Research Projects

Module categories are among the most fundamental objects in algebra. Their properties account for much of the idea behind a Grothendieck category: an abelian category with a generator in which filtered colimits are exact. If k is a field and A is a k-algebra, then A is a finitely generated projective generator in the category of (right) A-modules. Conversely, we know that any Grothendieck category that has a finitely generated projective generator is equivalent to a module category. This suggests that any ring may be understood through its category of modules. Indeed, several ring theoretic notions can be replaced by properties of the module category. For example, a ring is noetherian precisely when its category of modules is locally noetherian. On the other hand, there are comodule categories. If we have a coalgebra C over a field k, then the category of comodules is also a Grothendieck category. Some of the properties of comodule categories are dual to those of module categories. For instance, C is an injective cogenerator in the category of (right) C-comodules. Indeed the category of right C-comodules admits a fully faithful embedding into the category of (left) C* = Hom(C, k)-modules. Further, an entwining structure (A, C, ψ) consists of a k-algebra A, a k-coalgebra C and a morphism ψ : C ⊗ A −→ A ⊗ C satisfying conditions somewhat similar to those of a bialgebra, or a comodule algebra over an algebra. The notion was introduced in a famous paper of Brzezin ́ski and Majid. The original motivation was that an entwining structure would play a role similar to that of a noncommutative principal bundle. The algebra A replaces a space, while the coalgebra C with the entwining ψ : C ⊗ A −→ A ⊗ C replaces the action of a group. Since then, entwining structures have appeared in a number of contexts and their theory has been developed widely by many authors. The modules over an entwining structure (A, C, ψ) are known as entwined modules and the category of the module is known as entwined module Category. The category of entwined modules provides an exciting framework which combines many of the properties of module categories and comodule categories. The main objective of this project is to study the entwined module category on its own, its torsion theories, injective objects, and related notions such as the Gabriel-Ziegler spectrum, study of functors between entwined module categories induced by more general morphisms known as measurings and cohomology theories.