Nil-generated algebras and group algebras whose units satify a Laurent polynomial identity

The LFED and LNED Conjectures for Laurent Polynomial Algebras

Let $R$ be an integral domain of characteristic zero, $x=(x_1, x_2, ..., x_n)$ $n$ commutative free variables, and ${\mathcal A}_n:=R[x^{-1}, x]$, i.e., the Laurent polynomial algebra in $x$ over $R$. In this paper we first classify all locally finite or locally nilpotent $R$-derivations and $R$-$\mathcal E$-derivations of ${\mathcal A}_n$, where by an $R$-$\mathcal E$-derivation of ${\mathcal A}_n$ we mean an $R$-linear map of the form $\operatorname{Id}_{{\mathcal A}_n}-ϕ$ for some $R$-algebra endomorphism $ϕ$ of ${\mathcal A}_n$. In particular, we show that ${\mathcal A}_n$ has no nonzero l…

代数 $K$-理论中强 $\mathbb{Z}$-分级环的‘基本定理’

The "fundamental theorem" for the algebraic $K$-theory of strongly $\mathbb{Z}$-graded rings

本文研究了强 $\mathbb{Z}$-分级环的代数 $K$-理论，证明了在经典情况下，Laurent多项式环的 $K$-群可以表示为原环 $L$ 的 $K$-群的直接和，以及某些‘nil’群；并进一步推广了这一结果，涉及与分级结构相关的 $L$-模的移位作用，以及将‘nil’群识别为同调 nilpotent 旋转变换的减少 $K$-理论，并建立了类似 Mayer-Vietoris 和局部化序列的类比。