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.

Key Words:

• Affine semigroup ring
• Constructible sheaf
• Gorenstein*
• Irreducible resolution
• Local cohomology
• Monomial ideal
• Sequentially Cohen–Macaulay
• Squarefree modules

2000 Mathematics Subject Classification:

• Primary 13D02
• Secondary 13H10, 13F55, 13D45

ACKNOWLEDGMENTS

The author is grateful to Professor Ngo Viet Trung for telling him the ring given in Example 2.5(2). He also thanks Professors Ezra Miller and Tim Römer for careful reading and comments on an earlier version of this article, and Professor Mitsuyasu Hashimoto for useful comment around Lemma 5.11.

Notes

Communicated by W. Bruns.