An algebraic approach to finite projective planes

David Cook; Juan Migliore; Uwe Nagel; Fabrizio Zanello

doi:10.1007/s10801-015-0644-8

An algebraic approach to finite projective planes

David Cook, Juan Migliore, Uwe Nagel, Fabrizio Zanello

Mathematics

Research output: Contribution to journal › Article › peer-review

3 Scopus citations

Abstract

A finite projective plane, or more generally a finite linear space, has an associated incidence complex that gives rise to two natural algebras: the Stanley–Reisner ring (Formula presented.) and the inverse system algebra (Formula presented.). We give a careful study of both of these algebras. Our main results are a full description of the graded Betti numbers of both algebras in the more general setting of linear spaces (giving the result for the projective planes as a special case), and a classification of the characteristics in which the inverse system algebra associated to a finite projective plane has the weak or strong Lefschetz Property.

Original language	English
Pages (from-to)	495-519
Number of pages	25
Journal	Journal of Algebraic Combinatorics
Volume	43
Issue number	3
DOIs	https://doi.org/10.1007/s10801-015-0644-8
State	Published - May 1 2016

Bibliographical note

Publisher Copyright:
© 2015, Springer Science+Business Media New York.

Funding

The work for this paper was done while J. Migliore was partially supported by the National Security Agency under Grant Number H98230-12-1-0204 and by a Simons Foundation grant (#309556), while U. Nagel was partially supported by the National Security Agency under Grant Number H98230-12-1-0247 and by the Simons Foundation under Grant #317096, and while F. Zanello was partially supported by a Simons Foundation Grant (#274577).

Funders	Funder number
Simons Foundation	274577, 317096, 309556, H98230-12-1-0247
National Security Agency	H98230-12-1-0204

Keywords

Finite projective plane
Inverse system
Level algebra
Linear space
Minimal free resolution
Monomial algebra
Stanley–Reisner ring
Strong Lefschetz property
Weak Lefschetz property

ASJC Scopus subject areas

Algebra and Number Theory
Discrete Mathematics and Combinatorics

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10.1007/s10801-015-0644-8

Cite this

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abstract = "A finite projective plane, or more generally a finite linear space, has an associated incidence complex that gives rise to two natural algebras: the Stanley–Reisner ring (Formula presented.) and the inverse system algebra (Formula presented.). We give a careful study of both of these algebras. Our main results are a full description of the graded Betti numbers of both algebras in the more general setting of linear spaces (giving the result for the projective planes as a special case), and a classification of the characteristics in which the inverse system algebra associated to a finite projective plane has the weak or strong Lefschetz Property.",

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AU - Nagel, Uwe

AU - Zanello, Fabrizio

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KW - Inverse system

KW - Level algebra

KW - Linear space

KW - Minimal free resolution

KW - Monomial algebra

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KW - Strong Lefschetz property

KW - Weak Lefschetz property

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An algebraic approach to finite projective planes

Abstract

Bibliographical note

Funding

Keywords

ASJC Scopus subject areas

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