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arXiv · 1502.05424

Euler class groups, and the homology of elementary and special linear groups

Marco Schlichting
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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.

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Marco Schlichting. 2015-02-18. Euler class groups, and the homology of elementary and special linear groups. https://arxiv.org/abs/1502.05424

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