Cohen Factorizations: Weak Functoriality and Applications

TL;DR

This paper explores Cohen factorizations of local ring homomorphisms, establishing a weak functoriality property, and applying these factorizations to analyze properties of local rings and their deformations.

Contribution

It introduces a weak functoriality result for Cohen factorizations and applies this to study properties and deformations of local rings.

Findings

Proved a weak functoriality theorem for Cohen factorizations.

Applied Cohen factorizations to analyze properties like Gorenstein and Cohen-Macaulay.

Investigated the structure of quasi-deformations and CI-dimension behavior.

Abstract

We investigate Cohen factorizations of local ring homomorphisms from three perspectives. First, we prove a "weak functoriality" result for Cohen factorizations: certain morphisms of local ring homomorphisms induce morphisms of Cohen factorizations. Second, we use Cohen factorizations to study the properties of local ring homomorphisms (Gorenstein, Cohen-Macaulay, etc.) in certain commutative diagrams. Third, we use Cohen factorizations to investigate the structure of quasi-deformations of local rings, with an eye on the question of the behavior of CI-dimension in short exact sequences.