Morphisms and Semilinear Maps

Abstract

We start with the theorem that every non-degenerate morphism g: P V - - → P W is of the form g = P f for some semilinear map f : V → W, where non-degenerate means that Im g contains three non-collinear points. This implies immediately the Fundamental Theorem of projective geometry: Every non-degenerate morphism g between arguesian geometries is described in homogeneous coordinates by a semi-linear map f. For given coordinates f is unique up to a non-zero constant factor. Moreover, the map f is quasilinear if and only if g is a homomorphism.