Notes on C-Graded Modules Over an Affine Semigroup Ring K[C

Abstract

Let C ⊂ ℕ d be an affine semigroup, and R = K[C] its semigroup ring. This article is a collection of various results on “C-graded” R-modules M = ⨁ c∈C M c , especially, monomial ideals of R. For example, we show the following: If R is normal and I ⊂ R is a radical monomial ideal (i.e., R/I is a generalization of Stanley–Reisner rings), then the sequentially Cohen–Macaulay property of R/I is a topological property of the “geometric realization” of the cell complex associated with I. Moreover, we can give a squarefree modules/constructible sheaves version of this result. We also show that if R is normal and I ⊂ R is a Cohen–Macaulay monomial ideal, then is Cohen–Macaulay again.