Related papers: The Unstable Slice Filtration
This paper is part of an endeavor to define an analogue of the slice filtration in the unstable motivic homotopy category. Our approach was inspired by the fact that the triangulated structures do not play a relevant role for the…
Let $X$ be a Noetherian separated scheme of finite Krull dimension. We show that the layers of the slice filtration in the motivic stable homotopy category $\stablehomotopy$ are strict modules over Voevodsky's algebraic cobordism spectrum.…
Let $k$ be a field with resolution of singularities, and $X$ a separated $k$-scheme of finite type with structure map $g$. We show that the slice filtration in the motivic stable homotopy category commutes with pullback along $g$.…
We provide a description of Voevodsky's $\infty$-category of motivic spectra in terms of the subcategory of motives of smooth proper varieties. As applications, we construct weight filtrations on the Betti and \'{e}tale cohomologies of…
Colored operads were introduced in the 1970's for the purpose of studying homotopy invariant algebraic structures on topological spaces. In this paper we introduce colored operads in motivic stable homotopy theory. Our main motivation is to…
We prove a new convergence result for the slice spectral sequence, following work by Levine and Voevodsky. This verifies a derived variant of Voevodsky's conjecture on convergence of the slice spectral sequence. This is, in turn, a…
In this expository article, we give the foundations, basic facts, and first examples of unstable motivic homotopy theory with a view towards the approach of Asok-Fasel to the classification of vector bundles on smooth complex affine…
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…
Using the trivial fiber topology we describe motivic $\infty$-loop spaces and fibrant replacements in the motivic stable homotopy category $\mathbf{SH}_{\mathbb{A}^1,\mathrm{Nis}}(B)$ defined over one-dimensional base schemes $B$.
We study various triangulated motivic categories and introduce a vast family of aisles (these are certain classes of objects) in them. These aisles are defined in terms of the corresponding "motives" (or motivic spectra) of smooth varieties…
We introduce a new stable birational invariant, which takes the form of a functor sending a degenerating variety to the homotopy type of a chain complex. Our invariant is a categorification of the motivic volume of Nicaise and Shinder. From…
In this paper we explore the isotropic stable motivic homotopy category constructed from the usual stable motivic homotopy category, following the work of Vishik on isotropic motives (see [29]), by killing anisotropic varieties. In…
This is an expository paper providing an overview of the unstable motivic homotopy category using the theory of $(\infty,1)$-categories. In this paper, we examine two constructions in the literature and discuss their equivalence.
The classical Mahowald invariant is a method for producing nonzero classes in the stable homotopy groups of spheres from classes in lower stems. We study the Mahowald invariant in the setting of motivic stable homotopy theory over…
These notes, written version of a Bourbaki talk, survey Morel-Voevodsky's motivic homotopy theory over a field, with a focus on computations of motivic homotopy sheaves, both stable and unstable. We also describe Isaksen-Wang-Xu's…
Let k be a perfect field of characteristic different from two. We show that the filtration on the Grothendieck-Witt group GW(k) induced by the slice filtration for the sphere spectrum in the motivic stable homotopy category is the I-adic…
A paper by Haynes Miller shows that there is a filtration on the unitary groups that splits in the stable homotopy category, where the stable summands are certain Thom spaces over Grassmannians. We give an algebraic version of this result…
We consider Voevodsky's slice tower for a finite spectrum E in the motivic stable homotopy category over a perfect field k. In case k has finite cohomological dimension (in characteristic two, we also require that k is infinite), we show…
We show that the slice filtration introduced by Voevodsky is compatible in a suitable sense with the symmetric monoidal structure in the category of motivic symmetric T-spectra constructed by Jardine. It follows from this compatibility that…
In this paper we study a model structure on a category of schemes with a group action and the resulting unstable and stable equivariant motivic homotopy theories. The new model structure introduced here samples a comparison to the one by…