Remarks on the non-Cohen-Macaulay locus of Noetherian schemes

In this paper we give a notion of polynomial type $p(X)$ of a Noetherian scheme $X$ and define the function $dp:\, X\longrightarrow \mathbb{Z}$ by $dp(x)=\dim O_{X,x} -p(O_{X,x} )$ for all $x\in X.$ Then we show that if $X$ admits a dualizing complex and $X$ is equidimensional, $dp$ is (lower) semicontinuous; moreover, in that case, the non-Cohen-Macaulay locus nCM$(X)=\{ x\in X\mid O_{X,x}$ is not Cohen-Macaulay} is biequidimensional iff $dp$ is constant on nCM$(X).$