Federico Scavia
Combining work of Peyre, Colliot-Thélène and Voisin, we give the first example of a finite group $G$ such that the motivic class of its classifying stack $BG$ in Ekedahl's Grothendieck ring of stacks over $\mathbb{C}$ is non-trivial and $BG$ has trivial unramified Brauer group.
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Unramified Motivic Multiple Mixed Values
研究多重混合值何时处于一级的问题，利用Brown等人的下降理论，确定了深度小于四的所有非分歧动机多重混合值及一些无界深度的族，并提出关于大于三深度的非分歧MMV的猜想。
On Some Unramified Families of Motivic Euler Sums
本文应用 Glanois 的无分歧判据构造两类无分歧动机 Euler 和族，并在解析恒等式成立的假设下证明其中一族与动机 MZV 的具体关系。
Let BG be a classifying variety for an exceptional simple simply connected algebraic group G. We compute the degree 3 unramified Galois cohomology of BG with values in Q/Z(2) over an arbitrary field F. Combined with a paper by Merkurjev, this completes the computation of these cohomology groups for G semisimple simply connected over all fields. These computations provide another example of a simple simply connected group G such that BG is not stably rational.
We compute the class of the classifying stack of the special orthogonal group in the Grothendieck ring of stacks, and check that it is equal to the multiplicative inverse of the class of the group.
Unramified Motivic Alternating Multiple Mixed Values
本文研究多重混合值的交错版本，利用Brown和Glanois的下降理论识别五族非分歧交错多重混合值，并猜想所有此类值均已被给出。
We prove that the class of the classifying stack $B PGL_n$ is the multiplicative inverse of the class of the projective linear group $PGL_n$ in the Grothendieck ring of stacks for $n = 2$ and $n = 3$ under mild conditions on the base field $k$. In particular, although it is known that the multiplicativity relation $\{T\} = \{S\}\cdot\{PGL_n\}$ does not hold for all $PGL_n$-torsors $T \to S$, it holds for the universal $PGL_n$-torsors for said $n$.
Let $G$ be a finite group, and let $\{B_{\mathbb{C}}G\}$ the class of its classifying stack $B_{\mathbb{C}}G$ in Ekedahl's Grothendieck ring of algebraic $\mathbb{C}$-stacks $K_0(\operatorname{Stacks}_{\mathbb{C}})$. We show that if $B_{\mathbb{C}}G$ has the mixed Tate property, the invariants $H^i(\{B_{\mathbb{C}}G\})$ defined by T. Ekedahl are zero for all $i\neq 0$. We also extend Ekedahl's construction of these invariants to fields of positive characteristic.
We compute the class of the classifying stack of the exceptional algebraic group $G_2$ and of the spin groups $\mathrm{Spin}_7$ and $\mathrm{Spin}_8$ in the Grothendieck ring of stacks, and show that they are equal to the inverse of the class of the corresponding group. Furthermore, we show that the computation of the motivic classes of the stacks $\mathscr{B}\mathrm{Spin}_n$ can be reduced to the computation of the classes of $\mathscr{B} Δ_n$, where $Δ_n\subset \mathrm{Pin}_n$ is the "extraspecial $2$-group", the preimage of the diagonal matrices under the projection $\mathrm{Pin}_n\to \math…
Higher Chow groups with finite coefficients and refined unramified cohomology
本文证明域上拟射影等维概形的Bloch有限系数高阶圈类映射自然嵌入涉及Schreieder精细未分歧上同调的长正合序列，证明后者满足局部化序列，并猜想其为动机上同调理论。
Auel-Bigazzi-Böhning-Graf von Bothmer proved that if a proper smooth variety $X$ over a field $k$ of characteristic $p>0$ has universally trivial Chow group of $0$-cycles, the cohomological Brauer group of $X$ is universally trivial as well. In this paper, we generalize their argument to arbitrary unramified mod $p$ étale motivic cohomology groups. We also see that the properness assumption on the variety $X$ can be dropped off by using the Suslin homology together with a certain tame subgroup of the unramified cohomology group.
For $k$ a perfect field of characteristic $p>0$ and $G/k$ a split reductive group with $p$ a non-torsion prime for $G,$ we compute the mod $p$ motivic cohomology of the geometric classifying space $BG_{(r)}$, where $G_{(r)}$ is the $r$th Frobenius kernel of $G.$ Our main tool is a motivic version of the Eilenberg-Moore spectral sequence, due to Krishna. For a flat affine group scheme $G/k$ of finite type, we define a cycle class map from the mod $p$ motivic cohomology of the classifying space $BG$ to the mod $p$ étale motivic cohomology of the classifying stack $\mathcal{B}G.$ This also gives …
Let $G$ be a reductive group defined over an algebraically closed field of characteristic $0$ such that the Dynkin diagram of $G$ is the disjoint union of diagrams of types $G_{2}, F_{4}, E_{6}, E_{7}, E_{8}$. We show that the degree $3$ unramified cohomology of the classifying space of $G$ is trivial. In particular, combined with articles by Merkurjev \cite{Mer17} and the author \cite{Baek}, this completes the computations of degree $3$ unramified cohomology and reductive invariants for all split semisimple groups of a homogeneous Dynkin type.
Divisibility phenomena in motivic Bloch--Ogus theory
本文研究了光滑项目ive变种上的动机Bloch-Ogus理论中的可除现象，证明了函数域的MilnorK群中无分歧类的子群包含在n可除元素子群中，并推广到任意双度数的动机上同调，进一步证明当k为有限或可分闭域且l为k中可逆素数时，X的动机上同调的Bloch-Ogus滤波中除了最后一步外都是l可除的，且推广到任意准项目ivek-方案。
Let k be a field, let G be a finite group and let T be a split k-torus on which G acts multiplicatively, and for every m greater than 1 denote by T[m] the m-torsion subgroup of T. Under a suitable assumption on m, we show that the motivic class of the classifying stack of the semi-direct product of T[m] and G in K_0(Stacks_k) is trivial. As a consequence, we prove that the motivic class of BW is trivial for a large class of complex reflection groups W.
If $G$ is a presheaf of groupoids on a small site, and $A$ is a sheaf of abelian groups, we prove that the sheaf cohomology group $H^2 (BG, A)$ is in bijection with a set of central extensions of $G$ by $A$. We use this result to study the motivic cohomology of the Nisnevich classifying space $BG$, when $G$ is a presheaf of groups on the smooth Nisnevich site over a field, and particularly when $G = PGL_{n}$. Finally, we show that, when $p$ is an odd prime, the Chow ring of the classifying space of $PGL_{p}$ injects into the motivic cohomology of the Nisnevich classifying space $BPGL_{p}$, ove…
Odd rank vector bundles in eta-periodic motivic homotopy theory
本文研究eta-周期motivic稳定同伦范畴中奇秩向量丛的类截面行为，探讨了motivic环谱的SLc-定向与代数群étale分类空间的推论，并计算了对角化群的分类空间。
Motivic cohomology of the Nisnevich classifying space of even Clifford groups
本文计算了分裂偶 Clifford 群的 Nisnevich 分类空间的模 2 动机上同调环，并指出其与 spin 群情形的主要差异在于第二个精细 Stiefel-Whitney 类非平凡。
Integral Motivic Cohomology of $BSO_{4}$
本文计算了 $BSO_{4}$ 的积分动机上同调，借鉴 Guillot 和 Yagita 的方法，为 $BG_{2}$ 的类似计算奠定基础。
For a linear algebraic group $G$ over a field $k$, we define an equivariant version of the Voevodsky's motivic cobordism $MGL$. We show that this is an oriented cohomology theory with localization sequence on the category of smooth $G$-schemes and there is a natural transformation from this functor to the functor of equivariant motivic cohomology. We give several applications. In particular, we use this equivariant motivic cobordism to study the cobordism ring of the classifying spaces and the cycle class maps from the algebraic to the singular cohomology of such spaces. This theory of motivic…
In this paper, we study the Nisnevich sheafification $\mathcal{H}^1_{\acute{e}t}(G)$ of the presheaf associating to a smooth scheme the set of isomorphism classes of $G$-torsors, for a reductive group $G$. We show that if $G$-torsors on affine lines are extended, then $\mathcal{H}^1_{\acute{e}t}(G)$ is homotopy invariant and show that the sheaf is unramified if and only if Nisnevich-local purity holds for $G$-torsors. We also identify the sheaf $\mathcal{H}^1_{\acute{e}t}(G)$ with the sheaf of $\mathbb{A}^1$-connected components of the classifying space ${\rm B}_{\acute{e}t}G$. This establishe…
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 $F$ be a field of characteristic zero admitting a biquadratic field extension. We give an example of a torus $G$ over $F$ whose classifying stack $BG$ is stably rational and such that $\{BG\}\{G\}\neq 1$ in the Grothendieck ring of algebraic stacks over $F$. We also give an example of a finite étale group scheme $A$ over $F$ such that $BA$ is stably rational and $\{BA\}\neq 1$.
In the present article we investigate ordinary and equivariant Rost motives. We provide an equivariant motivic decomposition of the variety X of full flags of a split semisimple algebraic group over a smooth base scheme, study torsion subgroup of the Chow group of twisted forms of X, define some equivariant Rost motives over a field and ordinary Rost motives over a general base scheme, and relate equivariant Rost motives with classifying spaces of some algebraic groups.
The Chow Ring of the 2-Sylow Subgroup of $\mathrm{GL}(4,2)$
本文计算$\mathrm{Syl}_2(\mathrm{GL}(4,2))$的motivic étale分类空间的模2 Chow环，并提出一种源自Totaro且已自动化的通用策略，以快速计算更多同类Chow环。
Torsion birational motives of surfaces and unramified cohomology
该研究探讨了曲面的torsion birational motives与unramified cohomology的关系，并扩展了Kahn的结果，同时给出了违反积分Hodge猜想的例子。
This is the first of a series of papers, where we investigate hierarchies of generalized {L}üroth problems on the hierarchy of rationality, starting with the obvious hierarchy between the rationality and the ruledness. Our primary goal here was to construct very general necessary conditions for a smooth, not necessary proper, scheme of finite type over the perfect base field $k$ to be "retract $(-i)$-rational". We achieve this goal by constructing "stably birationalized Nisnevich subsheaf" $S_{sb}$ inside any Morel's unramified sheaf $S,$ where $S_{sb}$ coincides with $S$ on proper smooth $k$-…
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…
Extending [14], we obtain a complete description of the motivic cohomology with ${\mathbb Z}/2$-coefficients of the Nisnevich classifying space of the spin group $Spin_n$ associated to the standard split quadratic form. This provides us with very simple relations among subtle Stiefel-Whitney classes in the motivic cohomology of Čech simplicial schemes associated to quadratic forms from $I^3$, which are closely related to $Spin_n$-torsors over the point. These relations come from the action of the motivic Steenrod algebra on the second subtle Stiefel-Whitney class. Moreover, exploiting the rela…
A Serre-type spectral sequence for motivic cohomology
本文构造了动机构上同调的Serre型谱序列，并将其应用于射影一般线性群Nisnevich分类空间的动机构上同调计算，从而给出Severi-Brauer簇动机的显式描述。
We give an alternate formulation of pseudo-coherence over an arbitrary derived stack X. The full subcategory of pseudo-coherent objects forms a stable sub-infinity-category of the derived category associated to X. Using relative Tor-amplitude we define a derived stack classifying pseudo-coherent objects. For reasonable base schemes, this classifies the bounded derived category. In the case that X is a projective derived scheme flat over the base, we show the moduli is locally geometric and locally of almost finite type. Using this result, we prove the existence of a derived motivic Hall algebr…
A coproduct obstruction for derived unramified cohomology
本文证明在指数特征可逆后，第一个导出非分支函子与任意小直和交换，并描述第二个导出非分支函子的范畴障碍，应用于曲面的扭阶界和对应检测。
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.
This paper sets out basic properties of motivic twisted K-theory with respect to degree three motivic cohomology classes of weight one. Motivic twisted K-theory is defined in terms of such motivic cohomology classes by taking pullbacks along the universal principal BG_m-bundle for the classifying space of the multiplicative group scheme. We show a Kuenneth isomorphism for homological motivic twisted K-groups computing the latter as a tensor product of K-groups over the K-theory of BG_m. The proof employs an Adams Hopf algebroid and a tri-graded Tor-spectral sequence for motivic twisted K-theor…
Motivic homotopy theory with ramification filtrations
本文推广Morel–Voevodsky动机同伦理论，发展了带分歧滤子的同伦理论，刻画野生分歧等非𝔸¹同伦不变现象，证明相关上同调理论可在该范畴中表示并恢复已知构造。
Motivic Pontryagin classes and hyperbolic orientations
本文提出了motivic环谱的双曲定向概念，证明其对应Pontryagin类理论，并构造了万有双曲定向eta-周期交换motivic环谱，为研究向量丛示性类提供了新框架。
Motivic homotopy theory of the classifying stack of finite groups of Lie type
本文将Lie型有限群分类栈的Tate动机与旗流形的等变Tate动机相关联，将Edidin-Graham的经典结果推广至高阶等变Chow群，并计算了G-zips栈的动机。
The 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 $DM^{eff}_{gm}[1/p]$ (of effective geometric Voevodsky's motives with $Z[1/p]$-coefficients). It follows that $DM^{eff}_{gm}[1/p]$ could be endowed with a Chow weight structure $w_{Chow}$ whose heart is $Chow^{eff}[1/p]$ (weight structures were introduced in a preceding paper, where the existence of $w_{Chow}$ for $DM^{eff}_{gm}Q$ was also proved). As sho…
Galois descent for motivic theories
本文给出了纤维化加性范畴成为stack的条件，适用于多种motive理论，并简化证明了motivic Galois群序列的正合性及Chow-Lefschetz motives的Galois群计算。
Using symmetrized Grassmannians we give an algebraic geometric presentation, in the level of classifying spaces, of the Chern character and its relation to Chern classes. This allows one to define, for any projective variety $X$, a Chern character map from the "holomorphic K-theory of X to its morphic cohomology (introduced by Friedlander and Lawson). The holomorphic K-theory of $X$, introduced by Lawson, Lima-Filho and Michelsohn and also by Friedlander and Walker, is defined in terms a group-completion of the space of algebraic morphisms from X into BU. It has been further studied by the aut…
Following [14], we compute the motivic cohomology ring of the Nisnevich classifying space of the unitary group associated to the standard split hermitian form of a quadratic extension. This provides us with subtle characteristic classes which take value in the motivic cohomology of the Čech simplicial scheme associated to a hermitian form. Comparing these new classes with subtle Stiefel-Whitney classes arising in the orthogonal case, we obtain relations among the latter ones holding in the motivic cohomology of the Čech simplicial scheme associated to a quadratic form divisible by a 1-fold Pfi…
We construct geometric models for classifying spaces of linear algebraic groups in G-equivariant motivic homotopy theory, where G is a tame group scheme. As a consequence, we show that the equivariant motivic spectrum representing the homotopy K-theory of G-schemes (which we construct as an E-infinity-ring) is stable under arbitrary base change, and we deduce that homotopy K-theory of G-schemes satisfies cdh descent.
Polylogarithm Variations and Motivic Extensions of $\mathbb{Q}$ by $\mathbb{Q}(m)$
本文通过在 Tate 椭圆曲线限制的幂单基本群 Lie 代数上实现导子，将 Deligne、Goncharov 等人关于动形 Galois 群深度 2 二次关系的结果从 $N=1$ 推广至所有 $N \ge 1$，并揭示了其与 $K$-群和 Eisenstein 级数的相容性。
Motivic Chern classes of Schubert cells, Hecke algebras, and applications to Casselman's problem
本文计算了旗流形中 Schubert 胞的 motivic Chern 类，证明其由 Hecke 代数中的 Demazure-Lusztig 算子递归决定，并利用此递归与 Langlands 对偶性证明了关于标准基与 Casselman 基过渡矩阵系数分解及全纯性质的两个猜想。
A main theme of the paper is a conjecture of Bloch-Kato on the image of $p$-adic regulator maps for a proper smooth variety $X$ over an algebraic number field $k$. The conjecture for a regulator map of particular degree and weight is related to finiteness of two arithmetic objects: One is the $p$-primary torsion part of the Chow group in codimension 2 of $X$. Another is an unramified cohomology group of $X$. As an application, for a regular model ${\mathscr X}$ of $X$ over the integer ring of $k$, we show an injectivity result on torsion of a cycle class map from the Chow group in codimension …
Cluster construction of the second motivic Chern class
本文通过簇坐标构造了任意分裂单连通单代数群 G 的第二泛动机陈类的显式余循环，并给出其分量构造、不同簇间的上边缘关系及在量子形变等方面的应用。
Chow groups with twisted coefficients
本文推广Rost的Chow群定义至非挠性系数，计算多个有限群分类空间的扭曲Chow群，并揭示其与扭曲动机上同调之间的满射但非同构关系。
We apply the machinery developed by the first-named author to the K-theory of coherent G-sheaves on a finite type G-scheme X over a field, where G is a finite group. This leads to a definition of G-equivariant higher Chow groups (different from the Chow groups of classifying spaces constructed by Totaro and generalized to arbitrary X by Edidin-Graham) and an Atiyah-Hirzebruch spectral sequence from the G-equivariant higher Chow groups to the higher K-theory of coherent G-sheaves on X. This spectral sequence generalizes the spectral sequence from motivic cohomology to K-theory constructed by Bl…
A quotient of the Bruhat-Tits building classifying maximal matrix orders over curves
本文将布吕阿-蒂茨树的商图构造推广到高维，研究曲线上极大矩阵序的商建筑，明确分裂极大序顶点的结构及与尖点区域的关联，为算术群和序的研究提供几何框架。
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