Compactly generated homotopy categories (Q865283)

Given a class \(\mathbb X\) of \(R\)-modules closed under coproducts and direct summands, \({\mathcal K}(\mathbb X)\) denotes the triangulated category of chain complexes of modules in \(\mathbb X\) where the morphisms are chain maps modulo homotopy equivalences. The second author proved that \({\mathcal K}(\operatorname{Proj} R)\) is compactly generated under certain conditions and \textit{H. Krause} proved [Compos. Math. 141, 1128--1162 (2005; Zbl 1090.18006)] that \({\mathcal K}(\operatorname{Inj} R)\) is so as well if \(R\) is left Noetherian. The main result of this article isolates sufficient conditions for \({\mathcal K}(\mathbb X)\) to be compactly generated, namely that every finitely presented module has a right \(\mathbb X\)-resolution, and that every pure exact sequence of modules in \(\mathbb X\) is split exact. This theorem is illustrated by many examples.