论文标题
多项式代数的投影模块的拆分标准
Splitting criteria for projective modules over polynomial algebras
论文作者
论文摘要
本文研究了多项式代数$ a [t] $在各种基本环上的有限生成的投影模块$ p $的分裂问题，其中$ \ text {rank}（p）= \ dim（a）$。我们的主要方法是（1）根据\ emph {Generic部分}和（2）用\ emph {monic倒置原理}。我们证明，如果$ p $具有完整的交点通用部分，那么它将脱离排名第一的免费求和，其中$ a $是一个代数封闭的特征性$ \ neq 2 $的仿射代数。由于Roitman在Monic倒置原则上，我们对一个旧问题给出了部分答案。每当$ a $都是$ \ overline {\ mathbb {f}} _ p $上的仿射代数时，我们证明了理想的一元反演原则。我们进一步展示了一些应用。
This article investigates the splitting problem for finitely generated projective modules $P$ over polynomial algebras $A[T]$ on various base rings, where $\text{rank}(P) = \dim(A)$. Our main approaches are (1) in terms of \emph{generic sections}, and (2) in terms of \emph{monic inversion principles}. We prove that if $P$ has a complete intersection generic section, then it splits off a free summand of rank one, where $A$ is an affine algebra over an algebraically closed field of characteristic $\neq 2$. We give a partial answer to an old question due to Roitman on monic inversion principle for projective modules over affine $\mathbb{Z}$-algebras. Whenever $A$ is an affine algebra over $\overline{\mathbb{F}}_p$, we prove a monic inversion principle for ideals. We further exhibit some applications.
