Notes onC-Graded Modules Over an Affine Semigroup RingK[C]

Abstract

Abstract. Let C ⊂ N d be an affine semigroup, and R = K[C] its semigroup ring. This paper 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 √ I is Cohen-Macaulay again. IntroductionFirst, we fix the notation used throughout this paper. Let C ⊂ Z d ⊂ R d be an affine semigroup (i.e., C is a finitely generated additive submonoid of Z d ), andOf course, R = c∈C Kx c is a Z d -graded ring. We say a Z d -graded ideal of R is a monomial ideal. Let *Mod R be the category of Z d -graded R-modules and their degree preserving R-homomorphisms, and *mod R its full subcategory consisting of finitely generated modules. As usual, for M ∈ *Mod R and a ∈ Z d , M a denotes the degree a component of M, and M(a) denotes the shifted module of M with M(a) b = M a+b . We say M ∈ *Mod R is C-graded, if M a = 0 for all a ∈ C. A monomial ideal I ⊂ R and the quotient ring R/I are C-graded modules. Let *mod C R be the full subcategory of *mod R consisting of C-graded modules.Miller [14] proved that *mod C R has enough injectives and any object has a minimal injective resolution in this category, which is unique up to isomorphism and has finite length. This resolution is called a minimal irreducible resolution, since an indecomposable injective in *mod C R corresponds to a monomial irreducible ideal.In §2, under the assumption that R is Cohen-Macaulay and simplicial, we show that information on M ∈ *mod C R such as depth and Cohen-Macaulay property can be read off from numerical invariants of the minimal irreducible resolution of M (something analogous to "Bass numbers"). One might think these results should be 2000 Mathematics Subject Classification. Primary 13D02; Secondary 13H10, 13F55, 13D45.

Dualizing Complex of a Toric Face Ring

“…When R is a normal semigroup ring, the second author showed in [18,Lemma 3.8] that there is a natural isomorphism between D and RHom(−, D • R ). The next result generalizes this to toric face rings.…”

Linearity defects of face rings

“…But, even if V = M a for some M ∈ * mod R and a ∈ Z n , we set the degree of V * to be 0. [19,Lemma 3.8]…”

“…which is a direct summand of I i , and the differential lin l (I • ) i → lin l (I • ) i+1 is the corresponding component of the differential [19,Theorem 3.9].) With the above notation, we have…”

Lyubeznik Numbers of Local Rings and Linear Strands of Graded Ideals

“…However, we cannot drop this assumption, since we have no idea whether the condition of being sequentially Cohen-Macaulay is preserved after taking radicals. What is known is that if R/I is Cohen-Macaulay then so is R/ √ I (see [30,Theorem 6.1]). Hence if R/I is Cohen-Macaulay then the Lyubeznik table of R/I is trivial.…”