Resumen de Extended modules over skew PBW extensions

William Alfredo Fajardo Cardenas

• In the book “Serre’s problem on projective modules” [42], Tsit Yuen Lam defines the class E of extended rings; these rings satisfy the extended version of the Quillen-Suslin theorem. In this thesis we investigate extended modules and rings for skew P BW extensions from a matrix-constructive approach. We determine conditions on the parameters that define a skew extension A = σ(R)hx1, . . . , xni in order to A be projective-free (PF), or more generally, extended (E). Under such particular conditions we prove Vaserstein’s, Quillen’s patching, Horrocks’ and Quillen-Suslin’s theorems for this type of non-commutative rings of polynomial type. Complementary, but as a very important part of the thesis, a computational package has been developed not only for the computations involved in the matrix-constructive proofs related to the PF and E properties, but also for many homological applications of the Gr¨obner theory of skew P BW extensions developed recently in many papers.