Cohen-Macaulayness of non-affine normal semigroups

In this paper, we study the Cohen-Macaulayness of non-affine normal semigroups in $\mathbb{Z}^n$. We do this by establishing the following four statements each of independent interest: 1) a Lazard type result on $I$-supported elements of $\prod_{\mathbb{N}}\mathbb{Q}_{\geq0}$ for an index set $I\subset\mathbb{N}$; 2) a criterion of regularity of sequences of elements of the ring via projective dimension; 3) a direct limit of polynomial rings with toric maps; 4) any direct summand of rings of the third item is Cohen-Macaulay. To illustrate the idea, we give many examples.