Cohen-Macaulayness of non-affine normal semigroups

Abstract

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 ${\prod }_{\mathbb{N}}{\mathbb{Q}}_{\ge 0}$ 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.

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