Let $p$ be a prime number, and let $A$ be a ring in which $p$ is nilpotent. In this paper, we consider the maps $$K_{q+1}(A[x]/(x^m), (x))\to K_{q+1}(A[x]/(x^{mn}), (x)),$$induced by the ring homomorphism $A[x]/(x^{m})\to A[x]/(x^{mn})$, $x\mapsto x^n$. We evaluate these maps, up to extension, for general $A$ in terms of topological Hochschild homology, and for regular $\mathbb{F}_p$-algebras $A$, in terms of groups of de Rham-Witt forms. After the evaluation, we give a calculation of the relative $K$-group of $\mathcal{O}_{K}/p\mathcal{O}_{K}$ for certain perfectoid fields $K$.

中文翻译：

---

截断多项式代数的 K 群之间的 Verschiebung 映射

令 $p$ 为质数，令 $A$ 为 $p$ 幂零的环。在本文中，我们考虑映射 $$K_{q+1}(A[x]/(x^m), (x))\to K_{q+1}(A[x]/(x^{ mn}), (x)),$$由环同态引起 $A[x]/(x^{m})\to A[x]/(x^{mn})$, $x\mapsto x ^n$。我们评估这些映射，直到扩展，根据拓扑 Hochschild 同源性的一般 $A$ 和常规 $\mathbb{F}_p$-algebras $A$，根据 de Rham-Witt 形式的群。评估后，我们对某些完美体$K$ 的$\mathcal{O}_{K}/p\mathcal{O}_{K}$ 的相对$K$-group 进行计算。