Trace ideals of syzygies

Abstract

We study the behavior of trace ideals under taking syzygies. In particular, if $R$ is a Noetherian local ring and if $I$ is an ideal in $R$ of positive grade, we show that the trace ideal of $I$ is contained in that of its syzygy. When $R$ is numerical semigroup ring and $I$ is homogeneous, we also obtain an upper estimate for $\operatorname{tr}_R(Ω^1_R(I))$, and we show this estimate is sharp when $I$ is the conductor ideal $\mathfrak{c}_R$ of $R$. Using these results, we characterize the numerical semigroup rings for which $\operatorname{tr}_R(M) \subseteq \operatorname{tr}_R(Ω^1_R(M))$ holds for every finitely generated $R$-module $M$. In a similar vein, we characterize numerical semigroup rings for which $\operatorname{tr}_R(Ω^1_R(\mathfrak{c}_R))=\mathfrak{c}_R$.