A bird's eye view of the theory of graded modules over a graded ring might give the impression that it is nothing but ordinary module theory with all its statements decorated with the adjective “graded”. Once the grading is considered to be trivial, the graded theory reduces to the usual module theory. From this perspective, the theory of graded modules can be considered as an extension of module theory. However, this simplistic overview might conceal the point that graded modules come equipped with a shift, thanks to the possibility of partitioning the structures and then rearranging the partitions. This adds an extra layer of structure (and complexity) to the theory. This monograph focuses on the theory of the graded Grothendieck group K0gr, that provides a sparkling illustration of this idea. Whereas the usual K0 is an abelian group, the shift provides K0gr with a natural structure of a ℤ [Γ]-module, where Γ is the group used for the grading and ℤ [Γ] its group ring. As we will see throughout this book, this extra structure carries substantial information about the graded ring.

Let Γ and Δ be abelian groups and f : Γ → Δ a group homomorphism. Then for any Γ-graded ring A, one can consider a natural Δ-grading on A (see §1.1.2); in the same way, any Γ-graded A-module can be viewed as a Δ- graded A-module. These operations induce functors

Uf : GrΓ-A → GrΔ-A,

(−)Ω : GrΓ-A → GrΩ-AΩ,

(see §1.2.8), where GrΓ-A is the category of Γ-graded right A-modules, GrΔ-A that of Δ-graded right A-modules, and GrΩ-A the category of Ω-graded right AΩ-module with Ω = ker(f).

One aim of the theory of graded rings is to investigate the ways in which these categories relate to one another, and which properties of one category can be lifted to another. In particular, in the two extreme cases when the group Δ = 0 or f : Γ → Δ is the identity, we obtain the forgetful functors

U : GrΓ-A → Mod-A,

(−)0 : GrΓ-A → Mod-A0.

The category PgrΓ-A of graded finitely generated projective A-modules is an exact category.