Artinian level algebras of socle degree 4

Masuti, Shreedevi K.; Rossi, Maria Evelina

doi:10.1016/j.jalgebra.2018.03.043

Cited by 5 publications

(4 citation statements)

References 32 publications

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“…Non-canonically graded algebras with Hilbert function (1, 3, 4, 3, 1) are investigated independently in [MR16], where more generally Hilbert functions (1, n, m, n, 1) are considered.…”

Section: Discussionmentioning

confidence: 99%

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Classifying local Artinian Gorenstein algebras

Jelisiejew

2016

Collect. Math.

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The classification of local Artinian Gorenstein algebras is equivalent to the study of orbits of a certain non-reductive group action on a polynomial ring. We give an explicit formula for the orbits and their tangent spaces. We apply our technique to analyse when an algebra is isomorphic to its associated graded algebra. We classify algebras with Hilbert function (1, 3, 3, 3, 1), obtaining finitely many isomorphism types, and those with Hilbert function (1, 2, 2, 2, 1, 1, 1). We consider fields of arbitrary, large enough, characteristic.

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“…Non-canonically graded algebras with Hilbert function (1, 3, 4, 3, 1) are investigated independently in [MR16], where more generally Hilbert functions (1, n, m, n, 1) are considered.…”

Section: Discussionmentioning

confidence: 99%

“…We also discuss the case of (1, 3, 4, 3, 1), show that there are infinitely many isomorphism types and classify algebras which are not canonically graded, see Example 3.28. Non-canonically graded algebras with Hilbert function (1, 3, 4, 3, 1) are investigated independently in [MR16].…”

Section: Introductionmentioning

confidence: 99%

Classifying local Artinian Gorenstein algebras

Jelisiejew

2016

Collect. Math.

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“…If it actually occurs as the complete Hilbert function decomposition for an AG algebra, we call D(A) realizable. S. Masuti and M. Rossi showed that all socle degree four potential decomposition sequences D satisfying the three conditions are realizable [MasR,§3.6,3.7].…”

Section: Realizabilitymentioning

confidence: 99%

Reducibility of a family of local Artinian Gorenstein algebras

Iarrobino¹,

Marques²

2021

Preprint

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The Jordan type of an Artinian algebra is the Jordan block partition associated to multiplication by a generic element of the maximal ideal. We study the Jordan type for Artinian Gorenstein (AG) local (non-graded) algebras A, and the interaction of Jordan type with the symmetric decomposition of the Hilbert function H(A). We give examples of Gorenstein sequences H for which the family Gor(H) of AG algebras having Hilbert function H has three irreducible components, each corresponding to a symmetric decomposition of H. The component structure results from the intersection of two opposing filtrations of the family Gor(H) of AG algebras: that by Jordan type satisfies the usual dominance property; the second filtration, by symmetric decomposition, satisfies a known semicontinuity property. Our examples are in codimension three -the lowest codimension of such an example, as Gor(H) is irreducible in codimension two. We also compare the Jordan type of the AG algebra A and of its associated graded algebra A * , showing that they can be different. We explore a problem of realizability of partial symmetric decompositions.

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“…It is natural to ask if there are further conditions besides the symmetry conditions (1.4), the Macaulay conditions (1.5) and the condition that H A (0) is a graded Gorenstein sequence (1.6), on the symmetric decompositions D(A) or on the component Hilbert functions H(A) for AG algebras. S. Masuti and M. Rossi show that all socle degree four decomposition sequences D satisfying the three conditions are realizable -occur as some D(A) for an AG algebra A [MasR,§3.6,3.7]. Lists of realizable decompositions are proposed in [I6, §5F] for lengths n ≤ 16, and n ≤ 21 for embedding dimension at most three.…”

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confidence: 99%

Symmetric Decomposition of the Associated Graded Algebra of an Artinian Gorenstein Algebra

Iarrobino,

Marques

2018

Preprint

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We study the symmetric subquotient decomposition of the associated graded algebras A * of a non-homogeneous commutative Artinian Gorenstein (AG) algebra A. This decomposition arises from the stratification of A * by a sequence of ideals A * = C A (0) ⊃ C A (1) ⊃ • • • whose successive quotients Q(a) = C(a)/C(a + 1) are reflexive A * modules. These were introduced by the first author [I4, I5], developed in the Memoir [I6], and have been used more recently by several groups, especially those interested in short Gorenstein algebras, and in the scheme length (cactus rank) of forms.For us a Gorenstein sequence is an integer sequence H occurring as the Hilbert function H = H(A) for an AG algebra A, that is not necessarily homogeneous. Such a Hilbert function H(A) is the sum of symmetric non-negative sequences H A (a) = H Q A (a) , each having center of symmetry (j − a)/2 where j is the socle degree of A: we call these the symmetry conditions, and the decomposition D(A) = H A (0), H A (1), . . . the symmetric decomposition of H(A).We here study which sequences may occur as the summands H A (a): in particular we construct in a systematic way examples of AG algebras A for which H A (a) can have interior zeroes, as H A (a) = (0, s, 0, . . . , 0, s, 0). We also study the symmetric decomposition sets D(A), and in particular determine which sequences H A (a) can be non-zero when the dual generator is linear in a subset of the variables (Theorem 1.38).Several groups have studied "exotic summands" of the Macaulay dual generator F : these are summands that involve more successive variables than would be expected from

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Artinian level algebras of socle degree 4

Cited by 5 publications

References 32 publications

Classifying local Artinian Gorenstein algebras

Classifying local Artinian Gorenstein algebras

Reducibility of a family of local Artinian Gorenstein algebras

Symmetric Decomposition of the Associated Graded Algebra of an Artinian Gorenstein Algebra

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