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

TL;DR

This paper proves homology stability for elementary and special linear groups over rings with many units, improving known ranges, and explores implications for algebraic K-theory, Euler classes, and Milnor-Witt K-theory.

Contribution

It advances stability results for linear groups, confirms a conjecture of Bass, and links Euler classes with Milnor-Witt K-theory in a novel way.

Findings

Homology stability for linear groups over rings with many units.

Stability results imply stability for unstable Quillen K-groups.

Vanishing of Euler classes characterizes splitting of projective modules.

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.