[Paper Review] Normal Sally modules of rank one

Phuong Tran Thi|arXiv (Cornell University)|Jun 17, 2015

Commutative Algebra and Its Applications12 references3 citations

TL;DR

This paper investigates normal Sally modules of rank one in analytically unramified Nagata reduced local rings, establishing that extremal bounds on the first normal Hilbert coefficient $\overline{e}_1(I)$ imply depth($\overline{\mathcal{G}}$) $\geq \dim R - 1$, with $\overline{\mathcal{G}}$ becoming Cohen-Macaulay when the third normal Hilbert coefficient vanishes. The key contribution is a characterization of rank one normal Sally modules via module-theoretic and cohomological conditions on the associated graded ring.

ABSTRACT

In this paper, we explore the structure of the normal Sally modules of rank one with respect to an $m$-primary ideal in a Nagata reduced local ring which is not necessary Cohen-Macaulay. As an application of this result, when the base ring is Cohen-Macaulay analytically unramified, the extremal bound on the first normal Hilbert coefficient leads to the Cohen-Macaulayness of the associated graded rings with respect to a normal filtration.

Motivation & Objective

• To extend the theory of normal Sally modules to non-Cohen-Macaulay local rings.
• To establish a precise relationship between the first normal Hilbert coefficient $\overline{e}_1(I)$ and the depth of the associated graded ring $\overline{\mathcal{G}}$.
• To determine conditions under which $\overline{\mathcal{G}}$ becomes Cohen-Macaulay, particularly when $\overline{e}_3(I) = 0$.
• To generalize results from Cohen-Macaulay rings to reduced Nagata local rings satisfying conditions $(C_1)$, $(C_2)$, and $(C_3)$.

Proposed method

• Define the normal Sally module $\overline{S}_{Q}(I)$ as the cokernel of the inclusion $\overline{I}T \hookrightarrow \overline{\mathcal{R}}_+(1)$, where $T = R[Qt]$.
• Use the graded structure of the extended Rees algebra $\overline{\mathcal{R}}$ and the associated graded ring $\overline{\mathcal{G}}$ to analyze module-theoretic properties.
• Apply the conditions $(C_1)$, $(C_2)$, and $(C_3)$ to control the behavior of the filtration $\{\overline{I^n}\}$ and ensure the validity of depth bounds.
• Employ the equality $\overline{\mathcal{R}} = \oplus_{n \geq 0} \overline{I^n}t^n$ and the finite generation of $\overline{\mathcal{R}}$ over $T$ to derive cohomological constraints.
• Use the isomorphism $\overline{\mathcal{R}}_+(1) \cong \oplus_{n \geq 0} \overline{I^{n+1}}t^n$ to express $\overline{S}$ as $\oplus_{n \geq 1} \overline{I^{n+1}} / Q^n \overline{I}$.
• Leverage known results on Cohen-Macaulayness of Rees algebras and associated graded rings via the $a$-invariant and module-theoretic criteria.

Experimental results

Research questions

• RQ1 When does $\overline{e}_1(I) = e_0(I) + e_1(Q) - \ell_R(R/\overline{I}) + 1$ imply that $\overline{S}_{Q}(I)$ is a rank one module over $B = T/\mathfrak{m}T$?
• RQ2 What depth bounds on $\overline{\mathcal{G}}$ are implied by the extremal value of $\overline{e}_1(I)$ in non-Cohen-Macaulay rings?
• RQ3 Under what conditions is $\overline{\mathcal{G}}$ Cohen-Macaulay when $\overline{e}_3(I) = 0$?
• RQ4 How does the structure of $\overline{S}_{Q}(I)$ relate to the Cohen-Macaulayness of $\overline{\mathcal{R}}$ and $\overline{\mathcal{G}}$ in dimension $d \leq 2$?

Key findings

• The normal Sally module $\overline{S}_{Q}(I)$ is isomorphic to $B(-q)$ for some $q \geq 1$ if and only if $\overline{e}_1(I) = e_0(I) + e_1(Q) - \ell_R(R/\overline{I}) + 1$, under conditions $(C_1)$, $(C_2)$, and $(C_3)$.
• When $\overline{e}_1(I) = e_0(I) + e_1(Q) - \ell_R(R/\overline{I}) + 1$, the module $\overline{S}_{Q}(I)$ is Cohen-Macaulay as a $T$-module.
• The depth of $\overline{\mathcal{G}}$ satisfies $\mathrm{depth}\,\overline{\mathcal{G}} \geq \max\{d-1, t\}$, where $t = \mathrm{depth}\,R$, under the same coefficient condition.
• If $\overline{e}_3(I) = 0$ and $\overline{e}_1(I) = e_0(I) - \ell_R(R/\overline{I}) + 1$, then $\overline{\mathcal{G}}$ is Cohen-Macaulay.
• For $d=2$, $\overline{\mathcal{G}}$ may be Cohen-Macaulay even if $\overline{\mathcal{R}}$ is not, as shown by a counterexample with $\dim R=2$, $\mathrm{e}_0(\mathfrak{m})=3$, and $\ell_R(\mathfrak{m}^2/Q\mathfrak{m})=1$.
• In dimension one, $\overline{\mathcal{R}}$ is Cohen-Macaulay if and only if $R$ is a discrete valuation ring, under the assumption that $\overline{\mathcal{G}}$ is Cohen-Macaulay.