An algebraic approach to finite projective planes

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 $$R/I_\Lambda $$R/I? and the inverse system algebra $$R/I_\Delta $$R/IΔ. 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.