This note arises from an attempt to give some model-theoretic interpretation of the concept of flatness. It is well-known that a necessary condition for a ring morphism A → B to be faithfully flat, is that any linear system of equations with coefficients from A which has a solution over B, must have already a solution over A. In fact, if we strenghten this condition to any solution over B comes...

We prove a version of faithfully flat descent in rigid analytic geometry, for almost perfect complexes and without finiteness assumptions on the rings involved. This extends results Drinfeld vector bundles.

This paper studies the structure of U(g)-Galois extensions. In particular, we use a result of Bell to construct a “PBW-like” free basis for faithfully flat U(g)-Galois extensions. We then move to non-faithfully flat extensions and propose a possible equivalent condition for a U(g)-extension to be Galois. We get a partial result for this.

Properties of (most general) non-commutative torsors or A-B torsors are analysed. Starting with pre-torsors it is shown that they are equivalent to a certain class of Galois extensions of algebras by corings. It is then concluded that every faithfully flat unital pre-torsor admits a (left and right) flat (bimodule) connection. It is shown that a class of faithfully flat pre-torsors induces equi...

This paper continues our study, begun in [MS], of the relationship between the prime ideals of an algebra A and of a subalgebra R such that R ⊂A is a faithfully flat H -Galois extension, for some finite-dimensional Hopf algebra H . In that paper we defined three basic Krull relations, Incomparability (INC), t-Lying Over (t-LO), and Going Up (GU), analogous to the classical Krull relations for p...