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  • 1.
    Altafi, Nasrin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Lefschetz Properties of Monomial Ideals2018Licentiate thesis, comprehensive summary (Other academic)
    Abstract [en]

    This thesis concerns the study of the Lefschetz properties of artinian monomial algebras. An artinian algebra is said to satisfy the strong Lefschetz property if multiplication by all powers of a general linear form has maximal rank in every degree. If it holds for the first power it is said to have the weak Lefschetz property (WLP).

    In the first paper, we study the Lefschetz properties of monomial algebras by studying their minimal free resolutions. In particular, we give an afirmative answer to an specific case of a conjecture by Eisenbud, Huneke and Ulrich for algebras having almost linear resolutions. Since many algebras are expected to have the Lefschetz properties, studying algebras failing the Lefschetz properties is of a great interest. In the second paper, we provide sharp lower bounds for the number of generators of monomial ideals failing the WLP extending a result by Mezzetti and Miró-Roig which provides upper bounds for such ideals. In the second paper, we also study the WLP of ideals generated by forms of a certain degree invariant under an action of a cyclic group. We give a complete classication of such ideals satisfying the WLP in terms of the representation of the group generalizing a result by Mezzetti and Miró-Roig.

  • 2.
    Altafi, Nasrin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Lefschetz Properties of Monomial Ideals with Almost Linear ResolutionIn: Article in journal (Other academic)
    Abstract [en]

    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

    ideals with almost linear resolutions.

  • 3.
    Altafi, Nasrin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    The Weak Lefschetz Property of Equigenerated Monomial IdealsIn: Article in journal (Other academic)
    Abstract [en]

    We determine the sharp lower bound for the Hilbert function in degree d of a

    monomial algebra failing the WLP over a polynomial ring with n variables and generated in

    degree d. We consider artinian ideals in the polynomial ring with

    n variables generated by homogeneous polynomials of degree d invariant under an action of

    the cyclic group Z/dZ. We give a complete classification of

    such ideals in terms of the WLP depending on the action.

  • 4.
    Altafi, Nasrin
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Nemati, N.
    Fakhari, S. A. S.
    Yassemi, S.
    Free resolution of powers of monomial ideals and Golod rings2017In: Mathematica Scandinavica, ISSN 0025-5521, E-ISSN 1903-1807, Vol. 120, no 1, p. 59-67Article in journal (Refereed)
    Abstract [en]

    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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