Divisorial algebras and modules on schemes

TL;DR

This paper investigates modules over Cartier divisor algebras on schemes and introduces an inductive approach to analyze their finite generation, linking these properties to key conjectures in the minimal model program.

Contribution

It presents a novel inductive method for studying finite generation of algebras and modules, connecting these properties to the minimal model and abundance conjectures.

Findings

Finite generation of log canonical algebras is equivalent to the minimal model conjecture.

Finite generation of modules is linked to the abundance conjecture.

An inductive framework for analyzing algebraic structures on schemes.

Abstract

We study certain modules over the algebra of a Cartier divisor on a scheme. Using these modules, we present an inductive method for studying finite generation properties of algebras and modules. In the context of the minimal model program, we show that finite generation of log canonical algebras and modules is equivalent to the minimal model and abundance conjectures.