Search results for "motivic homotopy"

showing 3 items of 3 documents

Orientation theory in arithmetic geometry

2016

This work is devoted to study orientation theory in arithmetic geometric within the motivic homotopy theory of Morel and Voevodsky. The main tool is a formulation of the absolute purity property for an \emph{arithmetic cohomology theory}, either represented by a cartesian section of the stable homotopy category or satisfying suitable axioms. We give many examples, formulate conjectures and prove a useful property of analytical invariance. Within this axiomatic, we thoroughly develop the theory of characteristic and fundamental classes, Gysin and residue morphisms. This is used to prove Riemann-Roch formulas, in Grothendieck style for arbitrary natural transformations of cohomologies, and a …

Mathematics - Algebraic Geometryresiduescobordism14C40 14F42 14F20 19E20 19D45 19E15Mathematics::K-Theory and HomologyMathematics::Category Theory[MATH.MATH-AG] Mathematics [math]/Algebraic Geometry [math.AG][MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG]Orientation theorymotivic homotopyMathematics::Algebraic TopologyRiemann-Roch formulas
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On some aspects of Borel-Moore homology in motivic homotopy : weight and Quillen’s G-theory

2016

The theme of this thesis is different aspects of Borel-Moore theory in the world of motives. Classically, over the field of complex numbers, Borel-Moore homology, also called “homology with compact support”, has some properties quite different from singular homology. In this thesis we study some generalizations and applications of this theory in triangulated categories of motives.The thesis is composed of two parts. In the first part we define Borel-Moore motivic homology in the triangulated categories of mixed motives defined by Cisinski and Déglise and study its various functorial properties, especially a functoriality similar to the refined Gysin morphism defined by Fulton. These results…

Quillen’s K-theory and G-theoryStructure de poidsMixed motives[MATH.MATH-AG] Mathematics [math]/Algebraic Geometry [math.AG]Motivic homotopy theoryHomologie de Borel-MooreThéorie de l’homotopie motiviqueMotifs de ChowChow motives[MATH.MATH-KT] Mathematics [math]/K-Theory and Homology [math.KT]G-théorieFormalisme des six foncteursWeight structureSix functors formalismMotifs mixtesRefined Gysin morphismBorel-Moore homologyMorphisme de Gysin raffinéK-théorie de Quillen
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Voisinages tubulaires épointés et homotopie stable à l'infini

2022

We initiate a study of punctured tubular neighborhoods and homotopy theory at infinity in motivic settings. We use the six functors 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-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…

links of singularities[MATH.MATH-AG] Mathematics [math]/Algebraic Geometry [math.AG]Motivic homotopy theorypunctured tubular neighborhoods[MATH.MATH-AT] Mathematics [math]/Algebraic Topology [math.AT]stable homotopy at infinityMathematics::Algebraic TopologyMathematics - Algebraic Geometrylinks of singularities.Mathematics::Algebraic Geometryquadratic invariantsMathematics::K-Theory and HomologyFOS: MathematicsAlgebraic Topology (math.AT)14F42 19E15 55P42 14F45 55P57Mathematics - Algebraic TopologyAlgebraic Geometry (math.AG)qua- dratic invariants
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