We define a new version of $\mathbb A^1$-homology, called cellular $\mathbb A^1$-homology, for smooth schemes over a field that admit an increasing filtration by open subschemes with cohomologically trivial closed strata. We provide several explicit computations of cellular $\mathbb A^1$-homology and use them to determine the $\mathbb A^1$-fundamental group of a split reductive group over an arbitrary field, thereby obtaining the motivic version of Matsumoto's theorem on universal central extensions of split, semisimple, simply connected algebraic groups. As applications, we uniformly explain and generalize results due to Brylinski-Deligne and Esnault-Kahn-Levine-Viehweg, determine the isomorphism classes of central extensions of such an algebraic group by an arbitrary strictly $\mathbb A^1$-invariant sheaf and also reprove classical results of E. Cartan on homotopy groups of complex Lie groups.
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.
Let $k$ be a field, let $R$ be a commutative ring, and assume the exponential characteristic of $k$ is invertible in $R$. In this note, we prove that isomorphisms in Voevodsky's triangulated category of motives $\mathcal{DM}(k;R)$ are detected by motivic homology groups of base changes to all separable finitely generated field extensions of $k$. It then follows from previous conservativity results that these motivic homology groups detect isomorphisms between certain spaces in the pointed motivic homotopy category $\mathcal{H}(k)_*$.
We show that motivic homology, motivic Borel-Moore homology and higher Chow groups satisfy homological descent for hyperenvelopes, and l-hyperenvelopes after inverting l.
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 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 $\mathbb{F}_p$.
By defining and studying functorial properties of the Borel-Moore motivic homology, we identify the heart of Bondarko-Hébert's weight structure on Beilinson motives with Corti-Hanamura's category of Chow motives over a base, therefore answering a question of Bondarko.
We generalize some classical results on Chow group of an abelian variety to semiabelian varieties and to motivic (co)homology, using a result of Ancona--Enright-Ward--Huber on a decomposition of the motive of a semiabelian variety in the Voevodsky's category.
Motivic characteristic classes of possibly singular algebraic varieties are homology class versions of motivic characteristics, not classes in the so-called motivic (co)homology. This paper is a survey on them with more emphasis on capturing infinitude finitely and on the motivic nature, in other words, the scissor relation or additivity.
This note concerns a weak form of Parshin's conjecture, which states that the rational motivic Borel--Moore homology of a quasiprojective variety of dimension $m$ over a finite field in bidegree $(s,t)$ vanishes for $s>m+t$. It is shown that this conjecture holds if and only if the cyclic action on the motivic cohomology of an Artin--Schreier field extension in bidegree $(i,j)$ is trivial if $i<j$.
For a smooth proper variety over a $p$-adic field, the Brauer group and abelian fundamental group are related to the higher Chow groups by the Brauer-Manin pairing and the class field theory. We generalize this relation to smooth (possibly non-proper) varieties, using the motivic homology and the tame version of Wiesend's ideal class group. Several examples are discussed.
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 a triangulated motivic category admits categorical Thom isomorphisms for vector bundles with an additional structure if and only if the generalized motivic cohomology theory represented by the tensor unit object admits Thom classes. We also show that the stable $\mathbb{A}^1$-derived category does not admit Thom isomorphisms for oriented vector bundles and, more generally, for symplectic bundles. In order to do so we compute the first homology sheaves of the motivic sphere spectrum and show that the class in the coefficient ring of $\mathbb{A}^1$-homology corresponding to the seco…
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…
We describe certain criteria for a motif $M$ to be $r$-effective, i.e., to belong to the $r$th Tate twist $Obj DM^{eff}_{gm,R}(r)=Obj DM^{eff}_{gm,R} \otimes L^{\otimes r}$ of effective Voevodsky motives (for $r\ge 1$; $R$ is the coefficient ring). In particular, $M$ is 1-effective if and only if a complex whose terms are certain Chow groups of zero-cycles is acyclic. The dual to this statement checks whether an effective motif $M$ belongs to the subcategory of $DM^{eff}_{gm,R}$ generated by motives of varieties of dimension $\le r$. These criteria are formulated in terms of the Chow-weight (c…
We give an account of the construction of the Bhatt--Morrow--Scholze motivic filtration on topological cyclic homology and related invariants, focusing on the case of equal characteristic $p$ and the connections to crystalline and de Rham--Witt theory.
We establish a motivic version of the May Nilpotence Conjecture: if E is a normed motivic spectrum that satisfies $E \wedge HZ \simeq 0$, then also $E \wedge MGL \simeq 0$. In words, motivic homology detects vanishing of normed modules over the algebraic cobordism spectrum.
The paper is suspended. The reason: as was noted by prof. H. Esnault, Theorem 2.1.1 of the previous version (as well as the related Theorem 6.1.1 of http://arxiv.org/PS_cache/math/pdf/9908/9908037v2.pdf of D. Arapura and P. Sastry) is wrong unless one assumes H to be a generic hyperplane section. Hence the proofs of all results starting from 2.3 contain gaps. The author hopes to correct this (somehow) in a future version. At least, most of the results follow from certain "standard" motivic conjectures (see part 1 of Remark 3.2.4 in the previous version). If the author would not find a way to p…
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…
In the first half of this article we define a new weight homology functor on Voevodsky's category of effective motives, and investigate some of its properties. In special cases we recover Gillet-Soulé's weight homology, and Geisser's Kato-Suslin homology. In the second half, we consider the notions of "co-étale" and "reduced" motives, and use the notions to a prove a theorem comparing motivic homology to étale motivic homology. Due to the first author's Ph.D. thesis arXiv:1305.5349 we do not have to restrict to smooth schemes.
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…
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…
For an arithmetical scheme X, K. Kato introduced a certain complex of Gersten-Bloch-Ogus type whose component in degree a involves Galois cohomology groups of the residue fields of all the points of X of dimension a. He stated a conjecture on its homology generalizing the fundamental exact sequences for Brauer groups of global fields. We prove the conjecture over a finite field assuming resolution of singularities. Thanks to a recently established result on resolution of singularities for embedded surfaces, it implies the unconditional vanishing of the homology up to degree 4 for X projective …
In this paper we study some new theories of characteristic homology classes for singular complex algebraic (or compactifiable analytic) spaces. We introduce a motivic Chern class transformation mC_{*}: K_{0}(var/X)-> G_{0}(X)[y], which generalizes the total λ-class of the cotangent bundle to singular spaces. Here K_{0}(var/X) is the relative Grothendieck group of complex algebraic varieties over X as introduced and studied by Looijenga and Bittner in relation to motivic integration, and G_{0}(X) is the Grothendieck group of coherent sheaves of O_{X}-modules. We define a natural transformation …
We review some recent results and conjectures saying that, roughly speaking, periodic cyclic homology of a smooth non-commutative algebraic variety should carry all the additional "motivic" structures possessed by the usual de Rham cohomology of a smooth algebraic variety (specifically, an R-Hodge structure for varieties over R, and a filtered Dieudonne module structure for varieties over Z_p). To appear in Proc. ICM 2010.
In this paper we study some new theories of characteristic homology classes for singular complex algebraic varieties. First we introduce a natural transformation T_{y}: K_{0}(var/X) -> H_{*}(X,Q)[y] commuting with proper pushdown, which generalizes the corresponding Hirzebruch characteristic. Here K_{0}(var/X) is the relative Grothendieck group of complex algebraic varieties over X as introduced and studied by Looijenga and Bittner in relation to motivic integration. T_{y} is a homology class version of the motivic measure corresponding to a suitable specialization of the well known Hodge poly…
Motivic Chern and Hirzebruch classes are polynomials with K-theory and homology classes as coefficients, which specialize to Chern-Schwartz-MacPherson classes, K-theory classes, and Cappell-Shaneson L-classes. We provide formulas to compute the motivic Chern and Hirzebruch classes of Grassmannian and vexillary degeneracy loci. We apply our results to obtain the Hirzebruch $χ_y$-genus of classical and one-pointed Brill-Noether varieties, and therefore their topological Euler characteristic, holomorphic Euler characteristic, and signature.
We reformulate the construction of Kontsevich's completion and use Lawson homology to define many new motivic invariants. We show that the dimensions of subspaces generated by algebraic cycles of the cohomology groups of two $K$-equivalent varieties are the same, which implies that several conjectures of algebraic cycles are $K$-statements. We define stringy functions which enable us to ask stringy Grothendieck standard conjecture and stringy Hodge conjecture. We prove a projective bundle theorem in morphic cohomology for trivial bundles over any normal quasi-projective varieties.
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…
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…
We establish the analogue of the Friedlander-Mazur conjecture for Teh's reduced Lawson homology groups of real varieties, which says that the reduced Lawson homology of a real quasi-projective variety $X$ vanishes in homological degrees larger than the dimension of $X$ in all weights. As an application we obtain a vanishing of homotopy groups of the mod-2 topological groups of averaged cycles and a characterization in a range of indices of the motivic cohomology of a real variety as homotopy groups of the complex of averaged equidimensional cycles. We also establish an equivariant Poincare dua…
The moving lemma of Suslin states that a cycle on $X\times \mathbb{A} ^n$ meeting all faces properly can be moved so that it becomes equidimensional over $\mathbb{A}^n$. This leads to an isomorphism of motivic Borel-Moore homology and higher Chow groups. In this short paper we formulate and prove a variant of this. It leads to an isomorphism of Suslin homology with modulus and higher Chow groups with modulus, in an appropriate pro setting.
We prove versions of the Suslin and Gabber rigidity theorems in the setting of equivariant pseudo pretheories of smooth schemes over a field with an action of a finite group. Examples include equivariant algebraic $K$-theory, presheaves with equivariant transfers, equivariant Suslin homology, and Bredon motivic cohomology.
In this paper we compute the motivic Chern classes and homology Hirzebruch characteristic classes of (possibly singular) toric varieties, which in the complete case fit nicely with a generalized Hirzebruch-Riemann-Roch theorem. As special cases, we obtain new (or recover well-known) formulae for the Baum-Fulton-MacPherson Todd (or MacPherson-Chern) classes of toric varieties, as well as for the Thom-Milnor L-classes of simplicial projective toric varieties. We present two different perspectives for the computation of these characteristic classes of toric varieties. First, we take advantage of …
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…
In mixed characteristic and in equal characteristic $p$ we define a filtration on topological Hochschild homology and its variants. This filtration is an analogue of the filtration of algebraic $K$-theory by motivic cohomology. Its graded pieces are related in mixed characteristic to the complex $AΩ$ constructed in our previous work, and in equal characteristic $p$ to crystalline cohomology. Our construction of the filtration on $\mathrm{THH}$ is via flat descent to semiperfectoid rings. As one application, we refine the construction of the $AΩ$-complex by giving a cohomological construction o…
Let T be a torus, X a smooth quasi-compact separated scheme equipped with a T-action, and [X/T] the associated quotient stack. Given any localizing A1-homotopy invariant of dg categories E, we prove that the derived completion of E([X/T]) at the augmentation ideal I of the representation ring R(T) of T agrees with the Borel construction associated to the T-action on X. Moreover, for certain localizing A1-homotopy invariants, we extend this result to the case of a linearly reductive group scheme G. As a first application, we obtain an alternative proof of Krishna's completion theorem in algebra…
We give a new construction, based on categorical logic, of Nori's $\mathbb Q$-linear abelian category of mixed motives associated to a cohomology or homology functor with values in finite-dimensional vector spaces over $\mathbb Q$. This new construction makes sense for infinite-dimensional vector spaces as well, so that it associates a $\mathbb Q$-linear abelian category of mixed motives to any (co)homology functor, not only Betti homology (as Nori had done) but also, for instance, $\ell$-adic, $p$-adic or motivic cohomology. We prove that the $\mathbb Q$-linear abelian categories of mixed mot…
We compare various groups of 0-cycles on quasi-projective varieties over a field. As applications, we show that for certain singular projective varieties, the Levine-Weibel Chow group of 0-cycles coincides with the corresponding Friedlander-Voevodsky motivic cohomology. We also show that over an algebraically closed field of positive characteristic, the Chow group of 0-cycles with modulus on a smooth projective variety with respect to a reduced divisor coincides with the Suslin homology of the complement of the divisor. We prove several generalizations of the finiteness theorem of Saito and Sa…
