Related papers: Mixed Motives and Motivic Birational Covers

Following Nori's original idea we here provide certain motivic categories with a canonical tensor structure. These motivic categories are associated to a cohomological functor on a suitable base category and the tensor structure is induced…

In paper 0704.4003, Bondarko recently defined the notion of weight structure, and proved that the category $\DgM$ of geometrical motives over a perfect field k, as defined and studied by Voevodsky, Suslin and Friedlander, is canonically…

We study the triangulated subcategories of compact objects in stable homotopy categories such as the homotopy category of spectra, the derived categories of rings, and the stable module categories of Hopf algebras. In the first part of this…

The category of framed correspondences $Fr_*(k)$ was invented by Voevodsky in his notes in order to give another framework for SH(k) more amenable to explicit calculations. Based on that notes and on their JAMS paper Garkusha and the author…

We construct an algebraic-cycle based model for the motivic cohomology on the category of schemes of finite type over a field, where schemes may admit arbitrary singularities and may be non-reduced. We show that our theory is functorial on…

We construct a theory of (etale) Berkovich motives. This is closely related to Ayoub's theory of rigid-analytic motives, but works uniformly in the archimedean and nonarchimedean setting. We aim for a self-contained treatment, not relying…

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…

We show that associating the Euclidean cell decomposition due to Cooper and Long to each point of the moduli space of framed strictly convex real projective structures of finite volume on the once-punctured torus gives this moduli space a…

This paper introduces the trivial fiber topology on schemes. For one-dimensional base schemes, we use it to describe fibrant replacements in the stable motivic homotopy category and motivic infinite loop spaces. We also extend the…

In a previous work, the author have built two families of distinguished algebraic cycles in Bloch-Kriz cubical cycle complex over the projective line minus three points. The goal of this paper is to show how these cycles induce well-defined…

Following an idea of Kontsevich, we introduce and study the notion of formal completion of a compactly generated (by a set of objects) enhanced triangulated category along a full thick essentially small triangulated subcategory. In…

We prove the equivalence between the categories of motives of rigid analytic varieties over a perfectoid field $K$ of mixed characteristic and over the associated (tilted) perfectoid field $K^{\flat}$ of equal characteristic. This can be…

Let $G$ be a reductive group over $\mathbb{F}_{p}$ with associated finite group of Lie type $G^{F}$. Let $T$ be a maximal torus contained inside a Borel $B$ of $G$. We relate the (rational) Tate motives of $\text{B}G^{F}$ with the…

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 construct an upgrade of the motivic volume by keeping track of dimensions in the Grothendieck ring of varieties. This produces a uniform refinement of the motivic volume and its birational version introduced by Kontsevich and Tschinkel…

By the Telescope Conjecture for Module Categories, we mean the following claim: "Let R be any ring and (A, B) be a hereditary cotorsion pair in Mod-R with A and B closed under direct limits. Then (A, B) is of finite type." We prove a…

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…

Making use of the recent theory of noncommutative motives, we prove that Schur-finiteness in the setting of Voevodsky's mixed motives is invariant under homological projective duality. As an application, we show that the mixed motives of…

The motivic nearby fiber is an invariant obtained from degenerating a complex variety over a disc. It specializes to the Euler characteristic of the original variety but also contains information on the variation of Hodge structure…

We show that the cellular objects in the module category over a motivic E infinity ring spectrum E can be described as the module category over a graded topological spectrum if E is strongly periodizable in our language. A similar statement…