On the rational motivic homotopy category

2015

We introduce in this work the notion of the category of pure E-Motives, where E is a motivic strict ring spectrum and construct twisted E-cohomology by using six functors formalism of J. Ayoub. In particular, we construct the category of pure Chow-Witt motives CHW (k) Q over a field k and show that this category admits a fully faithful embedding into the geometric stable A 1-derived category D A 1 ,gm (k) Q .

Journal de l’École polytechnique — Mathématiques

Cet article est mis à disposition selon les termes de la licence CREATIVE COMMONS ATTRIBUTION-PAS DE MODIFICATION 3.

2021

The present paper is a continuation of earlier work by Gunnar Carlsson and the first author on a motivic variant of the classical Becker-Gottlieb transfer and an additivity theorem for such a transfer by the present authors. Here, we establish a motivic variant of the classical Segal-Becker theorem relating the classifying space of a 1-dimensional torus with the spectrum defining (algebraic) K-theory.

2013

Vladimir Voevodsky constructed a triangulated category of motives to “universally linearize the geometry” of algebraic varieties. In this thesis, I show that the geometry of a bigger class of objects, called Deligne-Mumford stacks, can be universally linearized using Voevodsky's triangulated category of motives. Also, I give a partial answer to a conjecture of Fabien Morel related to the connected component sheaf in motivic homotopy theory. Vladimir Voevodsky hat eine triangulierte Kategorie von Motiven konstruiert, um die Geometrie algebraischer Varietaten "universell zu linearisieren". In dieser Dissertation zeige ich, dass auch die Geometrie einer umfangreicheren Klasse von Objekten, namlich von Deligne-Mumford stacks, mit Hilfe der triangulierten Kategorie Voevodskys universell linearisiert werden kann. Ausserdem gebe ich eine partielle Antwort auf eine Vermutung von Fabien Morel in Bezug auf die Zusammenhangskomponenten-Garbe in motivischer Homotopie-Theorie.

2021

The aim of this paper is to extend the definition of motivic homotopy theory from schemes to a large class of algebraic stacks and establish a six functor formalism. The class of algebraic stacks that we consider includes many interesting examples: quasi-separated algebraic spaces, local quotient stacks and moduli stacks of vector bundles. We use the language of ∞-categories developed by Lurie. Morever, we use the so-called ’enhanced operation map’ due to Liu and Zheng to extend the six functor formalism from schemes to our class of algebraic stacks. We also prove that six functors satisfy properties like homotopy invariance, localization and purity.

2021

In this paper, we initiate a study of motivic homotopy theory at infinity. We use the six functor formalism to give an intrinsic definition of the stable motivic homotopy type at infinity of an algebraic variety. Our main computational tools include cdh-descent for normal crossing divisors, Euler classes, Gysin maps, and homotopy purity. Under l-adic realization, the motive at infinity recovers a formula for vanishing cycles due to Rapoport-Zink; similar results hold for Steenbrink’s limiting Hodge structures and Wildeshaus’ boundary motives. Under the topological Betti realization, the stable motivic homotopy type at infinity of an algebraic variety recovers the singular complex at infinity of the corresponding analytic space. We coin the notion of homotopically smooth morphisms with respect to a motivic ∞-category and use it to show a generalization to virtual vector bundles of Morel-Voevodsky’s purity theorem, which yields an escalated form of Atiyah duality with compact support....

Advances in Mathematics, 2017

The aim of this work is to construct certain homotopy tstructures on various categories of motivic homotopy theory, extending works of Voevodsky, Morel, Déglise and Ayoub. We prove these t-structures possess many good properties, some analogous to those of the perverse t-structure of Beilinson, Bernstein and Deligne. We compute the homology of certain motives, notably in the case of relative curves. We also show that the hearts of these t-structures provide convenient extensions of the theory of homotopy invariant sheaves with transfers, extending some of the main results of Voevodsky. These t-structures are closely related to Gersten weight structures as defined by Bondarko.

This paper initiates the incorporation of factorization algebra techniques to study motivic homotopy theory. We define a version of the Ran space of an algebraic variety and prove that it is contractible in the unstable motivic homotopy category. To do so, we prove that its A 1-homotopy sheaves vanish, the most difficult being the sheaf of A 1-connected components where we use an interpretation of this sheaf, available in the setting of infinite fields and a Riemann-Roch argument.

2011

In this paper, we develop the theory of equivariant motivic homotopy theory, both unstable and stable. While our main interest is the case when the group is pro-finite, we discuss our results in a more general setting so as to be applicable to other contexts, for example when the group is in fact a smooth group scheme. We also discuss how A 1-localization behaves with respect to ring and module spectra and also with respect to mod-l-completion, where l is a fixed prime. In forthcoming papers, we apply the theory developed here to produce a theory of Spanier-Whitehead duality and Becker-Gottlieb transfer in this framework, and explore various applications of the transfer.

Rendiconti del Seminario Matematico della Università di Padova

We develop the foundations of commutative algebra objects in the category of motives, which we call "motivic dga's". Work of White and of Cisinski-Déglise provides us with a suitable model structure. This enables us to reconstruct the unipotent fundamental group of a pointed scheme from the associated augmented motivic dga, and provides us with a factorization of Kim's relative unipotent section conjecture into several smaller conjectures with a homotopical flavor.

In this paper we define an E1-structure, i.e. a coherently homotopy associative and commutative product on chain complexes defining (integral and mod-l) motivic cohomology as well as mod -letale cohomology. We also discuss several applications.

Preprint. Available at http://www. math. uiuc. edu/K- …, 2000

Journal of the European Mathematical Society, 2021

We develop the theory of fundamental classes in the setting of motivic homotopy theory. Using this we construct, for any motivic spectrum, an associated twisted bivariant theory, extending the formalism of Fulton and MacPherson. We import the tools of Fulton's intersection theory into this setting: (refined) Gysin maps, specialization maps, and formulas for excess of intersection, selfintersections, and blow-ups. We also develop a theory of Euler classes of vector bundles in this setting. For the Milnor-Witt spectrum recently constructed by Déglise-Fasel, we get a bivariant theory extending the Chow-Witt groups of Barge-Morel, in the same way the higher Chow groups extend the classical Chow groups. As another application we prove a motivic Gauss-Bonnet formula, computing Euler characteristics in the motivic homotopy category.

preprint, 1995

Cornell University - arXiv, 2001

In this paper we define an explicit E∞-structure, i.e. a coherently homotopy associative and commutative product on chain complexes defining (integral and mod-l) motivic cohomology as well as mod-létale cohomology. We also discuss several applications.