Trace ideals of syzygies

Abstract

We study the behavior of trace ideals under taking syzygies. In particular, when $R$ is numerical semigroup ring and $I$ is a homogeneous ideal in $R$, we obtain an upper estimate for ${\mathrm{tr}}_R({\Omega }_R^1(I))$, and we show this estimate is sharp when $I$ is the conductor ideal ${\mathfrak{c}}_R$ of $R$. Using this result, we characterize the numerical semigroup rings for which ${\mathrm{tr}}_R(M)\subseteq {\mathrm{tr}}_R({\Omega }_R^1(M))$ holds for every finitely generated $R$-module $M$. In a similar vein, we characterize numerical semigroup rings for which ${\mathrm{tr}}_R({\Omega }_R^1({\mathfrak{c}}_R))={\mathfrak{c}}_R$.

16 pages