Graded gorenstein artin algebras whose hilbert functions have a large number of valleys

Boij, Mats

doi:10.1080/00927879508825208

Cited by 33 publications

(60 citation statements)

References 5 publications

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“…(1,8,16,24,36). Since the first difference is (1,7,8,8,12), which is not an O-sequence (since 12 > (8 (3) ) 1 1 = 10), we obtain our desired example. For socle degree 5 we make a similar construction using the pure O-sequences h = (1,3,6,10,15,21) and h = (1,3,5,5,3,1) to obtain the pure O-sequence h = (1,6,11,15,18,22).…”

Section: Differentiability and Unimodalitymentioning

confidence: 98%

“…R/I has Hilbert function (1,3,6,10,13,13,10,6,2), while for a general linear form L, R/(I, (L 3 )) has Hilbert function (1,3,6,9,10,7,1). Thus the multiplication by L 3 from degree 3 to degree 6 fails to be an isomorphism, and the SLP fails.…”

Section: Lemma 67 Given Two Pairs Of Non-negative Integersmentioning

confidence: 99%

“…(By this we mean that there is at most one "valley" in the sense of [7].) In [9], a monomial example is constructed (which can be lifted to reduced sets of points -cf.…”

Section: Moreover If Equality Holds In Some Degree J Then It Holds mentioning

confidence: 99%

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On the shape of a pure 𝑂-sequence

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Section: Differentiability and Unimodalitymentioning

confidence: 98%

Section: Lemma 67 Given Two Pairs Of Non-negative Integersmentioning

confidence: 99%

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On the shape of a pure 𝑂-sequence

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“…Recall that an h-vector is said to be unimodal if it is never strictly increasing after a strict decrease. Later on, in each codimension ≥ 5, examples of h-vectors were found that are not unimodal (see [3], [6], [5]). Since then the problem has been open whether non-unimodal Gorenstein h-vectors of codimension 4 exist (see, e.g., [3], [14], [21], [28,Problem 2.19], or [27, p. 66]).…”

Section: Introductionmentioning

confidence: 99%

A characterization of Gorenstein Hilbert functions in codimension four with small initial degree

Migliore

Nagel

Zanello

2008

Mathematical Research Letters

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Abstract. The main goal of this paper is to characterize the Hilbert functions of all (artinian) codimension 4 Gorenstein algebras that have at least two independent relations of degree four. This includes all codimension 4 Gorenstein algebras whose initial relation is of degree at most 3. Our result shows that those Hilbert functions are exactly the so-called SI-sequences starting with (1, 4, h 2 , h 3 , ...), where h 4 ≤ 33. In particular, these Hilbert functions are all unimodal.We also establish a more general unimodality result, which relies on the values of the Hilbert function not being too big, but is independent of the initial degree.

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“…In codimension r ≥ 4, it is known that the WLP may fail even for Gorenstein algebras (see Ikeda [Ik,Example 4.4], and the first author [Bo2,Theorem 3.6], for r = 4; see also [St,Example 4.3], [BI], [BL] and [Bo1], which supply examples of Gorenstein algebras of codimension r ≥ 5 that are nonunimodal, and therefore, a fortiori, without the WLP).…”

Section: Introductionmentioning

confidence: 99%

Level algebras with bad properties

Boij

Zanello

2007

Proc. Amer. Math. Soc.

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Abstract. This paper can be seen as a continuation of the works contained in the recent article (J. Alg., 305 (2006), 949-956) of the second author, and those of Juan Migliore (math. AC/0508067). Our results are: 1). There exist codimension three artinian level algebras of type two which do not enjoy the Weak Lefschetz Property (WLP). In fact, for e 0, we will construct a codimension three, type two h-vector of socle degree e such that all the level algebras with that h-vector do not have the WLP. We will also describe the family of those algebras and compute its dimension, for each e 0. 2). There exist reduced level sets of points in P 3 of type two whose artinian reductions all fail to have the WLP. Indeed, the examples constructed here have the same h-vectors we mentioned in 1).3). For any integer r ≥ 3, there exist non-unimodal monomial artinian level algebras of codimension r. As an immediate consequence of this result, we obtain another proof of the fact (first shown by Migliore in the abovementioned preprint, Theorem 4.3) that, for any r ≥ 3, there exist reduced level sets of points in P r whose artinian reductions are non-unimodal.

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Graded gorenstein artin algebras whose hilbert functions have a large number of valleys

Cited by 33 publications

References 5 publications

On the shape of a pure 𝑂-sequence

On the shape of a pure 𝑂-sequence

A characterization of Gorenstein Hilbert functions in codimension four with small initial degree

Level algebras with bad properties

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