On the rank of incidence matrices in projective Hjelmslev spaces

Let $$R$$R be a finite chain ring with $$|R|=q^m$$|R|=qm, $$R/{{\mathrm{Rad}}}R\cong \mathbb {F}_q$$R/RadR≅Fq, and let $$\Omega ={{\mathrm{PHG}}}({}_RR^n)$$Ω=PHG(RRn). Let $$\tau =(\tau _1,\ldots ,\tau _n)$$τ=(τ1,…,τn) be an integer sequence satisfying $$m=\tau _1\ge \tau _2\ge \cdots \ge \tau _n\ge 0$$m=τ1≥τ2≥⋯≥τn≥0. We consider the incidence matrix of all shape $$\varvec{m}^s=(\underbrace{m,\ldots ,m}_s)$$ms=(m,…,m⏟s) versus all shape $$\tau $$τ subspaces of $$\Omega $$Ω with $$\varvec{m}^s\preceq \tau \preceq \varvec{m}^{n-s}$$ms⪯τ⪯mn-s. We prove that the rank of $$M_{\varvec{m}^s,\tau }(\Omega )$$Mms,τ(Ω) over $$\mathbb {Q}$$Q is equal to the number of shape $$\varvec{m}^s$$ms subspaces. This is a partial analog of Kantor’s result about the rank of the incidence matrix of all $$s$$s dimensional versus all $$t$$t dimensional subspaces of $${{\mathrm{PG}}}(n,q)$$PG(n,q). We construct an example for shapes $$\sigma $$σ and $$\tau $$τ for which the rank of $$M_{\sigma ,\tau }(\Omega )$$Mσ,τ(Ω) is not maximal.