Initially Cohen-Macaulay Modules

TL;DR

This work introduces initially Cohen–Macaulay modules by replacing Krull dimension with the initial dimension $indim$, defined via the coheights of associated primes, and proves that $depth N \le indim N$ with equality characterizing initial CM. It shows that the first nonzero piece of the dimension filtration governs depth, linking initial CM to sequential CM, and develops both algebraic and combinatorial characterizations, including a generalized Reisner criterion and a degree-resolution criterion for Alexander duals. The theory is then transported to combinatorial settings, where Stanley–Reisner rings and edge ideals yield analogues of Reisner–Murai, Eagon–Reiner, and Duval-type results, and where graphs (notably chordal and co-chordal) are classified with respect to the IC M property and cases where $pdim$ equals the maximum degree. Overall, the work broadens the CM landscape, providing robust tools for analyzing rings associated with simplicial complexes and graphs and offering new structural insights and classifications with potential applications in combinatorial commutative algebra.

Abstract

In this paper, we introduce initially Cohen-Macaulay modules over a commutative Noetherian local ring $R$, a new class of $R$-modules that generalizes both Cohen-Macaulay and sequentially Cohen-Macaulay modules. A finitely generated $R$-module $N$ is initially Cohen-Macaulay if its depth is equal to its initial dimension, an invariant defined as the infimum of the coheights of the associated primes of $N$. We develop the theory of these modules, providing homological, combinatorial, and topological characterizations and confirming their compatibility with regular sequences, localization, and dimension filtrations. When this theory is applied to simplicial complexes, we establish analogues of Reisner's criterion, the Eagon-Reiner theorem, and Duval's characterization of sequentially Cohen-Macaulay complexes. Finally, we classify certain classes of initially Cohen-Macaulay graphs of interest and those whose projective dimension coincides with their maximum vertex degree.

Theorems & Definitions (106)

• Theorem 1.1: Proposition \ref{['depth-mdim']}; c.f. Matsumura1989
• Theorem 1.2: Proposition \ref{['ICM-unmixed']}
• Theorem 1.3: Proposition \ref{['ICM-filtration']}
• Theorem 1.4: Proposition \ref{['prop:ICM']}
• Theorem 1.5: Proposition \ref{['PCM degree resolution']}
• Theorem 1.6: Proposition \ref{['PCM CM']}
• Theorem 1.7: Proposition \ref{['SCM']}
• Theorem 1.8: Proposition \ref{['pro 1']}
• Theorem 1.9: Proposition \ref{['PCM Cn']}
• Theorem 1.10: Proposition \ref{['PCM co-chordal']}