On the Rees algebra and the conductor of an ideal

TL;DR

This paper introduces the conductor of an ideal as a new tool to analyze the Rees algebra, providing criteria for properties like being of linear type and comparing Rees and symmetric algebras.

Contribution

It defines the conductor of an ideal, generalizes existing results, and characterizes the conductor for specific classes of monomial ideals.

Findings

Conductor helps determine when Rees and symmetric algebras share properties.

Criteria established for when adding elements preserves linear type.

Explicit descriptions of conductors for certain monomial ideals.

Abstract

For an ideal $I$ in a Noetherian ring $R$, we introduce and study its conductor as a tool to explore the Rees algebra of $I$. The conductor of $I$ is an ideal $C(I)\subset R$ obtained from the defining ideals of the Rees algebra and the symmetric algebra of $I$ by a colon operation. Using this concept we investigate when adding an element to an ideal preserves the property of being of linear type. In this regard, a generalization of a result by Valla in terms of the conductor ideal is presented. When the conductor of a graded ideal in a polynomial ring is the graded maximal ideal, a criteria is given for when the Rees algebra and the symmetric algebra have the same Krull dimension. Finally, noting the fact that the conductor of a monomial ideal is a monomial ideal, the conductor of some families of monomial ideals, namely bounded Veronese ideals and edge ideals of graphs, are determined.