Almost Gorenstein Rees Algebras of pg-Ideals, Good Ideals, and Powers of the Maximal Ideals

Abstract

Let $(A,\mathfrak{m})$ be a Cohen–Macaulay local ring, and let $I$ be an ideal of $A$. We prove that the Rees algebra $\mathcal{R}\left(I\right)$ is an almost Gorenstein ring in the following cases:

(1) $(A,\mathfrak{m})$ is a two-dimensional excellent Gorenstein normal domain over an algebraically closed field $K\cong A/\mathfrak{m}$, and $I$ is a $p_g$-ideal;

(2) $(A,\mathfrak{m})$ is a two-dimensional almost Gorenstein local ring having minimal multiplicity, and $I={\mathfrak{m}}^{\ell }$ for all $\ell \ge 1$;

(3) $(A,\mathfrak{m})$ is a regular local ring of dimension $d\ge 2$, and $I={\mathfrak{m}}^{d-1}$. Conversely, if $\mathcal{R}\left({\mathfrak{m}}^{\ell }\right)$ is an almost Gorenstein graded ring for some $\ell \ge 2$ and $d\ge 3$, then $\ell =d-1$.