15 - Algebraically Graded Noncommutative Rings

Summary

This chapter, as well as Chapters 16 and 17, are on algebraically graded rings, which is our name for graded rings that have lower indices and do not involve the Koszul sign rule (in contrast with the cohomologically graded rings that underlie DG rings). Simply put, these are the usual graded rings that one encounters in textbooks on algebra. With few exceptions, the base ring K in the four final chapters of the book is a field. Let A be an algebraically graded ring. The category of algebraically graded A-modules is M(A,gr). Its morphisms are the A-linear homomorphisms of degree 0. We talk about finiteness in the algebraically graded context and about various kinds of homological properties, such as graded-injectivity. Special emphasis is given to connected graded rings. The category of complexes with entries in the abelian category M(A,gr) is the DG category C(A,gr) := C(M(A,gr)). Its objects are bigraded, by cohomological degree and algebraic degree. The strict subcategory of C(A,gr) is Cstr(A,gr). The derived category is the triangulated category D(A,gr) := D(M(A,gr)). We present the algebraically graded variants of K-injective resolutions and the relevant derived functors.