Components of the space parametrizing graded Gorenstein Artin algebras with a given Hilbert function

Boij, Mats

doi:10.2140/pjm.1999.187.1

Cited by 23 publications

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“…Once having one example of such a phenomena, we produce infinitely many by liaison (Remark 50). Even though this conjecture was open until now, Iarrobino and Boij have in a joint work already constructed other examples of reducible L(H ) whose general elements are type 2 level quotients, one with H = (1,3,6,10,14,18,20,20,12,6,2), and they have got a doubly infinite series of such components [5].…”

Section: (I) Q Is Smooth and Surjective With Connected Fibers Of Fibmentioning

confidence: 99%

Families of Artinian and one-dimensional algebras

Kleppe

2007

Journal of Algebra

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The purpose of this paper is to study families of Artinian or one-dimensional quotients of a polynomial ring R with a special look to level algebras. Let GradAlg H (R) be the scheme parametrizing graded quotients of R with Hilbert function H . Let B → A be any graded surjection of quotients of R with Hilbert function H B = (1, h 1 , . . . , h j , . . .) and H A , respectively. If dim A = 0 (respectively dim A = depth A = 1) and A is a "truncation" of B in the sense that H A = (1, h 1 , . . . , h j −1 , α, 0, 0, . . .) (respectively H A = (1, h 1 , . . . , h j −1 , α, α, α, . . .)) for some α h j , then we show there is a close relationship between GradAlg H A (R) and GradAlg H B (R) concerning e.g. smoothness and dimension at the points (A) and (B), respectively, provided B is a complete intersection or provided the Castelnuovo-Mumford regularity of A is at least 3 (sometimes 2) larger than the regularity of B. In the complete intersection case we generalize this relationship to "non-truncated" Artinian algebras A which are compressed or close to being compressed. For more general Artinian algebras we describe the dual of the tangent and obstruction space of graded deformations in a manageable form which we make rather explicit for level algebras of Cohen-Macaulay type 2. This description and a linkage theorem for families allow us to prove a conjecture of Iarrobino on the existence of at least two irreducible components of GradAlg H (R), H = (1, 3, 6, 10, 14, 10, 6, 2), whose general elements are Artinian level algebras of type 2.

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Section: (I) Q Is Smooth and Surjective With Connected Fibers Of Fibmentioning

confidence: 99%

Families of Artinian and one-dimensional algebras

Kleppe

2007

Journal of Algebra

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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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“…In particular, the above-mentioned Gorenstein algebras of codimension r 5 having a non-unimodal h-vector do not enjoy the WLP. In codimension 4 (even if, as we said before, no non-unimodal h-vector is known), examples of Gorenstein algebras without the WLP have been provided by Ikeda [Ik,Example 4.4] and Boij [Bo2,Theorem 3.6].…”

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confidence: 99%

A non-unimodal codimension 3 level h-vector

Zanello

2006

Journal of Algebra

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1, 3, 6, 10, 15, 21, 28, 27, 27, 28) is a level h-vector! This example answers negatively the open question as to whether all codimension 3 level h-vectors are unimodal.Moreover, using the same (simple) technique, we are able to construct level algebras of codimension 3 whose h-vectors have exactly N "maxima," for any positive integer N .These non-unimodal h-vectors, in particular, provide examples of codimension 3 level algebras not enjoying the Weak Lefschetz Property (WLP). Their existence was also an open problem before.In the second part of the paper we further investigate this fundamental property, and show that there even exist codimension 3 level algebras of type 3 without the WLP.

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Components of the space parametrizing graded Gorenstein Artin algebras with a given Hilbert function

Cited by 23 publications

References 7 publications

Families of Artinian and one-dimensional algebras

Families of Artinian and one-dimensional algebras

Level algebras with bad properties

A non-unimodal codimension 3 level h-vector

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