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