3 - Differential Graded Algebra

Summary

A good understanding of DG (differential graded) algebra is essential in our approach to derived categories. By DG algebra, we mean DG rings, DG modules, DG categories and DG functors. The first section is on cohomologically graded rings and modules, with a discussion of the monoidal braiding (i.e. the Koszul sign rule). After that we study DG rings, DG modules and operations on them. We go on to discuss DG categories, DG functors between them and morphisms between DG functors. We recall the DG category C(M) of complexes in an abelian category M.A new feature we introduce is the DG category C(A,M) of DG A-modules in M, where A is a DG ring and M is an abelian category. This includes as special cases the category C(M) mentioned above, and the category C(A) of DG A-modules over a DG ring A. Another new feature is the distinction between the DG category C(A,M) and its strict subcategory Cstr(A,M), whose morphisms are the degree 0 cocycles, and it is an abelian category.