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arXiv · 2407.06922

On the Rees algebra and the conductor of an ideal

Oleksandra Gasanova·Jürgen Herzog·Filip Jonsson Kling·Somayeh Moradi
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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.

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Oleksandra Gasanova, Jürgen Herzog, Filip Jonsson Kling, Somayeh Moradi. 2024-07-09. On the Rees algebra and the conductor of an ideal. https://arxiv.org/abs/2407.06922

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