On torsion-free modules and semi-hereditary rings

Abstract

A semi-hereditary ring is one of the key classes in commutative ring theory in which the Noetherian condition is not necessarily assumed. In recent years, within the realm of homological-algebraic commutative algebra, arguments that treat genuinely non-Noetherian rings such as the perfectoid rings—have become increasingly important. Because researchers in these areas have historically focused mainly on Noetherian rings, many challenges still remain in commutative algebra without the Noetherian hypothesis.

In this talk, I will present several results on the structure theory of semi-hereditary rings, with particular emphasis on the relationship between semi-hereditary rings and the flatness of torsion-free modules. I will also examine Shimomoto’s problem [Shi] concerning the flatness of the Frobenius map. This presentation is based on the preprint And24.

Based on paper:

• On torsion-free modules and semi-hereditary rings

Links

• handout (in Japanese)