论文标题
关于正交和自旋组的同源稳定性
On the Homological Stability of Orthogonal and Spin Groups
论文作者
论文摘要
我们证明了正交组，特殊正交组，基础正交组和自旋组相对于双曲线形式的同源稳定性结果。我们证明了与无限残留场的交换本地环$ r $相同的同源稳定性，因此$ 2 \ in r^{*} $。 在正交的情况下，这改善了Mirzaii给出的同源稳定性范围，并概括了Sprehn和Wahl获得的结果。在特殊的正交情况下，这将ESSERT获得无限场所获得的结果概括为本地环的情况，这是特殊正交组在本地环上获得的第一个同源稳定性结果。对于基础正交组，这是第一个已知的同源稳定性结果。对于自旋组，这与$ h_ {1} $ - 稳定性和$ h_ {2} $ - 在Hahn-o'meara中所述的稳定性结果，这是第一个占所有同源组的同源稳定性结果。
We prove homological stability results for the orthogonal group, special orthogonal group, elementary orthogonal group and the spin group with respect to the hyperbolic form. We prove homological stability over a commutative local ring $R$ with infinite residue field such that $2 \in R^{*}$. In the orthogonal case, this improves the range for homological stability given by Mirzaii by 1 and generalises the result obtained by Sprehn and Wahl to the case of local rings. In the special orthogonal case, this generalises the result obtained by Essert for infinite fields to the case of local rings, and is the first homological stability result for the special orthogonal group over a local ring. For the elementary orthogonal group, this is the first known homological stability result. For the spin group, this coincides with $H_{1}$-stability and $H_{2}$-stability results stated in Hahn-O'Meara, and is the first homological stability result that accounts for all homology groups.
