Search results for "Motivic cohomology"
showing 6 items of 6 documents
Schubert calculus and singularity theory
2010
Abstract Schubert calculus has been in the intersection of several fast developing areas of mathematics for a long time. Originally invented as the description of the cohomology of homogeneous spaces, it has to be redesigned when applied to other generalized cohomology theories such as the equivariant, the quantum cohomology, K -theory, and cobordism. All this cohomology theories are different deformations of the ordinary cohomology. In this note, we show that there is, in some sense, the universal deformation of Schubert calculus which produces the above mentioned by specialization of the appropriate parameters. We build on the work of Lerche Vafa and Warner. The main conjecture these auth…
Algebraic de Rham Cohomology
2017
Let k be a field of characteristic zero. We are going to define relative algebraic de Rham cohomology for general varieties over k, not necessarily smooth.
Motivic Complexes and Relative Cycles
2019
This part is based on Suslin and Voevodsky’s theory of relative cycles that we develop in categorical terms, in the style of EGA. The climax of the theory is obtained in the study of a pullback operation for suitable relative cycles which is the incarnation of intersection theory in this language. Properties of this pullback operation, and on the conditions necessary to its definition, are made again inspired by intersection theory. We study the compatibility of this pullback operation with projective limits of schemes. In Section 9, the theory of relative cycles is exploited to introduce Voevodsky’s category of finite type schemes over an arbitrary base with morphisms finite correspondence…
Functional equations of the dilogarithm in motivic cohomology
2009
We prove relations between fractional linear cycles in Bloch's integral cubical higher Chow complex in codimension two of number fields, which correspond to functional equations of the dilogarithm. These relations suffice, as we shall demonstrate with a few examples, to write down enough relations in Bloch's integral higher Chow group CH^2(F,3) for certain number fields F to detect torsion cycles. Using the regulator map to Deligne cohomology, one can check the non-triviality of the torsion cycles thus obtained. Using this combination of methods, we obtain explicit higher Chow cycles generating the integral motivic cohomology groups of some number fields.
𝔸1-contractibility of affine modifications
2019
We introduce Koras–Russell fiber bundles over algebraically closed fields of characteristic zero. After a single suspension, this exhibits an infinite family of smooth affine [Formula: see text]-contractible [Formula: see text]-folds. Moreover, we give examples of stably [Formula: see text]-contractible smooth affine [Formula: see text]-folds containing a Brieskorn–Pham surface, and a family of smooth affine [Formula: see text]-folds with a higher-dimensional [Formula: see text]-contractible total space.
Beilinson Motives and Algebraic K-Theory
2019
Section 12 is a recollection on the basic results of stable homotopy theory of schemes, after Morel and Voevodsky. In particular, we recall the theory of orientations in a motivic cohomology theory. Section 13 is a recollection of the fundamental results on algebraic K-theory which we translate into results within stable homotopy theory of schemes. In particular, Quillen’s localization theorem is seen as an absolute purity theory for the K-theory spectrum. In Section 14, we introduce the fibred category of Beilinson motives as an appropriate Verdier quotient of the motivic stable homotopy category. Using the Adams filtration on K-theory, we prove that Beilinson motives have the properties o…