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Almost finitely generated inverse systems and reduced k-algebras

Joan Elias, Maria Evelina Rossi

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.

Almost finitely generated inverse systems and reduced k-algebras

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 -dimensional schemes.

Paper Structure

This paper contains 5 sections, 10 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Inverse systems of local domains
  4. Inverse system of numerical semigroup rings
  5. Inverse systems of reduced rings

Key Result

Theorem 2.2

There is a one-to-one correspondence $\mathcal{C}$ between the following sets: (i) one-dimensional Gorenstein ${\mathbf{k}}$-algebras $A=R/I$, (ii) non-zero $G$-admissible $R$-submodules $M=\langle H_l, l\in \mathbb N_+\rangle$ of $\Gamma$. In particular, given an ideal $I\subset R$ with $A= R/I$ sa is $G$-admissible. Conversely, given an $R$-submodule $M$ of $\Gamma$ satisfying (ii), then

Theorems & Definitions (28)

  • Definition 2.1
  • Theorem 2.2: ER17, Theorem 3.8
  • Definition 3.1
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • proof
  • Remark 3.4
  • Lemma 3.5
  • proof
  • ...and 18 more