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PI Artin--Schelter regular algebras of dimension 3 are unique factorization rings

Silu Liu, Quanshui Wu

TL;DR

This work addresses the noncommutative analogue of unique factorization by proving that all noetherian PI AS-regular algebras of dimension $3$ are noetherian UFRs, aligning with Auslander–Buchsbaum phenomena for regular rings. The authors develop a divisor-class-group framework and a graded reduction to height-one primes, showing these primes are left and right projective, which forces principal height-one primes and yields the UFR property. Key contributions include a complete $3$-dimensional PI case, a connection between normal and graded class groups, and a direct demonstration that skew polynomial algebras are UFRs, with implications for dimension-$2$ AS-regular algebras and avenues for higher dimensions. The results enhance understanding of factorization in noncommutative AS-regular algebras and provide tools for studying reflexive discriminants and automorphism groups.

Abstract

We prove that all noetherian PI Artin--Schelter regular algebras of dimension $3$ are unique factorization rings. In a certain sense, this result is a noncommutative analogue to the fact that regular local rings of dimension 3 are UFDs. The fact constitutes a crucial component in the proof of the assertion that all regular local rings are UFDs, known as the Auslander--Buchsbaum theorem.

PI Artin--Schelter regular algebras of dimension 3 are unique factorization rings

TL;DR

This work addresses the noncommutative analogue of unique factorization by proving that all noetherian PI AS-regular algebras of dimension are noetherian UFRs, aligning with Auslander–Buchsbaum phenomena for regular rings. The authors develop a divisor-class-group framework and a graded reduction to height-one primes, showing these primes are left and right projective, which forces principal height-one primes and yields the UFR property. Key contributions include a complete -dimensional PI case, a connection between normal and graded class groups, and a direct demonstration that skew polynomial algebras are UFRs, with implications for dimension- AS-regular algebras and avenues for higher dimensions. The results enhance understanding of factorization in noncommutative AS-regular algebras and provide tools for studying reflexive discriminants and automorphism groups.

Abstract

We prove that all noetherian PI Artin--Schelter regular algebras of dimension are unique factorization rings. In a certain sense, this result is a noncommutative analogue to the fact that regular local rings of dimension 3 are UFDs. The fact constitutes a crucial component in the proof of the assertion that all regular local rings are UFDs, known as the Auslander--Buchsbaum theorem.
Paper Structure (5 sections, 25 theorems, 34 equations)

This paper contains 5 sections, 25 theorems, 34 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Unique factorization property of graded algebras
  4. Some noetherian PI AS-regular algebras are UFRs
  5. Some noetherian AS-regular UFRs beyond PI cases

Key Result

Theorem 1.1

(Theorem thm 3-dim UFR) All noetherian PI AS-regular algebras of dimension (no more than) $3$ are UFRs.

Theorems & Definitions (53)

  • Theorem 1.1
  • Proposition 1.2
  • Lemma 1.3
  • Proposition 1.4
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Example 2.5
  • Definition 2.6
  • ...and 43 more