The Projectivization Matroid of a q-Matroid

In this paper, we investigate the relation between a q-matroid and its associated matroid called the projectivization matroid. The latter arises by projectivizing the groundspace of the q-matroid and considering the projective space as the groundset of the associated matroid on which is defined a rank function compatible with that of the q-matroid. We show that the projectivization map is a functor from categories of q-matroids to categories of matroids, which allows us to prove new results about maps of q-matroids. We furthermore show the characteristic polynomial of a q-matroid is equal to that of the projectivization matroid. We use this relation to establish a recursive formula for the characteristic polynomial of a q-matroid in terms of the characteristic polynomial of its minors. Finally we use the projectivization matroid to prove a q-analogue of the Critical Theorem in terms of Fqm-linear rank metric codes and q-matroids.