ﻻ يوجد ملخص باللغة العربية
We study the 0-th stable A^1-homotopy sheaf of a smooth proper variety over a field k assumed to be infinite, perfect and to have characteristic unequal to 2. We provide an explicit description of this sheaf in terms of the theory of (twisted) Chow-Witt groups as defined by Barge-Morel and developed by Fasel. We study the notion of rational point up to stable A^1-homotopy, defined in terms of the stable A^1-homotopy sheaf of groups mentioned above. We show that, for a smooth proper k-variety X, existence of a rational point up to stable A^1-homotopy is equivalent to existence of a 0-cycle of degree 1.
We prove that existence of a k-rational point can be detected by the stable A^1-homotopy category of S^1-spectra, or even a rationalized variant of this category.
We study generically split octonion algebras over schemes using techniques of ${mathbb A}^1$-homotopy theory. By combining affine representability results with techniques of obstruction theory, we establish classification results over smooth affine s
We construct many``low rank algebraic vector bundles on ``simple smooth affine varieties of high dimension. In a related direction, we study the existence of polynomial representatives of elements in the classical (unstable) homotopy groups of sphere
We prove a trace formula in stable motivic homotopy theory over a general base scheme, equating the trace of an endomorphism of a smooth proper scheme with the Euler characteristic integral of a certain cohomotopy class over its scheme of fixed point
We introduce and study the homotopy theory of motivic spaces and spectra parametrized by quotient stacks [X/G], where G is a linearly reductive linear algebraic group. We extend to this equivariant setting the main foundational results of motivic hom