2009
DOI: 10.1080/00927870802502753
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Generic Initial Ideals, Graded Betti Numbers, andk-Lefschetz Properties

Abstract: We introduce the k-strong Lefschetz property (k-SLP) and the k-weak Lefschetz property (k-WLP) for graded Artinian K-algebras, which are generalizations of the Lefschetz properties. The main results obtained in this paper are as follows:1. Let I be a graded ideal of R = K[x 1 , x 2 , x 3 ] whose quotient ring R/I has the SLP. Then the generic initial ideal of I is the unique almost revlex ideal with the same Hilbert function as R/I.2. Let I be a graded ideal of R = K[x 1 , x 2 , . . . , x n ] whose quotient ri… Show more

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Cited by 22 publications
(20 citation statements)
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“…At last we show that the Moreno-Socias conjecture is true if I is a complete intersection Artinian ideal of R which is generated by generic forms of degrees d i satisfying the same condition (Corollary 2.12). As a recent work Harima and Wachi [9] show similar results with ours independently.…”
Section: Conjecture 11 (Moreno-socias)supporting
confidence: 92%
“…At last we show that the Moreno-Socias conjecture is true if I is a complete intersection Artinian ideal of R which is generated by generic forms of degrees d i satisfying the same condition (Corollary 2.12). As a recent work Harima and Wachi [9] show similar results with ours independently.…”
Section: Conjecture 11 (Moreno-socias)supporting
confidence: 92%
“…Which b i actually occur is given by the Hilbert function of A/(g) which is λ bi . Call the pairs (b i , a i ) the basic invariants of (A, g) and we can prove that g is an SL element for A if and only if the basic invariants have the following property: Lemma 7] shows that this condition is the necessary and sufficient condition for A to have the SLP with an SL element g. We change the cooredinate if needed, and we can apply [6,Lemma 7] …”
Section: Lemma 5 Let a Be A Graded Artinian K-algebra Having The Slpmentioning
confidence: 99%
“…Proof. If one of a, b, c and d is equal to one, the theorem is reduced to the case of three variables, and follows from [6], [1], or [2]. We consider the case where a, b, c, d ≥ 2.…”
Section: (4)mentioning
confidence: 99%
See 1 more Smart Citation
“…After this paper was written we were informed that Tadahito Harima and Akihito Wachi have independently obtained some of the results on this paper by different methods (see [9]). …”
Section: Inductively We Say That H Is a M-times Weak Lefschetz O-seqmentioning
confidence: 99%