Upper Bounds for Local Cohomology for Rings With Given Hilbert Function

Sbarra, Enrico

doi:10.1081/agb-100107934

Cited by 30 publications

(33 citation statements)

References 11 publications

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“…The proof follows from a standard deformation argument making use of flat families; see [13,Chap.15] and [25] for details. We will just sketch it.…”

Section: Preliminariesmentioning

confidence: 99%

Koszul homology and extremal properties of Gin and Lex

Conca

2003

Trans. Amer. Math. Soc.

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Abstract. For every homogeneous ideal I in a polynomial ring R and for every p ≤ dim R we consider the Koszul homology H i (p, R/I) with respect to a sequence of p of generic linear forms. The Koszul-Betti number β ijp (R/I) is, by definition, the dimension of the degree j part of H i (p, R/I). In characteristic 0, we show that the Koszul-Betti numbers of any ideal I are bounded above by those of the gin-revlex Gin(I) of I and also by those of the Lex-segment Lex(I) of I. We show that β ijp (R/I) = β ijp (R/Gin(I)) iff I is componentwise linear and that and β ijp (R/I) = β ijp (R/Lex(I)) iff I is Gotzmann. We also investigate the set Gins(I) of all the gin of I and show that the Koszul-Betti numbers of any ideal in Gins(I) are bounded below by those of the gin-revlex of I. On the other hand, we present examples showing that in general there is no J is Gins(I) such that the Koszul-Betti numbers of any ideal in Gins(I) are bounded above by those of J.

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“…The proof follows from a standard deformation argument making use of flat families; see [13,Chap.15] and [25] for details. We will just sketch it.…”

Section: Preliminariesmentioning

confidence: 99%

Koszul homology and extremal properties of Gin and Lex

Conca

2003

Trans. Amer. Math. Soc.

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“…where the first estimate is a consequence of Lemma 3.8 and the second inequality follows from the fact that, by [19]…”

Section: Extended Degree Functions and Monomial Modules 3583mentioning

confidence: 99%

“…Consider first the case that M = F/U is sequentially CM. Sbarra proved in his thesis (see [19] for a published proof) that…”

Section: Proposition 53 Let M Be a Finitely Generated Graded S-modumentioning

confidence: 99%

Extended degree functions and monomial modules

Nagel

Römer²

2005

Trans. Amer. Math. Soc.

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Abstract. The arithmetic degree, the smallest extended degree, and the homological degree are invariants that have been proposed as alternatives of the degree of a module if this module is not Cohen-Macaulay. We compare these degree functions and study their behavior when passing to the generic initial or the lexicographic submodule. This leads to various bounds and to counterexamples to a conjecture of Gunston and Vasconcelos, respectively. Particular attention is given to the class of sequentially Cohen-Macaulay modules. The results in this case lead to an algorithm that computes the smallest extended degree.

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“…We shall need the following result of Sbarra [6], whose proof we indicate for the convenience of the reader. Proof.…”

Section: Acyclicitymentioning

confidence: 99%