Related papers: More examples of motivic cell structures
We prove that some of the classical homological stability results for configuration spaces of points in manifolds can be lifted to motivic cohomology.
We construct characteristic classes for singular algebraic varieties in motivic Borel-Moore homology, extending the motivic Euler class of the tangent bundle defined for smooth varieties. The two classes we define refine, in the setting of…
Smooth projective $\mathbb{G}_m$-varieties with isolated rational fixed points admit Tate Milnor-Witt motives. Over Euclidean fields, we give a splitting formula of such motives, which reduces the computation of their Chow-Witt groups to…
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…
We study the structure of the rational motivic stable homotopy category over general base schemes. Our first class of results concerns the six operations: we prove absolute purity, stability of constructible objects, and…
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…
We survey computations of stable motivic homotopy groups over various fields. The main tools are the motivic Adams spectral sequence, the motivic Adams-Novikov spectral sequence, and the effective slice spectral sequence. We state some…
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…
We prove formulas for the rational Chow motives of moduli spaces of semistable vector bundles and Higgs bundles of rank 3 and coprime degree on a smooth projective curve. Our approach involves identifying criteria to lift identities in (a…
This paper is the second one in a series of papers about operations in motivic cohomology. Here we show that in the context of smooth schemes over a field of characteristic zero all the bi-stable operations can be obtained in the usual way…
(This is an updated version; following an idea of Voevodsky, we have strengthened our results so all of them apply to one form of motivic homotopy theory). We give two general constructions for the passage from unstable to stable homotopy…
Fix a base field F, a finite field K and consider a sequence of central simple F-algebras A_1,...,A_n. In this note we provide some results toward a classification of the indecomposable motives lying in the motivic decompositions of…
The apparatus of motivic stable homotopy theory provides a notion of Euler characteristic for smooth projective varieties, valued in the Grothendieck-Witt ring of the base field. Previous work of the first author and recent work of…
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…
We show that if G is a finite constant group acting on a scheme X such that the order of G is invertible in the residue fields of X, then the G-equivariant motivic stable homotopy category of X is equivalent to the stabilization of the…
We construct smooth presentations of algebraic stacks that are local epimorphisms in the Morel-Voevodsky $\mathbb{A}^1$-homotopy category. As a consequence we show that the motive of a smooth stack (in Voevodsky's triangulated category of…
In this article, we give a construction of the (un-)stable motivic homotopy category of an algebraic stack in the spirit of Morel-Voevodsky. We prove that this new construction agrees with the stable motivic homotopy category defined by…
We study the stable motivic homotopy groups $\pi_{s,w}$ of the 2-completion of the motivic sphere spectrum over $\mathbb{C}$. When arranged in the $(s,w)$-plane, these groups break into four different regions: a vanishing region, an…
We link small modifications of projective varieties with a ${\mathbb C}^*$-action to their GIT quotients. Namely, using flips with centers in closures of Bia{\l}ynicki-Birula cells, we produce a system of birational equivariant…
This text gives a construction of a differential graded Lie algebra in Nori's category of effective homological motives. In fact the construction works in more a general setting than that of an Abelian category. This allows us to give the…