Trace ideals, normalization chains, and endomorphism rings

In this paper we consider reduced (non-normal) commutative noetherian rings $R$.With the help of conductor ideals and trace ideals of certain $R$-modules we deduce a criterion for a reflexive $R$-module to be closed under multiplication with scalars in an integral extension of $R$. Using results of Greuel and Knörrer this yields a characterization of plane curves of finite Cohen–Macaulay type in terms of trace ideals.

Further, we study one-dimensional local rings $(R, \mathfrak{m})$ such that that their normalization is isomorphic to the endomorphism ring $\operatorname{End}_R (\mathfrak{m})$: we give a criterion for this property in terms of the conductor ideal, and show that these rings are nearly Gorenstein. Moreover, using Grauert–Remmert normalization chains, we show the existence of noncommutative resolutions of singularities of low global dimensions for curve singularities.

2010 Mathematics Subject Classification

Primary 13C14. Secondary 13B22, 13H10, 14B05, 16E10.

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The author is a Marie Skłodowska-Curie fellow at the University of Leeds (funded by the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 789580).

Received 4 January 2019

Accepted 26 November 2019

Published 13 November 2020