On Cohen-Macaulay non-prime collections of cells

TL;DR

The paper extends the study of Cohen–Macaulayness and Gorensteinness to non-prime collections of cells by developing a combinatorial–algebraic framework centered on inner $2$-minors and coordinate rings $K[\mathcal{P}]$. It introduces zig-zag collections and proves their coordinate rings are Cohen–Macaulay, with Hilbert-Poincaré data encoded by switching rook polynomials. The main result establishes that closed path polyominoes yield Cohen–Macaulay coordinate rings of dimension $|V(\mathcal{P})|-|\mathcal{P}|$, normality when zig-zag walks are absent, and a precise Gorenstein criterion: $K[\mathcal{P}]$ is Gorenstein exactly when $\mathcal{P}$ decomposes into maximal blocks of rank three. When zig-zag walks are present, $K[\mathcal{P}]$ is not Gorenstein. The work provides explicit Hilbert-Poincaré descriptions in terms of rook polynomials, connecting algebraic invariants to polyomino geometry and enabling a unified treatment of non-prime cases via Gröbner basis techniques and decomposition into components.

Abstract

In this paper we investigate Cohen-Macaulayness, Gorensteinness and the Hilbert-Poincaré series for some classes of non-prime collections of cells. In particular, we show that all closed path polyominoes are Cohen-Macaulay and we characterize those that are Gorenstein.

Figures (16)

• Figure 1: A non-simple polyomino, a non-simple collection of cells, a simple thin polyomino and a simple thin collection of cells.
• Figure 2: Non-simple polyominoes.
• Figure 3: Examples of closed paths.
• Figure 4: Arrangements of intervals.
• Figure 5: Collections of cells containing a zig-zag walk.

Theorems & Definitions (52)

• Lemma 1.1
• proof
• Remark 1.2
• Proposition 1.3
• proof
• Definition 1.4
• Lemma 2.1
• proof
• Theorem 2.2
• proof