Azumaya Algebras as Galois Comodules

It is shown that Azumaya algebras (over commutative rings) are special examples of Galois comodules. This leads to a new characterization of Azumaya algebras.     

Descent in Locally Presentable Categories

Let \(\mathbb {V}=(VV, \otimes , I)\) be a symmetric monoidal category such that \(\mathcal {V}\) is locally presentable and that all functors \(V\otimes - : \mathcal {V} \rightarrow \mathcal {V}\) for \(V \in \mathcal {V}\) preserve reflexive coequalizers and directed colimits. It is proved that any pure morphism of commutative ??-monoids is an effective descent morphism with respect to the indexed category given by commutative ??-monoids and modules over them. As a by-product, we prove that pure morphisms in a locally presentable category are effective for codescent.     

Pure Morphisms are Effective for Modules

Yet another proof of the result asserting that a morphism of commutative rings is an effective descent morphism for modules if and only if it is pure is given. Moreover, it is shown that this result cannot be derived from Moerdijk’s descent criterion.     

A Bicategorical Version of Masuoka’s Theorem

We give a bicategorical version of the main result of Masuoka (Tsukuba J Math 13:353–362, 1989) which proposes a non-commutative version of the fact that for a faithfully flat extension of commutative rings R í SR \subseteq S, the relative Picard group Pic(S/R) is isomorphic to the Amitsur 1–cohomology group H ¹(S/R,U) with coefficients in the units functor U.     

Effective codescent morphisms in locally presentable categories

A necessary and sufficient condition for pure morphisms in locally presentable categories to be effective is given.     

Descent Theory for Schemes

We give a complete characterization of the class of quasi-compact morphisms of schemes that are stable effective descent morphisms for the SCHEMES-indexed category given by quasi-coherent sheaves of modules.     

Entwining structures in monoidal categories

Interpreting entwining structures as special instances of J. Beck's distributive law, the concept of entwining module can be generalized for the setting of arbitrary monoidal category. In this paper, we use the distributive law formalism to extend in this setting basic properties of entwining modules.     

On Rational Pairings of Functors

In the theory of coalgebras C over a ring R, the rational functor relates the category $_{C^*}{\mathbb{M}}$ of modules over the algebra C ^* (with convolution product) with the category $^C{\mathbb{M}}$ of comodules over C. This is based on the pairing of the algebra C ^* with the coalgebra C provided by the evaluation map ${\rm ev}:C^*\otimes_R C\to R$ . The (rationality) condition under consideration ensures that $^C{\mathbb{M}}$ becomes a coreflective full subcategory of $_{C^*}{\mathbb{M}}$ . We generalise this situation by defining a pairing between endofunctors T and G on any category ${\mathbb{A}}$ as a map, natural in $a,b\in {\mathbb{A}}$ , $$ \beta_{a,b}:{\mathbb{A}}(a, G(b)) \to {\mathbb{A}}(T(a),b), $$ and we call it rational if these all are injective. In case T?=?(T, m _T , e _T ) is a monad and G?=?(G, δ _G , ε _G ) is a comonad on ${\mathbb{A}}$ , additional compatibility conditions are imposed on a pairing between T and G. If such a pairing is given and is rational, and T has a right adjoint monad T ^???, we construct a rational functor as the functor-part of an idempotent comonad on the T-modules ${\mathbb{A}}_{T}$ which generalises the crucial properties of the rational functor for coalgebras. As a special case we consider pairings on monoidal categories.     

Descent in ∗ -autonomous categories

We extend the result of Joyal and Tierney asserting that a morphism of commutative algebras in the ∗-autonomous category of sup-lattices is an effective descent morphism for modules if and only if it is pure, to an arbitrary ∗-autonomous category V (in which the tensor unit is projective) by showing that any V-functor out of V is precomonadic if and only if it is comonadic.