We consider a symmetric monoidal closed category V = (V ,⊗, I, [−,−]) together with a regular injective object Q such that the functor [−, Q] : V → V op is comonadic and prove that in such a category, as in the monoidal category of abelian groups, a morphism of commutative monoids is an effective descent morphism for modules if and only if it is a pure monomorphism. Examples of this kind of mon...

We present sufficient conditions under which effective descent morphisms in a quasivariety of universal algebras are the same as regular epimorphisms and examples for which they are the same as regular epimorphisms satisfying projectivity. 1. Preliminaries A variety is a full subcategory of the category of structures for a first order (one sorted) language, closed under substructures, products ...

In this paper we study two types of descent in the category Berkovich analytic spaces: flat and with respect to an extension ground field. Quite surprisingly, deepest results direction seem be second type, including properties being a good space morphism without boundary.

Each full reflective subcategory X of a finitely-complete category C gives rise to a factorization system (E,M) on C, where E consists of the morphisms of C inverted by the reflexion I : C → X . Under a simplifying assumption which is satisfied in many practical examples, a morphism f : A → B lies in M precisely when it is the pullback along the unit ηB : B → IB of its reflexion If : IA → IB; w...

In the theory of schemes, faithfully flat descent is a very powerful tool. One wants a descent theory not only for quasi-coherent sheaves and morphisms of schemes (which is rather elementary), but also for geometric objects and properties of morphisms between them. In rigid-analytic geometry, descent theory for coherent sheaves was worked out by Bosch and Görtz [BG, 3.1] under some quasi-compac...

To a B-coring and a (B, A)-bimodule that is finitely generated and projective as a right A-module an A-coring is associated. This new coring is termed a base ring extension of a coring by a module. We study how the properties of a bimodule such as separability and the Frobenius properties are reflected in the induced base ring extension coring. Any bimodule that is finitely generated and projec...

To a B-coring and a (B, A)-bimodule that is finitely generated and projective as a right A-module an A-coring is associated. This new coring is termed a base ring extension of a coring by a module. We study how the properties of a bimodule such as separability and the Frobenius properties are reflected in the induced base ring extension coring. Any bimodule that is finitely generated and projec...