A Brief Survey on Pure Cohen–Macaulayness in a Fixed Codimension

Abstract

A concept of Cohen-Macaulay in codimension t is defined and characterized for arbitrary finitely generated modules and coherent sheaves by Miller, Novik, and Swartz in 2011. Soon after, Haghighi, Yassemi, and Zaare-Nahandi defined and studied CM t simplicial complexes, which is the pure version of the abovementioned concept and naturally generalizes both Cohen-Macaulay and Buchsbaum properties. The purpose of this paper is to survey briefly recent results of CM t simplicial complexes.