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CI-sequences and almost complete intersections

Giuseppe Zappalà

Abstract

We study the Hilbert function and the graded Betti numbers of almost complete intersection artinian algebras. We show that that every Hilbert function of a complete intersection artinian algebra is the Hilbert function of an almost complete intersection algebra. In codimension $3$ we focus on almost complete intersection artinian algebras whose Hilbert function coincides with that of a complete intersection defined by $3$ forms of the same degree. We classify all the possible graded Betti numbers of such algebras and we specify what cancellations are allowed in a minimal graded free resolution.

CI-sequences and almost complete intersections

Abstract

We study the Hilbert function and the graded Betti numbers of almost complete intersection artinian algebras. We show that that every Hilbert function of a complete intersection artinian algebra is the Hilbert function of an almost complete intersection algebra. In codimension we focus on almost complete intersection artinian algebras whose Hilbert function coincides with that of a complete intersection defined by forms of the same degree. We classify all the possible graded Betti numbers of such algebras and we specify what cancellations are allowed in a minimal graded free resolution.
Paper Structure (4 sections, 13 theorems, 25 equations)

This paper contains 4 sections, 13 theorems, 25 equations.

Table of Contents

  1. Introduction
  2. Preliminary facts
  3. Almost Complete Intersection sequences
  4. Graded Betti numbers

Key Result

Theorem 2.1

Let $(d_1,\ldots, d_{2n+1}),$$d_1 \le\ldots\le d_{2n+1},$ be a sequence of positive integers. It is the sequence of the degrees of the minimal generators of a Gorenstein ideal of codimension $3$ iff

Theorems & Definitions (31)

  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • Definition 2.3
  • Remark 2.4
  • Definition 3.1
  • Proposition 3.2
  • proof
  • Remark 3.3
  • ...and 21 more