Conditions for the existence of spreads in projective Hjelmslev spaces

Let R be a finite chain ring with $$|R|=q^m$$|R|=qm, and $$R/\text {Rad }R\cong \mathbb {F}_q$$R/RadR≅Fq. Denote by $$\varPi ={{\mathrm{PHG}}}({}_RR^n)$$Π=PHG(RRn) the (left) $$(n-1)$$(n-1)-dimensional projective Hjelmslev geometry over R. As in the classical case, we define a $$\lambda $$λ-spread of $$\varPi $$Π to be a partition of its pointset into subspaces of shape $$\lambda =(\lambda _1,\ldots ,\lambda _n)$$λ=(λ1,…,λn). An obvious necessary condition for the existence of a $$\lambda $$λ-spread $$\mathcal {S}$$S in $$\varPi $$Π is that the number of points in a subspace of shape $$\lambda $$λ divides the number of points in $$\varPi $$Π. If the elements of $$\mathcal {S}$$S are Hjelmslev subspaces, i.e., free submodules of $${}_RR^n$$RRn, this necessary condition is also sufficient. If the subspaces in $$\mathcal {S}$$S are not Hjelmslev subspaces this numerical condition is not sufficient anymore. For instance, for chain rings with $$m=2$$m=2, there is no spread of shape $$\lambda =(2,2,1,0)$$λ=(2,2,1,0) in $${{\mathrm{PHG}}}({}_RR^4)$$PHG(RR4). An important (and maybe difficult) question is to find all shapes $$\lambda $$λ, for which $$\varPi $$Π has a $$\lambda $$λ-spread. In this paper, we present a construction which gives spreads by subspaces that are not necessarily Hjelmslev subspaces. We prove the non-existence of spreads of shape $$2^{n/2}1^a$$2n/21a [cf. (2)], $$1\le a\le n/2-1$$1≤a≤n/2-1, in $${{\mathrm{PHG}}}({}_RR^n)$$PHG(RRn), where n is even and R is a chain ring of length 2.