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arXiv · 2108.00414

When are trace ideals finite?

Shinya Kumashiro
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Abstract

In this paper, we study Noetherian local rings $R$ having a finite number of trace ideals. We proved that such rings are of dimension at most two. Furthermore, if the integral closure of $R/H$, where $H$ is the zeroth local cohomology, is equi-dimensional, then the dimension of $R$ is at most one. In the one-dimensional case, we can reduce to the situation that rings are Cohen-Macaulay. Then, we give a necessary condition to have a finite number of trace ideals in terms of the value set obtained by the canonical module. We also gave the correspondence between trace ideals of $R$ and those of the endomorphism algebra of the maximal ideal of $R$ when $R$ has minimal multiplicity.

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Shinya Kumashiro. 2021-08-01. When are trace ideals finite?. https://arxiv.org/abs/2108.00414

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