Normality of Ideals and Modules
Naoki Endo, Shiro Goto, Jooyoun Hong, Bernd Ulrich
TL;DR
The paper identifies practical, verifiable conditions under which the Rees algebra of an integrally closed $\mathfrak m$-primary ideal in a regular local ring is a Cohen–Macaulay normal domain in dimensions $d\ge3$, supplementing the dimension-2 theory. It proves that when $\mu_R(I)\le d+2$ or $v(R/I)\le2$, the Rees algebra $\mathcal{R}(I)$ is CM normal, and extends the theory to zero-dimensional homogeneous ideals with $d+3$ generators in characteristic zero, as well as to modules via generic Bourbaki ideals. The results broaden the scope of known complete-intersection and almost complete-intersection cases and provide a framework for module-level Rees algebras, highlighting the roles of integral closure, reduction, and dimension-reduction techniques. These contributions improve understanding of when normality and Cohen–Macaulayness coincide for Rees algebras in higher dimensions and offer tools for deriving CM-normality in broader algebraic settings.
Abstract
We investigate when the Rees algebra of an integrally closed $\mathfrak{m}$-primary ideal in a regular local ring is a Cohen-Macaulay normal domain. While this property always holds in dimension two, it fails in general in higher dimensions, prompting a search for sufficient conditions on the ideal. We show that if an integrally closed ideal contains a part of regular system of parameters of length $d-2$, where $d$ is the dimension of the regular local ring, then its Rees algebra is Cohen-Macaulay and normal. We also extend results of Goto and Ciupercă by proving the same conclusion when the minimal number of generators of an ideal is at most $d+2$. Furthermore, we treat the case of integrally closed zero-dimensional ideals generated by $d+3$ homogeneous polynomials. Finally, using generic Bourbaki ideals, we generalize these results to integrally closed torsionfree modules of finite colength.