有限群的Nil K-群
The Nil K-groups of finite groups
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中文总结 AI 辅助
本文证明有限群整群环的Nil K-群具有有限指数，给出显式界，并推广到正则环和概形情形，进而推出几乎循环群的K-群结构。
中文摘要 AI 辅助
我们证明，对于每个有限群 $G$ 和每个整数 $n$，$G$ 的整群环的 Nil 群 $\mathrm{NK}_n(\mathbb{Z}[G])$ 具有有限指数，并给出一个依赖于 $n$ 和阶 $|G|$ 的界。当 $|G|$ 的每个素因子至少为 $3+n/2$ 时，该界是显式的。更一般地，我们的有限指数结果适用于 $\mathrm{NK}_n(R[G])$，只要 $R$ 是一个正则、无挠、诺特交换环，且对于整除 $|G|$ 的每个素数 $p$，$R/p$ 是正则的。在附录中，M. Morrow 提供了一种替代方法，并还证明了优良诺特概形 $X$ 的 Nil 群 $\mathrm{NK}_n(X)$（其中 $X[1/p]$ 正则）被 $p$ 的有限幂零化，前提是 $X$ 允许合适的奇点消解。通过将他的论证扩展到某些非交换环，我们还证明，对于上述 $R$ 和 $G$ 的任意自同构 $\alpha$，Farrell Nil 群 $\mathrm{NK}_n(R[G],\alpha)$ 具有有限指数。作为推论，对于每个几乎循环群 $\Gamma$，群 $\mathrm{K}_n(\mathbb{Z}[\Gamma])$ 是一个有限生成阿贝尔群与一个无限可数多个有限阿贝尔群副本的直和的直和。
英文摘要
We show that for every finite group $G$ and every integer $n$, the Nil group $\mathrm{NK}_n(\mathbb{Z}[G])$ of the integral group ring of $G$ has finite exponent, and give a bound depending on n and the order $|G|$. This bound is explicit when every prime divisor of $|G|$ is at least $3+n/2$. More generally, our finite exponent result applies to $\mathrm{NK}_n(R[G])$ whenever $R$ is a regular, torsion-free, Noetherian commutative ring such that $R/p$ is regular for every prime $p$ that divides $|G|$. In the Appendix, M. Morrow provides an alternative approach and also shows that the Nil group $\mathrm{NK}_n(X)$ of an excellent Noetherian scheme $X$, with $X[1/p]$ regular, is annihilated by a finite power of $p$, provided $X$ admits a suitable resolution of singularities. By extending his argument to certain noncommutative rings, we also show that, for $R$ as above and $α$ any automorphism of $G$, the Farrell Nil groups $\mathrm{NK}_n(R[G],α)$ have finite exponent. As a consequence, for every virtually cyclic group $Γ$, the group $\mathrm{K}_n(\mathbb{Z}[Γ])$ is a direct sum of a finitely generated abelian group and an infinite countable direct sum of copies of a finite abelian group.
