# Minimal Cohen-Macaulay Simplicial Complexes

## TL;DR

This paper introduces the concept of minimal Cohen-Macaulay simplicial complexes, demonstrating their foundational role and providing methods for their construction, with implications for combinatorial topology.

## Contribution

It defines minimal Cohen-Macaulay complexes, proves their universality as shelled over complexes, and offers conditions and methods for constructing such complexes.

## Key findings

- Any Cohen-Macaulay complex is shelled over a minimal one.
- Many known Cohen-Macaulay complexes are minimal.
- The paper provides construction techniques for minimal Cohen-Macaulay complexes.

## Abstract

We define and study the notion of a minimal Cohen-Macaulay simplicial complex. We prove that any Cohen-Macaulay complex is shelled over a minimal one in our sense, and we give sufficient conditions for a complex to be minimal Cohen-Macaulay. We show that many interesting examples of Cohen-Macaulay complexes in combinatorics are minimal, including Rudin's ball, Ziegler's ball, the dunce hat, and recently discovered non-partitionable Cohen-Macaulay complexes. We further provide various ways to construct such complexes.