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Euler class groups, and the homology of elementary and special linear groups
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In this paper, it was shown that the vanishing of the Euler class is necessary and sufficient for a projective module P of rank d to split off a rank 1 free direct summand.Abstract:
We prove homology stability for elementary and special linear groups over rings with many units improving known stability ranges. Our result implies stability for unstable Quillen K-groups and proves a conjecture of Bass. For commutative local rings with infinite residue fields, we show that the obstruction to further stability is given by Milnor-Witt K-theory. As an application we construct Euler classes of projective modules with values in the cohomology of the Milnor Witt K-theory sheaf. For d-dimensional commutative noetherian rings with infinite residue fields we show that the vanishing of the Euler class is necessary and sufficient for a projective module P of rank d to split off a rank 1 free direct summand. Along the way we obtain a new presentation of Milnor-Witt K-theory.read more
Citations
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Affine representability results in A1-homotopy theory, I: Vector bundles
TL;DR: In this article, a general affine representability result for vector bundles in A1-homotopy theory over a general base is established. But this result is restricted to vector bundles.
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Affine representability results in A^1-homotopy theory II: principal bundles and homogeneous spaces
TL;DR: In this paper, a relative affine representability theorem for generically trivial torsors under isotropic reductive groups over infinite fields analogous to the Bass-Quillen conjecture for vector bundles was established.
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Smooth Models of Motivic Spheres and the Clutching Construction
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On quadratic forms over semilocal rings
TL;DR: In this article, it was shown that the Arason and Elman presentation of the powers of the fundamental ideal of the Witt ring of a field extends to semilocal rings which contain an infinite field of characteristic not 2.
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Milnor–Witt K-groups of local rings
TL;DR: In this article, the Milnor-Witt K-groups of local rings were shown to be the pullback of the nth power of the fundamental ideal in the Witt ring.
References
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Cohomology of Groups
TL;DR: In this paper, an advanced textbook introduces students to cohomology theory and no knowledge of homological algebra is assumed beyond what is normally taught in a first course in algebraic topology.
Book
Homotopy Limits, Completions and Localizations
A. K. Bousfield,Daniel M. Kan +1 more
TL;DR: The main purpose of part I of these notes is to develop for a ring R a functional notion of R-completion of a space X, which coincides up to homotopy with the p-profinite completion of Quillen and Sullivan as mentioned in this paper.
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Simplicial objects in algebraic topology
TL;DR: Simplicial Objects in Algebraic Topology as discussed by the authors has been the standard reference for the theory of simplicial sets and their relationship to the homotopy theory of topological spaces.
Journal ArticleDOI
$A^1$-homotopy theory of schemes
Fabien Morel,Vladimir Voevodsky +1 more
TL;DR: In this article, the authors present a legal opinion on the applicability of commercial or impression systématiques in the context of the agreement of the publication mathématique de l'I.H.É.S.