Euler类群，初等线性群与特殊线性群的同调
Euler class groups, and the homology of elementary and special linear groups
摘要：
本文证明了环上多个单位的初等线性群和特殊线性群的同调稳定性，改进了已知的稳定范围。我们的结果暗示了不稳定Quillen K-群的稳定性，并证明了Bass的一个猜想。对于具有无限剩余域的交换局部环，我们证明了Milnor-Witt K-理论给出了进一步稳定性的障碍.作为应用，我们构造了取值于Milnor Witt K-理论层上同调的投射模的Euler类.对于具有无限剩余域的d维交换诺特环，我们证明了Euler类的消失是秩为d的投射模P分裂秩为1的自由直和项的充要条件.在此过程中，我们得到了Milnor-Witt K-理论的一个新的表述。
英文摘要：
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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