Stability in the high-dimensional cohomology of congruence subgroups
Stability in the high-dimensional cohomology of congruence subgroups
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DOI：
10.1112/s0010437x20007046
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发表时间：
2018-06
影响因子：
1.8
通讯作者：
Jeremy Miller;Rohit Nagpal;Peter Patzt
中科院分区：
文献类型：
--
作者：
Jeremy Miller;Rohit Nagpal;Peter Patzt
我们证明了$\mathbf{SL}_{n}(\mathbb{Z})$的三级同余子群的余维一上同调的表示稳定性结果。这是丘奇、法布和普特曼问题的一个特例，我们把这个问题说得更精确。我们的方法包括证明群$\mathbf{SL}_{n}(K)$对于$K$a域的Steinberg模的有限性质。这也给出了Steinberg模的Ash，Putman和Sam的同调消失定理的一个新的证明。我们还证明了群$\mathbf{SL}_{n}(\mathbb{Z})$的Steinberg模的Church和Putman同调消失定理的一个积分精化。
We prove a representation stability result for the codimension-one cohomology of the level-three congruence subgroup of $\mathbf{SL}_{n}(\mathbb{Z})$. This is a special case of a question of Church, Farb, and Putman which we make more precise. Our methods involve proving finiteness properties of the Steinberg module for the group $\mathbf{SL}_{n}(K)$ for $K$ a field. This also lets us give a new proof of Ash, Putman, and Sam’s homological vanishing theorem for the Steinberg module. We also prove an integral refinement of Church and Putman’s homological vanishing theorem for the Steinberg module for the group $\mathbf{SL}_{n}(\mathbb{Z})$.