Primary decompositions of three-generated ideals (Q1910726)

Let \(R\) be a three-generated height two ideal in a regular local ring \(R\) of dimension \(d\) greater than or equal to three and let \(J\) be the intersection of all primary components of \(I\) of height two. If the projective dimension of \(R\) modulo \(J\) is two then the depth of \(R\) modulo \(I\) is greater than or equal to \(d\) minus three. This result is derived from a general and more elementary setting. A negative answer is given to the question of Huneke whether depth\(R\) modulo \(I\) is greater than or equal to depth\(R\) modulo \(J\) minus one if depth\(R\) modulo \(J\) is greater than or equal to two.