Relative Cancellation

TL;DR

This work introduces a relative cancellation framework for associative algebras, using ${ m N}$-filtrations and associated invariants to study how cancellativity behaves relative to chosen algebra families. It defines ${ m R}_{1},{ m R}_{2},{ m R}_{3}$ (and ${ m C}_{3}$) classes built from GK-dimension additivity and Aut-stability, and proves cancellativity/bicancellativity results for these classes under suitable hypotheses (e.g., ${u_{good}$-maximal$}$ domains of finite GK-dimension, codimension-one ideals, and affine commutative centers). A Kraft-type characterization is established: in characteristic zero over an algebraically closed field, an affine, connected graded algebra with no nontrivial Aut-stable subspaces is a polynomial ring $ m k[z_1,\, ilde z_m]$ for some $m\,\ge 2$, linking Aut-stability to classical polynomial-ring structure. The results together provide a structured, noncommutative extension of cancellation phenomena and connect to longstanding questions in affine geometry such as Kraft’s problem and the Zariski cancellation problem.

Abstract

We introduce and study a relative cancellation property for associative algebras. We also prove a characterization result for polynomial rings which partially answers a question of Kraft.

Theorems & Definitions (47)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Definition 5
• Theorem 6: Theorem \ref{['xxthm2.4']}(1,2)
• Corollary 7
• Theorem 8: Theorem \ref{['xxthm2.5']}(1,2)
• Theorem 9: Theorem \ref{['xxthm2.6']}(1)
• Theorem 11