Related papers: On the weight zero motivic cohomology

Let $X$ be a smooth proper variety over a field $k$ and suppose that the degree map $\mathrm{CH}_0(X \otimes_k K) \to \mathbb{Z}$ is isomorphic for any field extension $K/k$. We show that $G(\mathrm{Spec} k) \to G(X)$ is an isomorphism for…

Let $\mathbb K$ be a field of characteristic zero. We prove that its motivic cohomology in degree $m-1$ and weight $m$ is rationally isomorphic to the cohomology of the polylogarithmic complex. This gives a partial extension of A. Suslin…

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:…

We construct a theory of motivic cohomology for quasi-compact, quasi-separated schemes of equal characteristic, which is related to non-connective algebraic $K$-theory via an Atiyah--Hirzebruch spectral sequence, and to \'etale cohomology…

Voevodsky outlined a conjectural programme that his slice filtration in motivic homotopy theory should give rise to a good theory of $\mathbb{A}^1$-invariant motivic cohomology. This paper achieves his vision in the generality of arbitrary…

Over a field of characteristic zero, we establish the homotopy invariance of the Nisnevich cohomology of homotopy invariant presheaves with oriented weak transfers, and the agreement of Zariski and Nisnevich cohomology for such presheaves.…

We describe explicitely the unique isomorphism $\mathrm{H}_{sin}^*(X,\underline{\mathbb{Z}})\xrightarrow{\sim} \check{\mathrm{H}}_{\mathcal{U}}^*(X,\underline{\mathbb{Z}})$ between the cohomologies computed with the singular and \v{C}ech…

V. Voevodsyky laid the groundwork of delooping motivic spaces in order to provide a new, more computation-friendly, construction of the stable motivic category $SH(k)$, G. Garkusha and I. Panin made that project a reality, while…

This brief article gives an alternative interpretation, based on a theorem of Berkovich, of the Eisenstein classes in the cohomology of Shimura varieties, used in forthcoming work of the author with K. W. Lan, R. Taylor, and J. Thorne.

We show that a representable motivic cohomology theory admits a unique normalized SL^c-orientation if the zeroth cohomology presheaf is a Zariski sheaf. We also construct Thom isomorphisms in SL-oriented cohomology for SL^c-bundles and…

Let X be a smooth variety over a field k and D an effective divisor whose support has simple normal crossings. We construct an explicit cycle map from the r-th Nisnevich motivic complex of the pair (X,D) to a shift of the r-th relative…

We prove that the motivic cohomology of mixed characteristic schemes, introduced in our previous work, satisfies various expected properties of motivic cohomology, including a motivic refinement of Weibel's vanishing in algebraic…

We prove a blow-up formula for cyclic homology which we use to show that infinitesimal $K$-theory satisfies $cdh$-descent. Combining that result with some computations of the $cdh$-cohomology of the sheaf of regular functions, we verify a…

Let $S$ be a finite dimensional noetherian scheme. For any proper morphism between smooth $S$-schemes, we prove a Riemann-Roch formula relating higher algebraic $K$-theory and motivic cohomology, thus with no projective hypothesis neither…

Let G be a split semisimple linear algebraic group over a field k0. Let E be a G-torsor over a field extension k of k0. Let h be an algebraic oriented cohomology theory in the sense of Levine-Morel. Consider a twisted form E/B of the…

We propose here a transcendantal proof of the coherence of the higher direct images of a coherent sheaf by a proper morphism of algebraic varieties, which does not use Chow's lemma nor any projective method. The main tool here are…

We study a noncommutative version of the infinitesimal site of Grothendieck. A theorem of Grothendieck establishes that the cohomology of the structure sheaf on the infinitesimal topology of a scheme of characteristic zero is de Rham…

Let $K$ be a finite extension of ${\mathbb Q}_p$ and let $X$ be Drinfel'd's symmetric space of dimension $d$ over $K$. Let $\Gamma\subset {\rm SL}_{d+1}(K)$ be a cocompact discrete (torsionfree) subgroup and let…

We define, for a regular scheme $S$ and a given field of characteristic zero $\KK$, the notion of $\KK$-linear mixed Weil cohomology on smooth $S$-schemes by a simple set of properties, mainly: Nisnevich descent, homotopy invariance,…

Motivated by Murre's work on universal regular homomorphisms on Chow groups in codimension $2,$ we generalize the algebraic equivalence relation and regular homomorphisms to the context of Voevodsky motives over a field. In the Nisnevich…