Conditions for the existence of spreads in projective Hjelmslev spaces

Abstract

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