On the rank of incidence matrices in projective Hjelmslev spaces

Let (R) be a finite chain ring with (|R|=q^m) , (R/{{mathrm{Rad}}}Rcong mathbb {F}_q) , and let (Omega ={{mathrm{PHG}}}({}_RR^n)) . Let (tau =(tau _1,ldots ,tau _n)) be an integer sequence satisfying (m=tau _1ge tau _2ge cdots ge tau _nge 0) . We consider the incidence matrix of all shape (varvec{m}^s=(underbrace{m,ldots ,m}_s)) versus all shape (tau ) subspaces of (Omega ) with (varvec{m}^spreceq tau preceq varvec{m}^{n-s}) . We prove that the rank of (M_{varvec{m}^s,tau }(Omega )) over (mathbb {Q}) is equal to the number of shape (varvec{m}^s) subspaces. This is a partial analog of Kantor’s result about the rank of the incidence matrix of all (s) dimensional versus all (t) dimensional subspaces of ({{mathrm{PG}}}(n,q)) . We construct an example for shapes (sigma ) and (tau ) for which the rank of (M_{sigma ,tau }(Omega )) is not maximal.