable C-motivic homotopy theory that uses no algebro-geometric foundations. We compute the Steenrod algebra in this context, and we construct a "motivic modular forms" spectrum over C.
Let F be a field of characteristic different than 2. We establish surjectivity of Balmer's comparison map rho^* from the tensor triangular spectrum of the homotopy category of compact motivic spectra to the homogeneous Zariski spectrum of Milnor-Witt K-theory. We also comment on the tensor triangular geometry of compact cellular motivic spectra, producing in particular novel field spectra in this category. We conclude with a list of questions about the structure of the tensor triangular spectrum of the stable motivic homotopy category.
A topological model for cellular motivic spectra
本文为任意动机E∞-环谱A构造了细胞动机A-模谱范畴与Z-分次谱中E1-代数Θ上模范畴之间的等价，并在复数基或E∞-定向条件下将其细化为对称幺半等价，一般情形则借助Sagave-Schlichtkrull范畴提升该等价。
The term "motivic Moore spectrum" refers to a cone of an element in the motivic stable homotopy groups of spheres. This article discusses some properties of motivic Moore spectra, among them the question whether the ring structure on the motivic sphere spectrum descends to a ring structure on a motivic Moore spectrum. This discussion requires an understanding of some Toda brackets in the motivic stable homotopy groups of spheres.
Motivic Hochschild homology of mod 2 motivic cohomology over algebraically closed fields
计算复数上 normed 动机谱中乘法群概形与 mod-2 动机上同调谱的张量，发现结果代数在双次数(2,1)上自由生成，给出了 Bökstedt 周期性的动机类比。
Polyhedral products in abstract and motivic homotopy theory
本文在$\infty$-范畴设定下引入多面体乘积，推广了确定稳定同伦型的分裂结果，并利用该结果计算了力矩角复形motivic细化后的胞腔$\mathbb{A}^1$-同调与$\mathbb{A}^1$-欧拉示性数。
On the splitting of surfaces in motivic stable homotopy category
针对完美域上带有理点的光滑射影曲面，讨论其动机稳定同伦型分裂顶层胞腔的条件，研究K3曲面等例外例子，在特征≠2的代数闭域下给出曲线分裂的新证明，并借助动机胡雷维茨定理和周-维特对应分解理解卡拉比-丘曲面分裂。
In this paper, we produce a cellular motivic spectrum of motivic modular forms over $\R$ and $\C$, answering positively to a conjecture of Dan Isaksen. This spectrum is constructed to have the appropriate cohomology, as a module over the relevant motivic Steenrod algebra. We first produce a $\G$-equivariant version of this spectrum, and then use a machinery to construct a motivic spectrum from an equivariant one. We believe that this machinery will be of independent interest.
Motivic Segre classes of Schubert cells and the connective formal group law
利用连接形式群律定义 d 步旗簇中 Schubert 胞腔的动机 Segre 类的单参数 β 形变，并给出 d=1 时的有理函数表示和组合公式。
We show that algebraic K-theory KGL, the motivic Adams summand ML and their connective covers acquire unique E-infinity structures refining naive multiplicative structures in the motivic stable homotopy category. The proofs combine Gamma-homology computations and work due to Robinson giving rise to motivic obstruction theory. As an application we employ a motivic to simplicial delooping argument to show a uniqueness result for E-infinity structures on the K-theory Nisnevich presheaf of spectra.
Synthetic equivariant spectra for finite abelian groups and motivic homotopy theory
本文证明复数域上有限阿贝尔群等变动机谱范畴在素数完备化后等价于合成等变谱范畴的完备化，后者是等变谱的形变并范畴化等变完美偶滤过，其计算依赖于等变形式群律描述的等变代数配边同伦群。
Splitting of abelian varieties in motivic stable homotopy category
本文探讨阿贝尔簇的 motive 稳定同伦型，证明完美域上带有理点的阿贝尔簇在对应范畴中分裂出最高维胞腔，给出实数域上由 motive 和实点决定的具体分裂，并从 Chow-Witt 对应视角展开讨论。
The zeroth stable homotopy groups of motivic spheres over the integers
本文确定了整数上动机球面谱的第零积分Milnor-Witt茎，并建立了多种动机谱的细胞性及混合特征Dedekind域上的绝对纯性结果。
We study thick ideals in the stable motivic homotopy category SH(k) and in its subcategories of compact and of finite cellular objects. If k is a subfield of the complex or even the real numbers, then using comparison functors we find thick ideals corresponding to thick ideals in classical or Z/2-equivariant stable homotopy theory, respectively. We also study motivic Morava K-theories AK(n), for which we prove the motivic analogue of the decomposition of the Bousfield class of E(n) into Bousfield classes of K(i)'s over the complex numbers if p>2. In that case we also prove that AK(n)-acyclicit…
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 ge…
We show that if G is a finite constant group acting on a scheme X such that the order of G is invertible in the residue fields of X, then the G-equivariant motivic stable homotopy category of X is equivalent to the stabilization of the category of motivic G-spaces with finite étale transfers over X at the trivial representation sphere. Along the way we obtain several results of independent interest, among them: we construct and study norms in the motivic homotopy theory of stacks, and we extend the homotopy t-structure to DM-stacks and establish some favorable properties.
In this paper we begin the study of the (dual) Steenrod algebra of the motivic Witt cohomology spectrum $H_W\mathbb{Z}$ by determining the algebra structure of ${H_W\mathbb{Z}}_{**}H_W\mathbb{Z}$ over fields $k$ of characteristic not $2$ which are extensions of fields $F$ with $K^M_2(F)/2=0$. For example, this includes all fields of odd characteristic, as well as fields that are extensions of quadratically closed fields of characteristic $0$. After inverting $η$, this computes the $HW:=H_W\mathbb{Z}[η^{-1}]$-algebra ${HW}_{**}HW$. In particular, for the given base fields, this implies the $HW$…
The Multiplicative Structures on Motivic Homotopy Groups
该研究调和了Voevodsky与Dugger所用动机环谱同伦环上的乘法定义，指出二者关联虽初等且类似现象见于超对称领域，但学界此前未察觉该定义冲突及潜在影响，故以短注阐明。
If $f:S' \to S$ is a finite locally free morphism of schemes, we construct a symmetric monoidal "norm" functor $f_\otimes: \mathcal H_*(S') \to\mathcal H_*(S)$, where $\mathcal H_*(S)$ is the pointed unstable motivic homotopy category over $S$. If $f$ is finite étale, we show that it stabilizes to a functor $f_\otimes: \mathcal{SH}(S') \to \mathcal{SH}(S)$, where $\mathcal{SH}(S)$ is the $\mathbb P^1$-stable motivic homotopy category over $S$. Using these norm functors, we define the notion of a normed motivic spectrum, which is an enhancement of a motivic $E_\infty$-ring spectrum. The main co…
In this article we further the study of noncommutative numerical motives. By exploring the change-of-coefficients mechanism, we start by improving some of our previous main results. Then, making use of the notion of Schur-finiteness, we prove that the category NNum(k)_F of noncommutative numerical motives is (neutral) super-Tannakian. As in the commutative world, NNum(k)_F is not Tannakian. In order to solve this problem we promote periodic cyclic homology to a well-defined symmetric monoidal functor HP on the category of noncommutative Chow motives. This allows us to introduce the correct non…
Norms and Transfers in Motivic Homotopy Theory
本文证明动机同伦论中范数与转移相容，构造带转移动机空间的范数函子，细化无限 P¹-去环路机器，并赋予极有效 Hermitian K-理论谱赋范代数结构。
In this paper we study a model structure on a category of schemes with a group action and the resulting unstable and stable equivariant motivic homotopy theories. The new model structure introduced here samples a comparison to the one by Voevodsky and Hu-Kriz-Ormsby. We show that it allows to detect equivariant motivic weak equivalences on fixed points and how this property leads to a topologically convenient behavior of stable equivalences. We also prove a negative result concerning descent for equivariant algebraic K-theory.
Making the motivic group structure on the endomorphisms of the projective line explicit
本文在 Jouanolou 装置到射影直线的带点朴素同伦类上构造显式群结构，并证明 Cazanave 幺半群可经自然态射群完备化到由简单态射生成的子群。
Filtrations in $\mathbb{C}$-motivic stable homotopy theory
本文在复动机稳定同伦范畴中利用滤子谱模型研究三类滤子，计算了多种谱的切片及有效切片谱序列，验证了相关猜想并得到新结果。
Colored operads were introduced in the 1970's for the purpose of studying homotopy invariant algebraic structures on topological spaces. In this paper we introduce colored operads in motivic stable homotopy theory. Our main motivation is to uncover hitherto unknown highly structured properties of the slice filtration. The latter decomposes every motivic spectrum into its slices, which are motives, and one may ask to what extend the slice filtration preserves highly structured objects such as algebras and modules. We use colored operads to give a precise solution to this problem. Our approach m…
It is shown that the K-theory of every noetherian base scheme of finite Krull dimension is represented by a strict ring object in the setting of motivic stable homotopy theory. The adjective `strict' is used to distinguish between the type of ring structure we construct and one which is valid only up to homotopy. Both the categories of motivic functors and motivic symmetric spectra furnish convenient frameworks for constructing the ring models. Analogous topological results follow by running the same type of arguments as in the motivic setting.
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 the stable motivic homotopy groups $π_{s,w}$ of the 2-completion of the motivic sphere spectrum over $\mathbb{C}$. When arranged in the $(s,w)$-plane, these groups break into four different regions: a vanishing region, an $η$-local region that is entirely known, a $τ$-local region that is identical to classical stable homotopy groups, and a region that is not well-understood.
In this paper we explore the isotropic stable motivic homotopy category constructed from the usual stable motivic homotopy category, following the work of Vishik on isotropic motives (see [29]), by killing anisotropic varieties. In particular, we focus on cohomology operations in the isotropic realm and study the structure of the isotropic Steenrod algebra. Then, we construct an isotropic version of the motivic Adams spectral sequence and apply it to find a complete description of the isotropic stable homotopy groups of the sphere spectrum, which happen to be isomorphic to the $Ext$-groups of …
We give a brief introduction to tensor triangulated geometry, a brief introduction to various motivic categories, and then make some observations about the conjectural structure of the tensor triangulated spectrum of the Morel-Voevodsky stable homotopy category over a finite field.
We identify the equivariant structure of the filtered pieces of the motivic filtration defined by Bhatt, Morrow and Scholze on the topological Hochschild cohomology spectrum of polynomial algebras over $\mathbb{F}_p$.
For each prime $p$, we define a $t$-structure on the category $\widehat{S^{0,0}}/τ\text{-}\mathbf{Mod}_{harm}^b$ of harmonic $\mathbb{C}$-motivic left module spectra over $\widehat{S^{0,0}}/τ$, whose MGL-homology has bounded Chow-Novikov degree, such that its heart is equivalent to the abelian category of $p$-completed $BP_*BP$-comodules that are concentrated in even degrees. We prove that $\widehat{S^{0,0}}/τ\text{-}\mathbf{Mod}_{harm}^b$ is equivalent to $\mathcal{D}^b({{BP}_*{BP}\text{-}\mathbf{Comod}}^{ev})$ as stable $\infty$-categories equipped with $t$-structures. As an application, for…
Coarse moduli of motivic augmentations
本文提出动机增广的粗模空间，建立可表示性准则，计算实例并与Selmer簇比较，为Chabauty-Kim理论和整点界提供新工具。
Let $F$ and $k$ be perfect fields. The main goal of this paper is to investigate algebraic models for the Morel-Voevodsky unstable motivic homotopy category $\mathrm{Ho}(F)$ after $\mathbf{H}^{\mathbb{A}^1}k$ localization. More specifically, we extend results of Goerss to the $\mathbb{A}^1$-algebraic topology setting: we study the homotopy theory of the category $s\mathrm{coCAlg}_k(Sm_F)$ of presheaves of simplicial coalgebras over a field $k$ and their $τ$ and $\mathbb{A}^1$-localizations. For $k$ algebraically closed, we show that the unit of the adjunction $k^δ[-]\dashv(-)^{gp}$ determines …
On the Steenrod module structure of $\mathbb{R}$-motivic Spanier-Whitehead duals
本文研究了 $\mathbb{R}$-motivic 谱的 Spanier-Whitehead 对偶的上同调作为 Steenrod 代数模的结构恢复方法，并证明了 $\mathcal{A}^{\mathbb{R}}(1)$ 的 128 种模结构中有 16 种是自对偶的。
In this paper, we initiate a study of motivic homotopy theory at infinity. We use the six functor formalism to give an intrinsic definition of the stable motivic homotopy type at infinity of an algebraic variety. Our main computational tools include cdh-descent for normal crossing divisors, Euler classes, Gysin maps, and homotopy purity. Under $\ell$-adic realization, the motive at infinity recovers a formula for vanishing cycles due to Rapoport-Zink; similar results hold for Steenbrink's limiting Hodge structures and Wildeshaus' boundary motives. Under the topological Betti realization, the s…
We argue that the very effective cover of hermitian $K$-theory in the sense of motivic homotopy theory is a convenient algebro-geometric generalization of the connective real topological $K$-theory spectrum. This means the very effective cover acquires the correct Betti realization, its motivic cohomology has the desired structure as a module over the motivic Steenrod algebra, and that its motivic Adams and slice spectral sequences are amenable to calculations.
Norms and Hermitian $\mathrm{K}$-Theory
本文研究了动机谱中规范代数结构，展示了ko谱在希伯特K-理论中的应用，并证明了导向映射与规范的相容性。
Given a small simplicial category $\C$ whose underlying ordinary category is equipped with a Grothendieck topology $τ$, we construct a model structure on the category of simplicially enriched presheaves on $\C$ where the weak equivalences are the local weak equivalences of the underlying (non-enriched) simplicial presheaves. We show that this model category is a $t$-complete model topos and describe the Grothendieck topology $[τ]$ on the homotopy category of $\C$ that corresponds to this model topos. After we first review a proof showing that the motivic homotopy theory is not a model topos, w…
Weil restriction, normal bundles and motivic Thom spaces
本文通过Weil限制研究法丛与动机Thom空间，证明Weil限制保持向量丛并扩展法丛结果，进而建立与动机Thom空间的相容性，并在有理系数动机上同调中证明Weil限制映射与范数映射一致且保持Thom类。
Under a certain normalization assumption we prove that the $\Pro^1$-spectrum $\mathrm{BGL}$ of Voevodsky which represents algebraic $K$-theory is unique over $\Spec(\mathbb{Z})$. Following an idea of Voevodsky, we equip the $\Pro^1$-spectrum $\mathrm{BGL}$ with the structure of a commutative $\Pro^1$-ring spectrum in the motivic stable homotopy category. Furthermore, we prove that under a certain normalization assumption this ring structure is unique over $\Spec(\mathbb{Z})$. For an arbitrary Noetherian scheme $S$ of finite Krull dimension we pull this structure back to obtain a distinguished …
We study symmetric powers in the homotopy categories of abstract closed symmetric monoidal model categories, in both unstable and stable settings. As an outcome, we prove that symmetric powers preserve the Nisnevich and etale homotopy type in the unstable and stable motivic homotopy theories of schemes over a base. More precisely, if f is a weak equivalence of motivic spaces, or a weak equivalence between positively cofibrant motivic spectra, with respect to the Nisnevich or etale topology on schemes, then all symmetric powers Sym^n(f) are weak equivalences too. This gives left derived symmetr…
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