For an oriented cohomology theory A and a relative cellular space X, we decompose the A-motive of X into a direct sum of twisted motives of the base spaces. We also obtain respective decompositions of the A-cohomology of X. Applying them, one can compute A(X), where X is an isotropic projective homogeneous variety and A means algebraic K-theory, motivic cohomology or algebraic cobordism MGL.
We prove that, over a perfect field, Bredon motivic cohomology can be computed by Suslin-Friedlander complexes of equivariant equidimensional cycles. Partly based on this result we completely identify Bredon motivic cohomology of a quadratically closed field and of a euclidian field in weights 1 and $σ$. We also prove that Bredon motivic cohomology of an arbitrary field in weight 0 with integer coefficients coincides (as abstract groups) with Bredon cohomology of a point.
We derive a version of the Rothenberg-Steenrod, fiber-to-base, spectral sequence for cohomology theories represented in model categories of simplicial presheaves. We then apply this spectral sequence to calculate the equivariant motivic cohomology of the general linear group with a general Gm-action, this coincides with the equivariant higher Chow groups. Some of the equivariant motivic cohomology of a Stiefel variety with a general Gm-action is deduced as a corollary.
We prove that for a finitely generated field over an infinite perfect field k, and for any integer n, the (n,n)-th MW-motivic cohomology group identifies with the n-th Milnor-Witt K-theory group of that field
We extend results of Colliot-Thélène and Raskind on the $\mathcal{K}_2$-cohomology of smooth projective varieties over a separably closed field $k$ to the étale motivic cohomology of smooth, not necessarily projective, varieties over $k$. Some consequences are drawn, such as the degeneration of the Bloch-Lichtenbaum spectral sequence for any field containing $k$.
The main result of this paper is a computation of the motivic cohomology of varieties of n \times m-matrices of of rank m, including both the ring structure and the action of the reduced power operations. The argument proceeds by a comparison of the general linear group-scheme with a Tate suspension of a space which is A1-equivalent to projective n - 1-space with a disjoint basepoint.
We construct an algebraic-cycle based model for the motivic cohomology on the category of schemes of finite type over a field, where schemes may admit arbitrary singularities and may be non-reduced. We show that our theory is functorial on the category, that it detects nilpotence, and that its restriction to the subcategory of smooth schemes agrees with the pre-existing motivic cohomology theory, which is the higher Chow theory of S. Bloch (Adv. Math., 1986). A few structures and applications are discussed.
BY using Green-Griffiths' results on tangent spaces to algebraic cycles [4], we study the tangent space to $CH^{2}(X,1)$, where $X$ is a nonsingular projective curve over a field $k$ of characteristic $0$.
For any cohomology theory $H$ that can be factorized through (the Morel-Voevodsky's triangulated motivic homotopy category) $SH^{S^1}(k)$ we establish the $SH^{S^1}(k)$-functoriality 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 $DM(k)$, 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 generalization of the results of a previous pape…
We discuss the structure of integral etale motivic cohomology groups of smooth and projective schemes over algebraically closed fields, finite fields, local fields, and arithmetic schemes.
We show that the constructions done in part I generalize their classical counterparts: firstly, the classical Beilinson regulator is induced by the abstract Chern class map from $BGL$ to the Deligne cohomology spectrum. Secondly, Arakelov motivic cohomology is a generalization of arithmetic $K$-theory and arithmetic Chow groups. Finally, we give a conceptual explanation of the height pairing: it is given by the natural pairing of motivic homology and Arakelov motivic cohomology.
A rigidity property for the homotopy invariant stable linear framed presheaves is established. As a consequence a variant of Gabber rigidity theorem is obtained for a cohomology theory representable in the motivic stable homotopy category by a $ϕ$-torsion spectrum with $ϕ\in\mathrm{GW}(k)$ of rank coprime to the (exponential) characteristic of the base field $k$. It is shown that the values of such cohomology theories at an essentially smooth Henselian ring and its residue field coincide. The result is applicable to cohomology theories representable by $n$-torsion spectra as well as to the one…
We give a presentation of the motivic cohomology ring of the complement of a hyperplane arrangement considered as algebra over the motivic cohomology of the ground field.
We discuss duality pairings on integral étale motivic cohomology groups of regular and proper schemes over algebraically closed fields, local fields, finite fields, and arithmetic schemes.
We build a ring spectrum representing Milnor-Witt motivic cohomology, as well as its étale local version and show how to deduce out of it three other theories: Borel-Moore homology, cohomology with compact support and homology. These theories, as well as the usual cohomology, are defined for singular schemes and satisfy the properties of their motivic analog (and more), up to considering more general twists. In fact, the whole formalism of these four theories can be functorially attached to any ring spectrum, giving finally maps between the Milnor-Witt motivic ones to the classical motivic one…
In this paper, we present a general approach to establish motivic cohomology and build part of its six operations formalism. Applying this together with symplectic orientation on MW-motivic cohomology, we discuss the embedding theorem of effective Chow-Witt motives.
Due to the work of many authors in the last decades, given an algebraic orbifold (smooth proper Deligne-Mumford stack with trivial generic stabilizer), one can construct its orbifold Chow ring and orbifold Grothendieck ring, and relate them by the orbifold Chern character map, generalizing the fundamental work of Chen-Ruan on orbifold cohomology. In this paper, we extend this theory naturally to higher Chow groups and higher algebraic K-theory, mainly following the work of Jarvis-Kaufmann-Kimura and Edidin-Jarvis-Kimura.
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.
In this paper we give a proof of the Bloch-Kato conjecture relating motivic cohomology and etale cohomology. It is a corrected version of the paper with the same title which posted earlier.
We examine the slice spectral sequence for the cohomology of singular schemes with respect to various motivic T-spectra, especially the motivic cobordism spectrum. When the base field k admits resolution of singularities and X is a scheme of finite type over k, we show that Voevodsky's slice filtration leads to a spectral sequence for MGL(X) whose terms are the motivic cohomology groups of X defined using the cdh-hypercohomology. As a consequence, we establish an isomorphism between certain geometric parts of the motivic cobordism and motivic cohomology of X. A similar spectral sequence for th…
In this paper we construct an analog of Steenrod operations in motivic cohomology and prove their basic properties including the Cartan formula, the Adem relations and the realtions to characteristic classes.
We introduce a new algebraic-cycle model for the motivic cohomology theory of truncated polynomials $k[t]/(t^m)$ in one variable. This approach uses ideas from the deformation theory and non-archimedean analysis, and is distinct from the approaches via cycles with modulus. We compute the groups in the Milnor range when the base field is of characteristic $0$, and prove that they give the Milnor $K$-groups of $k[t]/(t^m)$, whose relative part is the sum of the absolute Kähler differential forms.
We use an analogue of Karoubi's construction in the motivic situation to give some cohomology operations in motivic cohomology. We prove many properties of these operations, and we show that they coincide, up to some nonzero constants, with the reduced power operations in motivic cohomology originally constructed by Voevodsky. The relation of our construction to Voevodsky's is, roughly speaking, that of a fixed point set to its associated homotopy fixed point set.
This note addresses the motivic nature of some classical cohomological results due to Lefschetz, namely the primitive decomposition (for the cohomology of smooth projective varieties), and, secondly, the splitting of the cohomology of a complete intersection into the "fixed" and "variable part".
A smooth projective scheme $X$ over a field $k$ is said to satisfy the Rost nilpotence principle if any endomorphism of $X$ in the category of Chow motives that vanishes on an extension of the base field $k$ is nilpotent. We show that an étale motivic analogue of the Rost nilpotence principle holds for all smooth projective schemes over a perfect field. This provides a new approach to the question of Rost nilpotence and allows us to obtain an elegant proof of Rost nilpotence for surfaces, as well as for birationally ruled threefolds over a field of characteristic $0$.
We present a formal scheme based cycle model for the motivic cohomology of the fat points defined by the truncated polynomial rings $k[t]/(t^m)$ with $m \geq 2$, in one variable over a field $k$. We compute their Milnor range cycle class groups when the field has sufficiently many elements. With some aids from rigid analytic geometry and the Gersten conjecture for the Milnor $K$-theory resolved by M. Kerz, we prove that the resulting cycle class groups are isomorphic to the Milnor $K$-groups of the truncated polynomial rings, generalizing a theorem of Nesterenko-Suslin and Totaro.
This paper introduces a new cohomology theory for schemes of finite type over an arithmetic ring. The main motivation for this Arakelov-theoretic version of motivic cohomology is the conjecture on special values of $L$-functions and zeta functions formulated by the second author. Taking advantage of the six functors formalism in motivic stable homotopy theory, we establish a number of formal properties, including pullbacks for arbitrary morphisms, pushforwards for projective morphisms between regular schemes, localization sequences, $h$-descent. We round off the picture with a purity result an…
In this paper we prove the 2-local part of the Beilinson-Lichtenbaum conjectures on tosion in motivic cohomology. In particular we prove the Milnor conjecture relating Milnor's K-theory and the Galois cohomology with Z/2-coefficients. This paper is a new version of the previously distributed preprint "The Milnor Conjecture".
We perform Hochschild homology calculations in the algebro-geometric setting of motives. The motivic Hochschild homology coefficient ring contains torsion classes which arise from the mod-$p$ motivic Steenrod algebra and from generating functions on the natural numbers with finite non-empty support. Under the Betti realization, we recover Bökstedt's calculation of the topological Hochschild homology of finite prime fields.
We explicitly calculate some Gromov--Witten correspondences determined by maps of labeled curves of genus zero to the moduli spaces of labeled curves of genus zero. We consider these calculations as the first step towards studying the self--referential nature of motivic quantum cohomology.
Let S be an essentially smooth scheme over a field of characteristic exponent c. Let MGL and HZ denote the algebraic cobordism spectrum and the motivic cohomology spectrum over S, respectively. We show that the canonical map MGL/(a1, a2, ...) -> HZ induced by the additive orientation of motivic cohomology becomes an equivalence after inverting c. As an application, we prove the convergence of the Atiyah-Hirzebruch spectral sequence for all Z[1/c]-local Landweber exact motivic spectra.
We prove a mixed characteristic analog of the Beilinson-Lichtenbaum Conjecture for p-adic motivic cohomology. It gives a description, in the stable range, of p-adic motivic cohomology (defined using algebraic cycles) in terms of differential forms. This generalizes a result of Geisser from small Tate twists to all twists and uses as a critical new ingredient the comparison theorem between syntomic complexes and p-adic nearby cycles proved recently in Colmez-Niziol.
We present a version of projective bundle theorem in MW-motives (resp. Chow-Witt rings), which says that $\widetilde{CH}^*(\mathbb{P}(E))$ is determined by $\widetilde{CH}^*(X)$, $\widetilde{CH}^*(X,det(E)^{\vee})$, $CH^*(X)$ and $Sq^2$ for smooth quasi-projective schemes $X$ and vector bundles $E$ over $X$ with $e(E^{\vee})=0\in H^n(X,W(det(E)))$, provided that $_2CH^*(X)=0$. As an application, we compute the MW-motives of blow-ups with smooth centers. Moreover, we discuss the invariance of Chow-Witt cycles of projective bundles under automorphisms of vector bundles.
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 …
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.
These lecture notes cover four topics. There is a proof of the fact that the functors represented by the motivic Eilenberg-Maclane spaces on the motivic homotopy category coincide with the motivic cohomology defined in terms of the motivic complexes. There is a description of the equivariant motivic homotopy category for a finite flat group scheme (over a noetherian base) together with a new characterization of A^1-equivalences. There is a part where we introduce a class of sheaves called solid sheaves. Finally there is a part where we study functors of the form X -> X/G and X -> X^W and show …
We use Burgos theory of arithmetic Chow groups to exhibit a realization of the degree zero part of the polylogarithm on abelian schemes in Deligne-Beilinson cohomology.
We prove equivariant versions of the Beilinson-Lichtenbaum conjecture for Bredon motivic cohomology of smooth complex and real varieties with an action of the group of order two. This identifies equivariant motivic and topological invariants in a large range of degrees.
We study polynomials with complex coefficients which are nondegenerate in two senses, one of Kouchnirenko and the other with respect to its Newton polyhedron, through data on contact loci and motivic nearby cycles. Introducing an explicit description of these quantities we can answer in part to questions concerning the motivic nearby cycles of restriction functions and the integral identity conjecture in the context of Newton nondegenerate polynomials. Furthermore, in the nondegeneracy in the sense of Kouchnirenko, we give calculations on cohomology groups of the contact loci.
We present a geometric construction of push-forward maps along projective morphisms for cohomology theories representable in the stable motivic homotopy category assuming that the element corresponding to the stable Hopf map is inverted in the coefficient ring of the theory. The construction is parallel to the one given by A. Nenashev for derived Witt groups. Along the way we introduce cohomology groups twisted by a formal difference of vector bundles as cohomology groups of a certain Thom space and compute twisted cohomology groups of projective spaces.
Motivic homotopy theory with ramification filtrations
RIKEN iTHEMS（理化学研究所 iTHEMS）；Institute for Fundamental Mathematics, Communication Science Laboratories, NTT, Inc.（日本电信电话株式会社 通信科学基础数学研究所）；Graduate School of Mathematical Sciences, the University of Tokyo（东京大学大学院数理科学研究科）
