Set-Theoretically Perfect Ideals and Residual Intersections

This paper studies algebraic residual intersections in rings with Serre’s condition $S_s$. It demonstrates that residual intersections admit free approaches i.e. perfect subideal with the same radical. This fact leads to determining a uniform upper bound for the multiplicity of residual intersections. In positive characteristic, it follows that residual intersections are cohomologically complete intersection and, hence, their variety is connected in codimension one.