Related papers: Detecting motivic equivalences with motivic homolo…

Over a perfect field k, let G be an extension of an abelian variety by the multiplicative group $\G_m$. We compute the motive of G in Voevodsky's category of etale motivic complexes with rational coefficients. The result is a decomposition…

We prove the analog of the Morel-Voevodsky localization theorem for framed motivic spaces. We deduce that framed motivic spectra are equivalent to motivic spectra over arbitrary schemes, and we give a new construction of the motivic…

Let k be a base commutative ring, R a commutative ring of coefficients, X a quasi-compact quasi-separated k-scheme, A a sheaf of Azumaya algebras over X of rank r, and Hmo(R) the category of noncommutative motives with R-coefficients.…

Let X and Y be smooth complex projective varieties. Orlov conjectured that if X and Y are derived equivalent then their motives M(X) and M(Y) are isomorphic in Voevodsky's triangulated category of geometrical motives with rational…

Let $K$ be a perfect field and let $E$ be a homotopy commutative ring spectrum in the Morel-Voevodsky stable motivic homotopy category $\mathcal{SH}(K)$. In this work we investigate the relation between the $E$-homology localization and…

Based on homological algebra of Grothendieck categories of enriched functors, two models for Voevodsky's category of big motives with reasonable correspondences are given in this paper.

We prove that a motivic equivalence of objects of the form \begin{equation*} X/(X-x)\simeq X^\prime/(X^\prime-x^\prime) \end{equation*} in $\mathbf{H}^\bullet(B)$ or $\mathbf{DM}(B)$ over a scheme $B$, where $x$ and $x^\prime$ are closed…

We investigate certain categorical aspects of Voevodsky's triangulated categories of motives. For this, various recollements for Grothendieck categories of enriched functors and their derived categories are established. In order to extend…

We show that motivic homology, motivic Borel-Moore homology and higher Chow groups satisfy homological descent for hyperenvelopes, and l-hyperenvelopes after inverting l.

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…

In this article we further the study of non-commutative motives. Our main result is the construction of a symmetric monoidal structure on the localizing motivator Mot of dg categories. As an application, we obtain : (1) a computation of the…

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…

In this paper we prove that the intersections of the levels of the dimension filtration on Voevodsky's motivic complexes over a field $k$ with the levels of the slice one are "as small as possible", i.e., that $Obj d_{\le m}DM^{eff}_{-,R}…

We prove a coherence theorem for invertible objects in a symmetric monoidal category. This is used to deduce associativity, skew-commutativity, and related results for multi-graded morphism rings, generalizing the well-known versions for…

Let k be a separably closed field. Let K_i=[A_i \to B_i] (for i=1,2,3) be three 1-motives defined over k. We define the geometrical notions of extension of K_1 by K_3 and of biextension of (K_1,K_2) by K_3. We then compute the homological…

Let X be a smooth projective variety over a field k. For k separably closed, we prove that the subgroup of unramified classes in the Milnor K-group $K^M_i(k(X))$ of the function field of X is contained in the subgroup of n-divisible…

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,…

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…

In this paper we study the category of localizing motives $\operatorname{Mot}^{\operatorname{loc}}$ -- the target of the universal finitary localizing invariant of idempotent-complete stable categories as defined by Blumberg-Gepner-Tabuada.…

For each fs log scheme $(X,\mathcal M_X)$ over a field $k$ we construct a geometrical Voevodsky motive $[X]^{log}\in DM_{gm}(k,\mathbb Q)$. We prove that, for $k=\mathbb C$, the Betti realization of $[X]^{log}$ is the log Betti cohomology…