A Structure Theorem and the Graded Betti Numbers for Almost Complete Intersections

TL;DR

This paper establishes a structure theorem for almost complete intersection ideals of depth three, revealing their generators as pfaffians, and characterizes their graded Betti numbers in projective space.

Contribution

It introduces a new structure theorem for these ideals and provides a complete description of their graded Betti numbers, advancing understanding of their algebraic properties.

Findings

Generators are pfaffians of submatrices of an alternating matrix

Characterization of graded Betti numbers for 3-codimensional schemes

Applicable to any Noetherian local ring

Abstract

We provide a structure theorem for all almost complete intersection ideals of depth three in any Noetherian local ring. In particular, we find that the minimal generators are the pfaffians of suitable submatrices of an alternating matrix. From the graded version of the previous result, we characterize the graded Betti numbers of all 3-codimensional almost complete intersection schemes of P^r.