Affine semigroups and Cohen-Macaulay rings generated by monomials

Trung, Ngô Viêt; Hoa, Lê Tuâń

doi:10.1090/s0002-9947-1986-0857437-3

Cited by 66 publications

(23 citation statements)

References 24 publications

(22 reference statements)

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“…In particular, we will show that for every simplicial complex Δ there exists a seminormal affine monoid M such that the only obstruction to the Cohen-Macaulay property of K[M] is exactly the simplicial homology of Δ. Choosing Δ as a triangulation of the real projective plane we obtain an example whose Cohen-Macaulay property depends on K. A similar result was proved by Trung and Hoa [18]. Our construction has the advantage of yielding a seminormal monoid M, and is geometrically very transparent.…”

Section: Introductionsupporting

confidence: 59%

On seminormal monoid rings

Bruns

Römer

2006

Journal of Algebra

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Given a seminormal affine monoid M we consider several monoid properties of M and their connections to ring properties of the associated affine monoid ring K[M] over a field K. We characterize when K[M] satisfies Serre's condition (S 2 ) and analyze the local cohomology of K [M]. As an application we present criteria which imply that K[M] is Cohen-Macaulay and we give lower bounds for the depth of K [M]. Finally, the seminormality of an arbitrary affine monoid M is studied with characteristic p methods.

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Section: Introductionsupporting

confidence: 59%

On seminormal monoid rings

Bruns

Römer

2006

Journal of Algebra

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“…In the first part of this article we consider the S 2 -fication and give some general results on the Cohen-Macaulayness of the canonical module, one of them extends and improves Proposition 2.5 of Goto (1982). We also extend and improve to the lattice case the above results from Goto et al (1976) and Trung and Hoa (1986), given shorter proofs.…”

Section: Moralesmentioning

confidence: 87%

“…In Goto et al (1976) and Trung and Hoa (1986) the authors consider a semigroup S ⊂ S which contains S such that we have an exact sequence: 0 −→ K S −→ K S −→ K S \S −→ 0 and dim K S \S ≤ dim K S − 2 K S is the S 2 -fication of K S . When K S is a Cohen-Macaulay ring, the support of K S\S coincide with the non-CohenMacaulay locus of K S .…”

Section: Moralesmentioning

confidence: 99%

On theS₂-Fication of Some Toric Varieties

Morales

2007

Communications in Algebra

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“…If the canonical log structure on Spec k[P ] satisfies ( * ), it need not be Cohen-Macaulay nor normal. Consider the monoids (4, 0), (3, 1), (1, 3), (0, 4) ⊂ N 2 and 2, 3 ⊂ N. Information on when affine semigroup rings are Cohen-Macaulay and its dependence on the characteristic, can be found in [32,33]. Information on the local cohomology modules and dualizing complexes of affine semigroup rings is found in [34] and [35].…”

Section: Some Properties Of Toric Singularitiesmentioning

confidence: 99%

Toric singularities revisited

Thompson

2006

Journal of Algebra

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In [K. Kato, Toric singularities, Amer. J. Math. 116 (5) (1994) 1073-1099], Kato defined his notion of a log regular scheme and studied the local behavior of such schemes. A toric variety equipped with its canonical logarithmic structure is log regular. And, these schemes allow one to generalize toric geometry to a theory that does not require a base field. This paper will extend this theory by removing normality requirements. Conventions and notationAll monoids considered in this paper are commutative and cancellative. All rings considered in this paper are commutative and unital. See Kato [2] for an introduction to log schemes. There Kato defines pre-log structures and log structures on the étale site of X. However, we will use the Zariski topology throughout this paper. P * the unit group of the monoid P . P the sharp image of the monoid P , P = P /P * is the orbit space under the natural action of P * on P . P + P + = P \ P * is the maximal ideal of the monoid P .P gp the group generated by P , that is, image of P under the left adjoint of the inclusion functor from Abelian groups to monoids. P sat the saturation of P , that is, {p ∈ P gp | np ∈ P for some n ∈ N + }. R[P ] the monoid algebra of P over a ring R. The elements of R[P ] are written as "polynomials". That is, they are finite sums r p t p with coefficients in R and exponents in P . (K)the ideal β(K)A, where β : P → A is a monoid homomorphism with respect to multiplication on A and K is an ideal of P . We say such an ideal is a log ideal of A. R [[P ]] the (P + )-adic completion of R[P ]. P p the localization of the monoid P at the prime ideal p ⊆ P . dim P the (Krull) dimension of the monoid P .

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Affine semigroups and Cohen-Macaulay rings generated by monomials

Cited by 66 publications

References 24 publications

On seminormal monoid rings

On seminormal monoid rings

On theS₂-Fication of Some Toric Varieties

Toric singularities revisited

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Affine semigroups and Cohen-Macaulay rings generated by monomials

Cited by 66 publications

References 24 publications

On seminormal monoid rings

On seminormal monoid rings

On theS2-Fication of Some Toric Varieties

Toric singularities revisited

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On theS₂-Fication of Some Toric Varieties