[Paper Review] The almost Gorenstein Rees algebras over two-dimensional regular local rings

Shirô Gotô, Naoyuki Matsuoka+2 more|arXiv (Cornell University)|Jun 22, 2015

Commutative Algebra and Its Applications7 references3 citations

TL;DR

This paper proves that the Rees algebra $ρ(I) = \bigoplus_{n\geq 0} I^n$ of any $\mathfrak{m}$-primary integrally closed ideal $I$ in a two-dimensional regular local ring $(R,\mathfrak{m})$ with infinite residue field is an almost Gorenstein graded ring. The result relies on Verma's joint reduction theory and extends the class of known almost Gorenstein Rees algebras, showing they are prevalent in dimension two despite Gorenstein Rees algebras being rare.

ABSTRACT

Let $(R,\mathfrak{m})$ be a two-dimensional regular local ring with infinite residue class field. Then the Rees algebra $\mathcal{R} (I)= \bigoplus_{n \ge 0}I^n$ of $I$ is an almost Gorenstein graded ring in the sense of Goto-Takahashi-Taniguchi for every $\mathfrak{m}$-primary integrally closed ideal $I$ in $R$.

Motivation & Objective

• To determine when the Rees algebra of an ideal in a two-dimensional regular local ring is almost Gorenstein.
• To extend the class of known almost Gorenstein graded rings beyond Gorenstein examples.
• To investigate the prevalence of almost Gorenstein Rees algebras in low-dimensional regular rings.
• To clarify the role of joint reductions and Hilbert-Burch matrices in characterizing almost Gorenstein structures.

Proposed method

• Utilizes Verma's result on the existence of joint reductions with joint reduction number zero for $\mathfrak{m}$-primary ideals in two-dimensional regular local rings.
• Applies the definition of almost Gorenstein graded rings via a short exact sequence $0 \to R \to \mathrm{K}_R(-a) \to C \to 0$ where $\mu_R(C) = e^{0}_{\mathfrak{M}}(C)$.
• Employs Hilbert-Burch theory to describe the presentation of the Rees algebra $\mathcal{R}(I)$ and compute its canonical module $\mathrm{K}_{\mathcal{R}}$.
• Analyzes the structure of the defining matrix of the Rees algebra over a power series ring to determine the $a$-invariant and module conditions.
• Uses localization and base change to reduce the problem to a regular local ring of dimension five, applying [6, Theorem 7.8] to rule out almost Gorenstein structure.
• Applies the criterion from [6, Theorem 7.8] that a local ring is not almost Gorenstein if its defining matrix over a regular local ring of dimension ≥5 is not equivalent to a matrix with a regular system of parameters in the first row.

Experimental results

Research questions

• RQ1 Under what conditions is the Rees algebra of an $\mathfrak{m}$-primary ideal in a two-dimensional regular local ring almost Gorenstein?
• RQ2 Can the theory of joint reductions be used to establish almost Gorenstein properties in Rees algebras?
• RQ3 Are there structural obstructions to the almost Gorenstein property in Rees algebras over higher-dimensional rings?
• RQ4 How does the Hilbert-Burch matrix of the Rees algebra relate to the almost Gorenstein condition?
• RQ5 Is the almost Gorenstein property preserved for Rees algebras of socle ideals in two-dimensional regular local rings?

Key findings

• The Rees algebra $\mathcal{R}(I)$ is almost Gorenstein for every $\mathfrak{m}$-primary integrally closed ideal $I$ in a two-dimensional regular local ring with infinite residue field.
• The Rees algebra $\mathcal{R}(\mathfrak{m}^\ell)$ is almost Gorenstein for all $\ell > 0$, extending the result to powers of the maximal ideal.
• For the Rees algebra of $Q = (x^m, y^n)$, it is almost Gorenstein if and only if $n = 2$, showing a sharp restriction in monomial cases.
• If the generators of $Q$ lie in $\mathfrak{m}^3$, then $\mathcal{R}_{\mathfrak{M}}$ is not almost Gorenstein, indicating a strong obstruction.
• The Rees algebra of a socle ideal $Q:\mathfrak{m}$ is not necessarily almost Gorenstein, even in dimension two, as shown by counterexamples.
• The defining matrix of the Rees algebra over a regular local ring of dimension five cannot be transformed into a form with a regular system of parameters in the first row, which by [6, Theorem 7.8] implies non-almost Gorenstein structure.