2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/77286We develop the noncommutative geometry (bundles, connections etc.) associated to algebras that factorise into two subalgebras. An example is the factorisation of matrices $M_2(\C)=\C\Z_2\cdot\C\Z_2$. We also further extend the coalgebra version of theory introduced previously, to include frame resolutions and corresponding covariant derivatives and torsions. As an example, we construct $q$-monopoles on all the Podleś quantum spheres $S^2_{q,s}$.39 pages, LaTeX. Final version, to appear in Commun. Math. PhysQuantum AlgebraMathematical Physics81R50 58B30 16W30text