Systems of parameters and the Cohen--Macaulay property

Abstract

We recall a numerical criteria for Cohen--Macaulayness related to system of parameters, and introduce monomial ideals of K\"onig type which include the edge ideals of K\"onig graphs. We show that a monomial ideal is of K\"onig type if and only if its corresponding residue class ring admits a system of parameters whose elements are of the form $x_i-x_j$. This provides an algebraic characterization of K\"onig graphs. We use this special parameter systems for the study of the edge ideal of K\"onig graphs and the study of the order complex of a certain family of posets. Finally, for any simplicial complex $\Delta$ we introduce a system of parameters for $K[\Delta]$ with a universal construction principle, independent of the base field and only dependent on the faces of $\Delta$. This system of parameters is an efficient tool to test Cohen--Macaulayness of the Stanley--Reisner ring of a simplicial complex.