| 1. |
- Abdallah, Nancy, et al.
(författare)
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Lefschetz properties of some codimension three Artinian Gorenstein algebras
- 2023
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Ingår i: Journal of Algebra. - : Elsevier BV. - 0021-8693 .- 1090-266X. ; 625, s. 28-45
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Tidskriftsartikel (refereegranskat)abstract
- Codimension two Artinian algebras have the strong and weak Lefschetz properties provided the characteristic is zero or greater than the socle degree. It is open to what extent such results might extend to codimension three Artinian Gorenstein algebras. Despite much work, the strong Lefschetz property for codimension three Artinian Gorenstein algebra has remained largely mysterious; our results build on and strengthen some of the previous results. We here show that every standard-graded codimension three Artinian Gorenstein algebra A having maximum value of the Hilbert function at most six has the strong Lefschetz property, provided that the characteristic is zero. When the characteristic is greater than the socle degree of A, we show that A is almost strong Lefschetz, they are strong Lefschetz except in the extremal pair of degrees.
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| 2. |
- Altafi, Nasrin, et al.
(författare)
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Complete intersection Jordan types in height two
- 2020
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Ingår i: Journal of Algebra. - : Academic Press. - 0021-8693 .- 1090-266X. ; 557, s. 224-277
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Tidskriftsartikel (refereegranskat)abstract
- We determine every Jordan type partition that occurs as the Jordan block decomposition for the multiplication map by a linear form in a height two homogeneous complete intersection (CI) Artinian algebra A over an algebraically closed field k of characteristic zero or large enough. We show that these CI Jordan type partitions are those satisfying specific numerical conditions; also, given the Hilbert function H(A), they are completely determined by which higher Hessians of A vanish at the point corresponding to the linear form. We also show new combinatorial results about such partitions, and in particular we give ways to construct them from a branch label or hook code, showing how branches are attached to a fundamental triangle to form the Ferrers diagram.
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| 3. |
- Altafi, Nasrin, et al.
(författare)
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Free resolution of powers of monomial ideals and Golod rings
- 2017
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Ingår i: Mathematica Scandinavica. - : Mathematica Scandinavica. - 0025-5521 .- 1903-1807. ; 120:1, s. 59-67
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Tidskriftsartikel (refereegranskat)abstract
- Let S = Kdbl[x1,⋯, xn] be the polynomial ring over a field Kdbl. In this paper we present a criterion for componentwise linearity of powers of monomial ideals. In particular, we prove that if a squarefree monomial ideal I contains no variable and some power of I is componentwise linear, then I satisfies the gcd condition. For a square-free monomial ideal I which contains no variable, we show that S/I is a Golod ring provided that for some integer s ≥ 1, the ideal Is has linear quotients with respect to a monomial order.
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| 4. |
- Altafi, Nasrin
(författare)
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Hilbert functions of Artinian Gorenstein algebras with the strong Lefschetz property
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- We prove that a sequence h of non-negative integers is the Hilbert function of some Artinian Gorenstein algebra with the strong Lefschetz property if and only if it is an SI-sequence. This generalizes the result by T. Harima which characterizes the Hilbert functions of Artinian Gorenstein algebras with the weak Lefschetz property. We also provide classes of Artinian Gorenstein algebras obtained from the ideal of points in $\mathbb{P}^n$ such that some of their higher Hessians have non-vanishing determinants. Consequently, we provide families of such algebras satisfying the SLP.
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| 5. |
- Altafi, Nasrin
(författare)
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Hilbert Functions Of Artinian Gorenstein Algebras With The Strong Lefschetz Property
- 2022
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Ingår i: Proceedings of the American Mathematical Society. - : American Mathematical Society (AMS). - 0002-9939 .- 1088-6826. ; 150:2, s. 499-513
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Tidskriftsartikel (refereegranskat)abstract
- We prove that a sequence h of non-negative integers is the Hilbert function of some Artinian Gorenstein algebra with the strong Lefschetz property (SLP) if and only if it is an Stanley-Iarrobino-sequence. This generalizes the result by T. Harima which characterizes the Hilbert functions of Artinian Gorenstein algebras with the weak Lefschetz property. We also provide classes of Artinian Gorenstein algebras obtained from the ideal of points in P-n such that some of their higher Hessians have non-vanishing determinants. Consequently, we provide families of such algebras satisfying the SLP.
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| 6. |
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| 7. |
- Altafi, Nasrin
(författare)
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Jordan types with small parts for Artinian Gorenstein algebras of codimension three
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- We study Jordan types of linear forms for graded Artinian Gorenstein algebras having arbitrary codimension. We introduce rank matrices of linear forms for such algebras that represent the ranks of multiplication maps in various degrees. We show that there is a 1-1 correspondence between rank matrices and Jordan degree types. For Artinian Gorenstein algebras with codimension three we classify all rank matrices that occur for linear forms with vanishing third power. As a consequence, we show for such algebras that the possible Jordan types with parts of length at most four are uniquely determined by at most three parameters.
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| 8. |
- Altafi, Nasrin
(författare)
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Jordan types with small parts for Artinian Gorenstein algebras of codimension three
- 2022
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Ingår i: Linear Algebra and its Applications. - : Elsevier BV. - 0024-3795 .- 1873-1856. ; 646, s. 54-83
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Tidskriftsartikel (refereegranskat)abstract
- We study Jordan types of linear forms for graded Artinian Gorenstein algebras having arbitrary codimension. We introduce rank matrices of linear forms for such algebras that represent the ranks of multiplication maps in various degrees. We show that there is a 1-1 correspondence between rank matrices and Jordan degree types. For Artinian Gorenstein algebras with codimension three we classify all rank matrices that occur for linear forms with vanishing third power. As a consequence, we show for such algebras that the possible Jordan types with parts of length at most four are uniquely determined by at most three parameters.
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| 9. |
- Altafi, Nasrin
(författare)
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Lefschetz properties and Jordan types of Artinian algebras
- 2020
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This thesis contains six papers concerned with studying the Lefschetz properties and Jordan types of linear forms for graded Artinian algebras. Lefschetz properties and Jordan types carry information about the ranks of multiplication maps by linear forms on graded Artinian algebras. A graded Artinian algebra is said to have the weak Lefschetz property (WLP) if multiplication map by some linear form on has maximal rank in all degrees. If it holds for all powers of the algebra is said to have the strong Lefschetz property (SLP). Jordan type is a partition determining the Jordan block decomposition for the multiplication map by on . It is a finer invariant than the WLP and SLP, which determines the ranks of multiplication maps by all powers of on .Papers A and B concern the study of the Lefschetz properties of Artinian algebras quotient of a polynomial ring in variables over a field of characteristic zero by a monomial ideal generated in a single degree . Paper A studies the connection between the Lefschetz properties of such algebras and their minimal free resolutions, namely the number of linear steps in their resolutions. Paper B consists of two parts. The first part provides sharp lower bounds for the Hilbert function in degree of such algebras failing the WLP. The second part of Paper B deals algebras quotients of polynomial rings by ideals generated by forms of the same degree and invariant under action of a cyclic group. The main result of this part classifies such algebras satisfying the WLP in terms of the representation of the action. Papers C and D both deal with determining the Jordan type partitions of linear forms for graded Artinian algebras with codimension two. Paper C concerns the problem for complete intersection Artinian algebras over an algebraically closed field of characteristic zero or large enough. The results of Paper C classify partitions of an integer that occur as Jordan type partitions for Artinian complete intersection algebras and some linear forms. Such classifications are provided in terms of the numerical conditions of the partitions. Also for a given Hilbert function of such algebras, the Jordan type partitions are completely determined by which higher Hessians vanish at the point corresponding to the linear form. Some combinatorial invariants of such partitions, namely branch label or hook code, have been studied in this paper as well. Paper D concerns the generalization of the results of Paper C. The family of graded Artinian quotients of , having arbitrary Hilbert function has been studied. The cell corresponding to a partition having diagonal lengths is comprised of all ideals in whose initial ideal is the monomial ideal determined by . These cells give a decomposition of the variety into affine spaces. The main result of Paper D determines the generic number of generators for the ideals in each cell ; generalizing a result of Paper C. In particular, partitions having the generic number of generators for an ideal defining an algebra in are determined.Paper E concerns the SLP of Artinian Gorenstein algebras via studying the higher Hessians of dual generators. The main result of this paper characterizes the Hilbert functions of Artinian Gorenstein algebras having arbitrary codimension satisfying the SLP. It proves that a sequence of non-negative integers is the Hilbert function of some Artinian Gorenstein algebra with the SLP if and only if is an SI-sequence. This is done by studying the higher Hessians of the dual generator of an Artinian Gorenstein quotient of the coordinate ring of a set of points in the projective space. Using this approach, we provide families of Artinian Gorenstein algebras obtained by points in the projective plane satisfying the SLP. Paper F concerns the study of Jordan types of linear forms for graded Artinian Gorenstein algebras having arbitrary codimension. This paper introduces rank matrices of linear forms for such algebras. There is a 1-1 correspondence between rank matrices and Jordan degree types. The main result of this paper classifies all matrices that occur as the rank matrix for some Artinian Gorenstein algebra of codimension three and a linear form such that . As a consequence, we prove that Jordan types with parts of length at most four are uniquely determined by at most three parameters.
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| 10. |
- Altafi, Nasrin, et al.
(författare)
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Lefschetz properties of monomial algebras with almost linear resolution
- 2019
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Ingår i: Communications in Algebra. - : Taylor and Francis Inc.. - 0092-7872 .- 1532-4125.
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Tidskriftsartikel (refereegranskat)abstract
- We study the WLP and SLP of artinian monomial ideals in S = K[x1,...,xn] via studying their minimal free resolutions. We study the Lefschetz properties of such ideals where the minimal free resolution of S/I is linear for at least n–2 steps. We give an affirmative answer to a conjecture of Eisenbud, Huneke and Ulrich for artinian monomial algebras with almost linear resolution.
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