ABSTRACT
The Flat Cover Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240 Extending the Flat Cover Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
10.3 The n-flat cotorsion pair of chain complexes and model structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 The Flat Cover Conjecture for chain complexes . . . . . . . . . . . . . . . . . 249 Pure subcomplexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 Monoidality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
10.4 The homotopy category of differential graded model structures 252 10.5 Degreewise n-flat model structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
Monoidality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260 10.6 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
10.1 Introduction The goal of this chapter is to obtain two families of Abelian model struc-
tures on ChpRq from flat dimensions. In the first of these families, the class of differential graded n-flat chain complexes appears as the class of cofibrant objects, while in the other one a chain complex is cofibrant if, and only if, it is a complex of n-flat modules, i.e., degreewise n-flat complexes. The contents presented next are motivated by Gillespie’s work on the flat model structure on ChpRq [Gil04], and on degreewise homological model structures [Gil08].
