Almost finitely generated inverse systems and reduced k-algebras

TL;DR

This paper characterizes certain algebraic structures like one-dimensional local domains and reduced schemes using Macaulay's inverse systems, introduces almost finitely generated modules, and applies these results to specific algebraic examples.

Contribution

It provides new characterizations of algebraic structures via inverse systems and explicitly computes inverse systems for numerical semigroup rings.

Findings

Characterization of one-dimensional local domains using inverse systems

Explicit computation of inverse systems for numerical semigroup rings

Characterization of reduced arithmetically Gorenstein schemes

Abstract

The purpose of this paper is to characterize one-dimensional local domains, or more in general reduced, in terms of its Macaulay's inverse system. This leads to study almost finitely generated modules in the divided power ring. We specialize the results to a numerical semigroup ring by computing explicitly its inverse system. In the graded case we characterize reduced arithmetically Gorenstein $0$-dimensional schemes.