On torsion-free modules and semi-hereditary rings

概要

半遺伝環（semi-hereditary ring）は，必ずしも Noether 性を仮定しない可換環論において重要なクラスの１つである．近年，ホモロジー代数周辺の可換環論では，パーフェクトイド環論など非 Noether 環を本質的な道具として扱う議論の重要性が増してきている．歴史的には，これらの分野の研究者の主な興味の対象は Noether 環であったこともあり，Noether 性を仮定しない可換環論には依然様々な課題が存在する．本講演では，半遺伝環の構造論に関するいくつかの結果を，半遺伝環と torsion-free 加群の平坦性との関係を中心に議論する．また，フロベニウス写像の平坦性に関する下元 [Shi] の問題についても考察する．本稿はプレプリント [And24] に基づく．

Semi-hereditary rings form an important class in commutative algebra where the Noetherian property is not necessarily assumed. In recent years, within commutative algebra and its surrounding homological algebra, arguments that utilize non-Noetherian rings as essential tools—such as the theory of perfectoid rings—have grown increasingly important. Historically, the primary focus of researchers in these fields has been on Noetherian rings; as a result, various challenges still remain in commutative algebra without the Noetherian assumption. In this talk, we will discuss several results regarding the structure theory of semi-hereditary rings, focusing primarily on the relationship between semi-hereditary rings and the flatness of torsion-free modules. Furthermore, we will also consider a question posed by Shimomoto [Shi] concerning the flatness of the Frobenius map. This manuscript is based on the preprint [And24].