The Containment Problem: Background

The study of ideals underlies both algebra and geometry. For example, the study of homogeneous ideals in polynomial rings is an aspect of both commutative algebra and of algebraic geometry. In both cases, given an ideal, one wants to understand how the ideal behaves. One way in which algebra and geometry differ is in what it means to be “given an ideal”. For an algebraist it typically means being given generators of the ideal. For a geometer it often means being given a locus of points (or a scheme) in projective space, the ideal then being all elements of the polynomial ring which vanish on the given locus or scheme. Determining generators for the ideal defining a scheme sometimes requires significant effort, and if given generators a geometer will usually want to know what vanishing locus they cut out. Thus while both algebraists and geometers study ideals, their starting points are different.