Ideals and their Fitting ideals
David Eisenbud, Antonino Ficarra, Jürgen Herzog, Somayeh Moradi
TL;DR
The paper investigates when Fitting ideals $\operatorname{Fitt}_j(I)$ of an ideal $I$ coincide with $I$ or share radicals, connecting these questions to the Hilbert–Burch theorem and the canonical module $\omega_R$. It develops general bounds $I^{m-j}\subseteq \operatorname{Fitt}_j(I)$ (with equality for regular sequences) and establishes radical equalities $\sqrt{\operatorname{Fitt}_i(I)}=\sqrt{I}$ for a broad range of $i$, while also relating Fitt$_1(I)$ to the trace $\operatorname{tr}(I)$. The paper extends these ideas to canonical modules, showing $\sqrt{\operatorname{Fitt}_1(\omega_R)}=\sqrt{\operatorname{tr}(\omega_R)}$ and deriving conditions under which $\operatorname{Fitt}_1(\omega_R)=\omega_R$ forces Gorensteinness, with a special emphasis on CM type. A key contribution is a complete treatment for squarefree monomial ideals: if $\operatorname{grade} I\ge j$, then $\operatorname{Fitt}_{j-1}(I)=I$ is equivalent to $\operatorname{Fitt}_{j-1}(I)$ being squarefree and to $I$ having a regular sequence of length $j$ (for $j>2$) or being perfect of grade $2$ (for $j=2$). The section on edge ideals provides a graph-theoretic description of $\sqrt{\operatorname{Fitt}_j(I(G))}$ via independent sets and admissible covers, highlighting the combinatorial depth of Fitting radicals. Overall, the work clarifies when Fitt$_j(I)$ reproduces $I$ or has the same radical, and it links these phenomena to structural properties like being unmixed, regularity, and graph combinatorics.
Abstract
For an ideal $I$ in a Noetherian ring $R$, the Fitting ideals $\textrm{Fitt}_j(I)$ are studied. We discuss the question of when $\textrm{Fitt}_j(I)=I$ or $\sqrt{\textrm{Fitt}_j(I)}=\sqrt{I}$ for some $j$. A classical case is the Hilbert-Burch theorem when $j=1$ and $I$ is a perfect ideal of grade $2$ in a local ring.