Traces of powers of canonical ideals

Abstract

Let $(R,\m)$ be a one-dimensional Cohen--Macaulay local ring with a canonical fractional ideal $R\subseteq K\subseteq \overline{R}$, and let $v$ denote its embedding dimension. Put $σ(R)=\sup\{j\ge0\mid\tr(K^j)\supseteq\m\}$. By definition, $R$ is nearly Gorenstein if and only if $\sigma (R)>0$. We prove that $R$ is almost Gorenstein if and only if $\sigma (R)\ge v$, equivalently, $\sigma (R)=\infty$. Thus every finite value of $\sigma (R)$ is at most $v-1$. We then determine the structure of rings for which $\sigma (R)$ attains its largest finite value $v-1$, just below the threshold for the almost Gorenstein property. This occurs if and only if $K^{v-1}\cong\m$ and $K^v\not\cong\m$. When these conditions hold, $R$ has type two, with multiplicity $e$ greater than $v$ and $e\equiv 3(\mathrm{mod}v-1)$. As an application, we characterize the numerical semigroup rings $R$ with $\sigma (R)=v-1$ as those generated by an arithmetic progression, of type two and multiplicity greater than $v$.