Morphisms of projective spaces over rings

The fundamental theorem of projective geometry is generalized for projective spaces over rings. Let RM and SN be modules. Provided some weak conditions are satisfied, a morphism g : PðMÞnE ! PðNÞ between the associated projective spaces can be induced by a semilinear map f : M ! N. These conditions are satisfied for instance if S is a left Ore domain and if the image of g contains three independent free points. No assumptions are made on the module M, and both modules may have some torsion. Introduction Two di¤erent approaches to projective spaces associated to modules are usually considered. One may choose as set of points the set of all submodules generated by a unimodular element, as defined in [20], or one may choose the lattice of all submodules, as defined in [3]. In the first approach one avoids the pathology (?) of small points contained in big points. But the price to pay is important. Following [9] it would be desirable if one had a functor from the category of modules and semilinear maps to a category of projective spaces and morphisms. But this is impossible with the first approach. Consider the ring R :1⁄4 Z=4Z and the linear map f : R ! R defined by f ðx; y; zÞ 1⁄4 ðxþ y; xþ 3y; zÞ. One easily shows that f cannot induce a map PðRÞ ! PðRÞ that preserves the incidence relation. So with this first approach we must restrict our attention to semilinear maps that preserve unimodular elements, and this is not natural. In the present paper the projective space PðMÞ associated to a module M is defined as the set of all cyclic (i.e. one-generated) submodules. This is equivalent to the second approach. Using axioms of Faigle and Herrmann [5] we propose a definition of projective spaces based on a single operator4. Morphisms of projective spaces are defined in the second section. It is shown that one has a functor from the category of modules and semilinear maps to the category of projective spaces and morphisms (this implies that a morphism must be a partially defined map between the point sets). The main result of this paper is a generalization of the fundamental theorem of projective geometry. It is proved in Section 3 by following mainly the lines of the proof given in [6]. Let RM and SN be modules and g : PðMÞnE ! PðNÞ a morphism between the associated projective spaces. We suppose that the ring S is directly finite, and that the image of g contains three independent free points B1;B2;B3 satisfying a weak condition (C3). Then there exists a semilinear map f : M ! N which induces g. Moreover, the map f is unique up to multiplication with a unit. This condition (C3) requires that for any C1;C2 A PðNÞ, there exists a point Bi which is independent from all the points of the line C14C2. In Section 4 we show that this condition is satisfied provided S is a left Ore domain. In Section 5 we show that it is satisfied provided S is a right Bezout domain and B1;B2;B3 generate a direct summand. In the literature, most generalizations of the fundamental theorem deal with isomorphisms. See for instance [18], [13], [12], [4] and [15]. Several interesting results in that direction (and others) can be found in [10]. Closer to our theorem is the result of Brehm [2]. His triangle-property resembles condition (C3), but it applies to the module M, not to N. The reason is that Brehm’s homomorphisms preserve disjointness. Since we do not make such assumptions, our Theorem 3.2 generalizes Theorem 1 in [2]. On the other hand, Brehm’s result is very general, because homomorphisms do not preserve cyclic submodules. For classical projective spaces (over division rings), the present version of the fundamental theorem was first proved in [8] and independently by Havlicek [11]. It generalized a former version due to Brauner [1] on linear maps. In the case of projective lattice geometries, these linear maps are discussed in [14]. Recently, a further generalization of the fundamental theorem for classical projective spaces appeared in [7]. It is possible that this generalization also applies to the case of projective spaces associated to modules. The author wishes to thank Professor Lashkhi for several valuable discussions on projective geometry over rings.