Article ID Journal Published Year Pages File Type
5778320 Advances in Mathematics 2017 81 Pages PDF
Abstract
We improve homology stability ranges for elementary and special linear groups over rings with many units. 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 an oriented 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 and of symplectic K2 simplifying the classical Matsumoto-Moore presentation.
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Physical Sciences and Engineering Mathematics Mathematics (General)
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