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Lefschetz Properties of Monomial Ideals
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.). (Mathematics)
2018 (English)Licentiate 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.

Place, publisher, year, edition, pages
Kungliga Tekniska högskolan, 2018.
Keywords [en]
Weak Lefschetz property, monomial ideals, group actions, almost linear resolution
National Category
Algebra and Logic
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-223373ISBN: 978-91-7729-703-1 (electronic)OAI: oai:DiVA.org:kth-223373DiVA, id: diva2:1183790
Presentation
2018-03-16, F11, Lindstedtsvagen 24, KTH, Stockholm, 14:00 (English)
Opponent
Supervisors
Note

QC 20180220

Available from: 2018-02-20 Created: 2018-02-19 Last updated: 2018-02-20Bibliographically approved
List of papers
1. Lefschetz Properties of Monomial Ideals with Almost Linear Resolution
Open this publication in new window or tab >>Lefschetz Properties of Monomial Ideals with Almost Linear Resolution
(English)In: Article in journal (Other academic) Submitted
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.

Keywords
Weak Lefschetz property, Monomial ideal, Almost linear resolution
National Category
Algebra and Logic
Research subject
Mathematics
Identifiers
urn:nbn:se:kth:diva-223380 (URN)
Note

QC 20180220

Available from: 2018-02-19 Created: 2018-02-19 Last updated: 2018-02-20Bibliographically approved
2. The Weak Lefschetz Property of Equigenerated Monomial Ideals
Open this publication in new window or tab >>The Weak Lefschetz Property of Equigenerated Monomial Ideals
(English)In: Article in journal (Other academic) Accepted
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.

Keywords
The Weak Lefschetz property, Monomial ideal, Group action
National Category
Algebra and Logic
Research subject
Mathematics
Identifiers
urn:nbn:se:kth:diva-223381 (URN)
Note

QC 20180220

Available from: 2018-02-19 Created: 2018-02-19 Last updated: 2018-02-20Bibliographically approved

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