The $a$-invariant and Gorensteinness of graded rings associated to filtrations of ideals in regular local rings

Abstract:

Let $A$ be a regular local ring and let $\mathcal {F} = \{F_{n}\}_{n \in \mathbb {Z}}$ be a filtration of ideals in $A$ such that $\mathcal {R}(\mathcal {F}) = \bigoplus _{n \geq 0}F_{n}$ is a Noetherian ring with $\mathrm {dim} \mathcal {R}(\mathcal {F}) = \mathrm {dim} A + 1$. Let $\mathcal {G}(\mathcal {F}) = \bigoplus _{n \geq 0}F_{n}/F_{n+1}$ and let $\mathrm {a}(\mathcal {G}(\mathcal {F}))$ be the $a$-invariant of $\mathcal {G}(\mathcal {F})$. Then the theorem says that $F_{1}$ is a principal ideal and $F_{n} = F_{1}^{n}$ for all $n \in \mathbb {Z}$ if and only if $\mathcal {G}(\mathcal {F})$ is a Gorenstein ring and $\mathrm {a}(\mathcal {G}(\mathcal {F})) = -1$. Hence $\mathrm {a}(\mathcal {G}(\mathcal {F})) \leq -2$, if $\mathcal {G}(\mathcal {F})$ is a Gorenstein ring, but the ideal $F_{1}$ is not principal.