Injectivity and Gorenstein-injectivity under faithfully flat ring extensions
A main object of study in algebraic geometry is a quotient of a polynomial ring. Faithfully flat extensions of such rings also have applications in algebraic geometry and play an important role in commutative algebra. Modules are representation objects for rings and one way to study a ring is to study its modules. Some modules are distinguished by special properties (e.g. injective or Gorenstein injective modules). In homological algebra, we approximate a module by these special modules, and we derive information about the module by studying these approximations. Studying mathematical objects involves studying particular types of maps (morphisms) that respect the essential structure of the objects. We consider the objects and their morphisms as a category. A functor is a type of mapping between categories. Hom functors are among the most important since many functors can be represented in terms of these. In this thesis, we study faithfully flat ring extensions and their homological properties. In particular, we study the transfer of injectivity (Gorenstein injectivity) via hom functors in this setting.