[Paper Review] Sequentially Cohen--Macaulay modules with respect to an irrelevant ideal

Ahad Rahimi|arXiv (Cornell University)|Dec 16, 2011

Commutative Algebra and Its Applications10 references3 citations

TL;DR

This paper introduces the concept of sequentially Cohen–Macaulay modules with respect to the bigraded irrelevant ideal $ Q $ in a standard bigraded polynomial ring $ S $, defining a relative Cohen–Macaulay filtration $ \mathcal{F} $ for finitely generated bigraduated $ S $-modules. It characterizes all hypersurface rings that are sequentially Cohen–Macaulay with respect to $ Q $, and explicitly computes the length of such filtrations.

ABSTRACT

For a finitely generated bigraded $S$-module $M$ where $S$ is a standard bigraded polynomial ring we define the relative Cohen--Macaulay filtration $\mathcal{F}$ of $M$ with respect to the bigraded irrelevant ideal $Q$. We call $M$ to be sequentially Cohen-Macaulay with respect $Q$ if $M$ admits a relative Cohen-Macaulay filtration with respect to $Q$. We investigate the algebraic properties of these modules and compute the length of a relative Cohen-Macaulay filtration with respect to $Q$ explicitly. All hypersurface rings that are sequentially Cohen--Macaulay with respect to $Q$ are classified.

Motivation & Objective

• To define and study sequentially Cohen–Macaulay modules with respect to the bigraded irrelevant ideal $ Q $ in a standard bigraded polynomial ring.
• To introduce the relative Cohen–Macaulay filtration $ \mathcal{F} $ as a structural tool for analyzing such modules.
• To compute the length of the relative Cohen–Macaulay filtration explicitly for finitely generated bigraded $ S $-modules.
• To classify all hypersurface rings that are sequentially Cohen–Macaulay with respect to $ Q $.

Proposed method

• Define the relative Cohen–Macaulay filtration $ \mathcal{F} $ of a bigraded $ S $-module $ M $ with respect to the irrelevant ideal $ Q $, using graded components and associated primes.
• Use the structure of the bigraded ring $ S $ and the properties of $ Q $ to analyze the filtration's existence and length.
• Apply homological algebra techniques to study the Cohen–Macaulay property in the relative setting, focusing on the associated primes and dimension filtration.
• Characterize the condition of sequential Cohen–Macaulayness via the existence of a filtration where each quotient is Cohen–Macaulay with respect to $ Q $.
• Employ the structure of hypersurface rings to determine when they satisfy the sequential Cohen–Macaulay condition with respect to $ Q $.
• Compute the length of the relative Cohen–Macaulay filtration by analyzing the number of non-trivial filtration steps in the dimension filtration.

Experimental results

Research questions

• RQ1 When is a finitely generated bigraded $ S $-module sequentially Cohen–Macaulay with respect to the bigraded irrelevant ideal $ Q $?
• RQ2 What is the explicit length of the relative Cohen–Macaulay filtration for such modules?
• RQ3 Which hypersurface rings are sequentially Cohen–Macaulay with respect to $ Q $, and how can they be characterized?
• RQ4 How does the relative Cohen–Macaulay filtration relate to the dimension and associated prime structure of $ M $?
• RQ5 What algebraic invariants control the sequential Cohen–Macaulay property in the bigraded setting?

Key findings

• The relative Cohen–Macaulay filtration $ \mathcal{F} $ of a bigraded $ S $-module $ M $ with respect to $ Q $ exists and is uniquely determined by the associated primes and dimension filtration of $ M $.
• The length of the relative Cohen–Macaulay filtration is explicitly computable and corresponds to the number of distinct dimensions in the dimension filtration of $ M $.
• All hypersurface rings that are sequentially Cohen–Macaulay with respect to $ Q $ are completely classified in terms of their defining equations and associated prime structure.
• A bigraded $ S $-module $ M $ is sequentially Cohen–Macaulay with respect to $ Q $ if and only if it admits a filtration where each successive quotient is Cohen–Macaulay with respect to $ Q $.
• The relative Cohen–Macaulay filtration provides a structural decomposition that reflects the dimension and depth behavior of $ M $ with respect to $ Q $.