Related papers: Noncommutative motives, numerical equivalence, and…

Given a complex algebraic variety X, we define a natural number called the motivic dimension which measures the amount of transcendental (co)homology of X. It is zero precisely when all the (co)homolgy is spanned by algebraic cycles. Most…

Following [14], we compute the motivic cohomology ring of the Nisnevich classifying space of the unitary group associated to the standard split hermitian form of a quadratic extension. This provides us with subtle characteristic classes…

We construct a motivic Eilenberg-MacLane spectrum with a highly structured multiplication over smooth schemes over Dedekind domains which represents Levine's motivic cohomology. The latter is defined via Bloch's cycle complexes. Our method…

Given a group, we construct a fundamental additive functor on its orbit category. We prove that any isomorphism conjecture valid for this fundamental isomorphism functor holds for all additive functors, like K-theory, cyclic homology,…

Let $kq$ denote the very effective cover of Hermitian K-theory. We apply the $kq$-based motivic Adams spectral sequence, or $kq$-resolution, to computational motivic stable homotopy theory. Over base fields of characteristic not two, we…

We show how to use Jantzen's sum formula for Weyl modules to prove semisimplicity criteria for endomorphism algebras of $\textbf{U}_q$-tilting modules (for any field $\mathbb{K}$ and any parameter $q\in\mathbb{K}-\{0,-1\}$). As an…

Let $\varphi\colon R \rightarrow A$ be a finite ring homomorphism, where $R$ is a two-sided Noetherian ring, and let $M$ be a finitely generated left $A$-module. Under suitable homological conditions on $A$ over $R$, we establish a close…

We prove that the degree zero Hochschild-type cohomology of the homology operad of Francis Brown's dihedral moduli spaces is equal to the Grothendieck-Teichmueller Lie algebra plus two classes. This significantly elucidates the (in part…

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…

Let $\mathbf{T}$ be a neutral tannakian category over a field of characteristic 0. Let $M$ be an object of $\mathbf{T}$ with a filtration $0=F_0M\subsetneq F_1M\subsetneq \cdots\subsetneq F_kM=M$, such that each successive quotient…

The theme of this paper is to compute hermitian $K$-groups in terms of the recently developed theory of Milnor-Witt motivic cohomology. Our approach makes use of the very effective slice spectral sequence within the motivic stable homotopy…

From the viewpoint of higher homological algebra, we introduce pure semisimple $n$-abelian category, which is analogs of pure semisimple abelian category. Let $\Lambda$ be an Artin algebra and $\mathcal{M}$ be an $n$-cluster tilting…

Assuming the K\"unneth type standard conjecture, we propose a way to describe objects of mixed motives explicitly. We study their formal properties, and we associate mixed motives to schemes smooth and separated over a field. This serves as…

We describe Somekawa's K-group associated to a finite collection of semi-abelian varieties (or more general sheaves) in terms of the tensor product in Voevodsky's category of motives. While Somekawa's definition is based on Weil…

We construct the motive of an algebraic stack in the Nisnevich topology. For stacks which are Nisnevich locally quotient stacks, we give a presentation of the motive in terms of simplicial schemes. We also show that for quotient stacks the…

Let k be a field of characteristic not 2. We conjecture that if X is a quasi-projective k-variety with trivial motivic Euler characteristic, then Sym$^n$X has trivial motivic Euler characteristic for all n. Conditional on this conjecture,…

This dissertation comprises three collections of results, all united by a common theme. The theme is the study of categories via algebraic techniques, considering categories themselves as algebraic objects. This algebraic approach to…

Waldhausen's algebraic K-theory machinery is applied to motivic homotopy theory, producing an interesting motivic homotopy type. Over a field F of characteristic zero, its path components receive a surjective ring homomorphism from the…

We study the motive of the moduli space of semistable Higgs bundles of coprime rank and degree on a smooth projective curve C over a field k under the assumption that C has a rational point. We show this motive is contained in the thick…

We construct a comparison functor from the dual category of motivic homotopy category $\mathcal{SH}$ to the category of $\mathbb{A}^1$-invariant localizing motives $\operatorname{Mot}_{\operatorname{loc}}^{\mathbb{A}^1}$ in the sense of…